Invent. math. 102, 17 71 (1990)

[~lverltiorles

matbematicae

9 Springer-Verlag 1990

Extensions of nilpotent blocks* Burkhard Kiilshammer 1 and Lluis Puig 2 1 Universit/it Dortmund, Fachbereich Mathematik, Postfach 500 500, D-4600 Dortmund 50, Federal Republic of Germany 2 CNRS, Universit6 de Paris 7, DMI Ecole Normale Sup6rieure, 45 rue d'Ulm, F-75005 Paris, France Oblatum 16-IX-1988 & 27-X11-1989

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Notation, terminology and quoted results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Semicovering exomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Auxiliary results on group extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. On the local structure of block extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. On the p-extensions of nilpotent blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. The projective equivalence between nilpotent block extensions . . . . . . . . . . . . . . . 8. Source algebra extensions of extensions of nilpotent blocks . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 26 36 43 46 51 54 60 70

1. Introduction 1.1.

Let p be a prime n u m b e r . In [2] Brou6 a n d Puig i n t r o d u c e the so-called block b of a finite g r o u p H is said to be nilpotent if, for any ps u b g r o u p Q of H a n d a n y block f of Cn(Q) associated with b, the q u o t i e n t NH(Q,f)/Cn(Q) of the stabilizer of (Q,f) in H by the centralizer of Q is a p-group. In [18] Puig completes the study of such blocks giving the s t r u c t u r e of their source algebras. As the title a n n o u n c e s , the m a i n p u r p o s e of this p a p e r is to study the s t r u c t u r e - i.e. the source algebra, in the sense of [17] a n d [18] - of any block of a finite g r o u p G lying over a n i l p o t e n t block b of a n o r m a l s u b g r o u p H of G.

nilpotent blocks: a

1.2. Such a p u r p o s e has been m o t i v a t e d by the following situation, which occurs in any block a of a n y finite g r o u p G: consider a p - s u b g r o u p Q of G a n d a b l o c k f o f C~(Q) associated with a; if Q is "big e n o u g h " - precisely, if Q is self-centralizing in P for a n y B r a u e r pair (P, e) c o n t a i n i n g (Q,f) (cf. [1]) - t h e n Z(Q) is a defect g r o u p of f in t h a t case, f is clearly a n i l p o t e n t block of Ca(Q) a n d therefore o u r results * This work has been partially supported by the DFG and the CNRS

18

B. K(ilshammer and L. Puig

determine the structure of the block of N~(Q) lying over f The particular case where Q is a defect group of the full block a of G has been already considered by Reynolds in [21], by Kfilshammer in [10] and by Puig in Proposition 14.6 of [19], where the source algebra of the block of N~(Q) lying over f is completely determined. 1.3. On the other hand, if NG(Q) is p-solvable, the situation above could be handled employing Fong's reduction techniques in [8] - suitably extended to discuss algebra structures as described for instance in [13]. However, notice that our results here are more precise even in this case since they guarantee the "uniqueness" of the final reduction, which is important to connect the structures coming from the different pairs (Q, f). 1.4. Let us come back to our basic situation. Let ,( be an algebraically closed field of characteristic p, (_9 a complete discrete valuation ring with residue field ~, G a finite group, H a normal subgroup of G, b a primitive idempotent of Z(gH, N the stabilizer ofb in G, P a maximal p-subgroup of N such that Bre(b ) @ 0 in Z~Cn(P) and Q the intersection of P and H. Our main results state that, if the block b of H is nilpotent, the structure of any block of G lying over b is completely determined by the following two invariants

1.4.1. a full C-matrix interior P-algebra S which has a P-stable (9-basis containing the unity (i.e. a capped endo-permutation (gP-module, in Dade's terms cf. [6], S being its (9-endomorphism ring).

1.4.2. a twisted group algebra (9, L constructed from a group extension L of N/H by Q, which contains P as a Sylow p-subgroup. Actually, Theorem 1.8 below shows that the uniqueness of L is stronger than the uniqueness expected from a group extension of N/H by Q: the effort to obtain that uniqueness - which is very important for forthcoming applications - lengthens particularly this paper. Let us mention an easy case: when Q has an N-stable complement K in H -- which is the situation considered by Dade in [7] L is just the quotient N/K. 1.5. To state precisely these results, we will freely use notation and terminology introduced in Sect. 2. Set A = (gG, B = (gH, fl = {b} so that fl is a point of H on B and N = NG(Ha) (cf. 2.7). For any X c N denote by .( the image of X in N/H; in particular, N = N/H and we consider N, endowed with the action on the set /q given by left multiplication, as an N-permutation group (cf. 2.4). We consider A endowed with B as a G-algebra extension - an interior G-algebra endowed with a G-stable subalgebra containing the unity (cf. 2.9) but, as we explain below, we are mostly interested in the restriction Rest(A) (cf. 2.9). Notice that fl is still a point of N (and of any subgroup of N containing H) on B and that H a c N a (cf. 2.7). 1.6. Let ct be the unique point of G on B such that H a c N a c G , (i.e. ~ = {Try(b)}), y a local point of P on B such that P r c N a c G, and 6 a local point of Q on B such that Q0 c Pr. Notice that 7 and 6 exist since Bren(b) 4:0 and therefore Brg(b) ~ 0 (cf. 2.6 and [1], w Moreover, P~ and Q0 are respectively defect pointed

Extensions of nilpotent blocks

19

groups of G, and H~ since our maximal choice of P guarantees that P~. is a maximal local pointed group on B contained in N~, so that ~ c T r ~ ( B P . 7 . B r (cf. 2.7), and then Qa is a maximal local pointed group on B contained in Htj (see Proposition 5.3 below for more detail). Consequently, we get N = H . N~(Qa) by the Frattini argument (cf. [19], 2.10.2) (we have Na(Qo) c N since fl is the unique point of H on B such that Q~ c H~); in particular, if the block b of H is nilpotent, the inclusion NG(Q~ ) c N induces IV ~- NG(Q~)/Q'Cn(Q) ~ EN,~(Q~ )

(1.6.1)

since NH(Qa)= Q ' C n ( Q ) (cf. [18], 1.7.2) and CN(Q~,N)= Cn(Q) (cf. 2.7). A general remark: if H = Q then all the points above coincide with { 1} and we will omit to mention them. 1.7. We denote respectively by J~: A , ~ A , .~:A~--*Res~(A) and J~:A~--*Res~(A) embedded algebra extensions associated with G~, N~ and any local pointed group R~ on B such that Q a c R ~ c P ~ , (cf. 2.11). Notice that, as b x b = O for any x e G - N, the embedding J~ induces an N-algebra extension isomorphism A~ ~ 6.Nb where (gNb is endowed with (CHb (cf. 2.9), and therefore a G-algebra extension isomorphism (cf. 2.12.2) A, ~ Ind~((5'Nb)

(1.7.1)

which shows that, for our purposes, it suffices to consider Ap, and afortiori Rest(A). Actually, as we explain from 1.16 on, it suffices to study the P-algebra extension A~, and Theorem 1.12 below describes its structure, which depends on a suitable group extension L of N / H by Q as we said in 1.4. We first give our main result about L. Theorem 1.8. With the notation above, assume that the block b of H is nilpotent. There are a finite group L and group homomorphisms ~: L ~ N and ~: P ~ L fulfittin 9 the .following conditions: 1.8.1. We have I m ( ~ ) = / ~ , Ker(r0 = T(Q), K e r ( z ) = {1} and ~(T(u))= ti for any u ~ P. In particular, considering L endowed with the action on the set l~ given by n and left multiplication as an i~-permutation group, z induces a bOection TR: I~Iom~(R, P) -~ tZlom~(r(R), T(P)) for any subgroup R of P. 1.8.2. For any local pointed group R~ on B such that Q~ ~ R~ ~ P~ we have zR(E N, ~(R~, P;,)) = EL, ~(z( R), T(P)). Moreover, if L' is a f n i t e group and 7 r ' : L ' ~ l ~ and z ' : P ~ L ' are group homomorphisms fulfilling conditions 1.8.1 and 1.8.2, there is a unique group exomorphism ~: L ~- L' having a representative a such that n' o a = ~ and a o z = z'. Remark 1.9. Notice that z ( P ) is a Sylow p-subgroup of L (cf. 1.8.1, 1.8.2 and 2.16.2). In particular, any group automorphism co of L such that noco = n and o2 o z = z is an inner automorphism determined by an element of C o ( P ) (cf. Remark 4.12 below), which proves the uniqueness of ~- above. Remark 1.10. In Proposition 5.5 below we show that e is the unique local point of R on B such that R~ c P~, and therefore if T~ is another local pointed group on B

20

B. Kiilshammer and L. Puig

such that Qo c T, c P~ we have ZT,R(EN, g(T,, R,)) = EL,g(z(T), z(R))

(1.10.1)

where zr, R is the bijection from PIom~(T, R) onto FIom~(r(T), T(R)) induced by z. 1.11. From now on, we assume that the block b of H is nilpotent. To construct A~ from the group L above, we have to introduce a suitable central extension of L by ~e• obtained by restriction from the following central extension of ,~ by ~r • As b is nilpotent, Ba/J(Ba) is a simple k-algebra (cf. [18], 1.9.1); denote by s: B a -* Ba/J(Ba) the canonical map and, in particular, set Ba/J(Ba) = s(Ba). The natural action of N on s(Bp) defines a group homomorphism N -+ Aut(s(Ba)) and, considering s(Ba) • as a central extension of Aut(s(Ba)) by g• we set (cf. 2.1 and 2.4 below) ]V = Resu(s(Bp) • ) ;

(1.11.1)

that is, the elements of N are the pairs (s(c), x) where x e N and c~B~ fulfils s(d ~) = s(d x) for any de Ba. Moreover, there is a unique group homomorphism H ~ N mapping x ~ H upon (s(x'l), x), this homomorphism is injective and its image is a normal subgroup of N intersecting trivially the image of ~r215So, identifying H with its image in/Q, the quotient N / H is a central extension of ~t by ~• denoted by N. Recall that, ifL is a central extension of a finite group L by ~r215we denoteby do,L the corresponding twisted group algebra (cf. 2.1 and 2.4 below) and by do,L~ the opposite do-algebra. Theorem 1.12. With the notation above, assume that the block b of H is nilpotent. Let L be a finite 9roup and ~: L--* 1~ and z: P ~ L 9roup homomorphisms fulfilling conditions 1.8.1 and 1.8.2, set/~ = Res~(~) and consider do,L ~ endowed with the 9roup homomorphism P ~ (do,L~ • induced by z and with the imaye of doQ, as a P-algebra extension. Then there is an do-simple interior P-algebra S, unique up .to isomorphism, such that we have an isomorphism Ar ~- S | 1 6 3

~

(1.12.1)

as P-algebra extensions, and the determinant maps the image of P in Ind~te)L(S~) • where S t is the restriction orS to z( P), onto {1}. Moreover, S has a P-stable do-basis containin9 the unity as unique P-fixed element. Remark 1.13. If S' is an d0-simple interior P-algebra such that A~ ~ S ' Q e d o , s ~ there is a group homomorphism 0: L ~ (9 • such that S' ~ Resddo0) | where C o denotes the corresponding interior L-algebra. Indeed, it is clear that Co | do,s ~ do,/~o as P-algebra extensions and, since p divides neither ranke(S ) nor JL: r(P)] (cf. Remark 1.9), we may choose 0 such that the determinant maps the image of P in ((9o - 1 @eInd~te)(S'~)) • onto {1}. Remark 1.14. Since 1.12.1 maps By onto S | and P- 1 onto {(u" 1) | u},~e, this isomorphism induces a P-algebra extension isomorphism from ~,~eu'Bv onto S| Hence, since J~ induces an embedding from ~" e u ' B r to ~,~e u'B~ = do(P'H), S | doP becomes a source algebra of b as a block of P ' H . This shows that b is still a nilpotent block of P 9H (cf. [18], Th. 1.6) and therefore

Extensions of nilpotent blocks

21

proves already the uniqueness of the P-algebra S and the existence of a suitable Pstable C-basis of S (cf. [18], Th. 1.6). In particular, as Aut(A~) stabilizes ~,~e u'B~, we have a group isomorphism Aut(C,/~ ~ ~ Aut(S | C , s ~

(1.14.1)

6

mapping 0 ~ , u t ( C , s ~ upon i~ts @ 0 (cf. [18], Cor. 6.9 and [15], Prop. 2.1). 1.15. When Q is not normal in H, it is useful to know the following alternative definition of/V 1, which depends on a theorem of Dade (cf. 2.12.4). The action of N~(Qa) on the simple k-algebra B(Q~) (cf. 2.7) defines a group homomorphism NG(Qa)~ Aut(B(Q~)), and then, considering B(Qa) • as a central extension of Aut(B(Q~)) by ,(• we claim that (cf. 2.4). 1.15.1. there is a group extension isomorphism ^

ResN,;(Q~)(N ) ~- ResN,(Q~)(B( Q~) • ) mapping (s(vz" l), vz) upon (sa(z), vz) for any v 9 Q and any z 9 Cu(Q). This isomorphism provides another definition of N; indeed, we have N ~ N~(Q~)/Q "Cn(Q) (cf. 1.7.2) and, identifying Q ' C n ( Q ) with its image in ResN~.IQ,)(B(Qa) ~ ) through the injective group homomorphism mapping vz upon (s~(z), vz), 1.15.1 induces a group extension isomorphism IV ~- ResNdQ~)(B(Q~) •

(1.15.2)

Set ResN~tQ,)(N)=/VG(Qo).; to prove 1.15.1 it suffices to exhibit a group extension homomorphism Na(Qa ) --* B(Qo) • extending the group homomorpbism Cn(Q)--* B(Q~) • induced by s~. We have already a group extension homomorphism ]V~(Q~) --* s(B~) •

(1.15.3)

extending the group homomorphism Cn(Q) ~ s(BB) induced by s (cf.l.l 1). Moreover, the canonical homomorphism ,( | B~ ~ s(B~) determines a strict covering exomorphism of interior H-algebras over ~ (cf. [18], 4.t4); indeed, setting also s(Bo) = Bo/J(B~), it follows from Theorem 1.6 and Example 4.25 in [18] that the homomorphism ~r | s(Bo) determines a strict covering exomorphism of interior Q-algebras over k (since b is a nilpotent block of H), and therefore it follows from Theorem 3.8 and Corollary 4.23 in [18] that the induced homomorphism Ind~(~f | Bo) ~ Indg(s(B~)) determines a strict covering exomorphism of interior H-algebras over ~' (since H is locally controlled by Q on ~ | B~, so on s(B~)); now the statement follows from Theorem 3.4 in [15]. and Proposition 4.17 in [18] 2. Consequently, s(6) is contained in a local point 6 of Q on s(Bo) (cf. [18], 4.14 and 4.19), and we have an NG(Q~)-algebra isomorphism (cf. [18], 4.14.2) B(Q~) ~ s(Bp)(Q'3)

(1.15.4)

This alternative definition is only employed here in 1.20 below to show a consequence of Theorem 1.12 2 This argument reproduces line by line the first part of the proof of Proposition 7.2 in [18], and the reader may refer himself there for more detail.

22

B. Kfilshammer and L. Puig

and a group extension isomorphism (cf. 2.12.4, [18], Th. 1.6 and [19], 6.22) f ,~B,)(Q~) = :~IB~)(Q~) ~- 'e• • F~,)( Q~) .

(1.15.5)

Now it suffices to compose the group extension homomorphism 1.15.3, whose image is contained in N~(B,)'((~3) where Q is the image of Q in s(Br • with the group extension homomorphism Ns(B#)~(O,6)~B(Q6) • obtained from 1.15.5, 2.12.3 and 1.15.4. 1.16. We will explain now why Theorem 1.12 provides complete information on the blocks of G lying over b, as announced in 1.7. First of all, by 2.14.1 there is a unique interior G-algebra embedding ~ : A, ~ IndeG(A~)

(1.16.1)

-,: oft ~~ = d~(A~), and therefore the local pointed groups on A~ are such that ReseG(g,) just the G-conjugates of local pointed groups on A~ (cf. [17], 1.12). Moreover, as A~ -~ S| (5~,L~ as interior P-algebras (cf.l.12.1), for any subgroup R of P there is a bijection ~.-CP~A(R) _~,.~i~r {o(R ) mapping e , ~ on the unique z(e~ ~ : ~ , s that m ~ ( : ) #^0 (cf. 2.3 applied t o ' l | (~9,/~~ R ~ (A~)R, and [18], Th. 5.3), and considering (9,L ~ as an L-algebra we set z(g)~(~o) = r ( R : ) .

(t.16.2)

So the local categories of A, and C,/~ ~ (cf. 2.8), and the local fusion category of A~ (cf. 2.12) have, up to isomorphism, the same objects and the following statement shows that they have also the same morphisms. 1.16.3. If R : and T~o are local pointed groups on A~ then E~(Tq,o , R : ) = FA(Tr and r T R(FA(Tq, o , R : ) ) = EL(r(Tr r(R~o)) where TT. R is the bijection from ITIom(T, R) onto Iqom(z(T), z( R )) induced by r. Indeed, by Theorem 3.1 and Proposition 2.14 in [17] we know already that Ea(Tr

R : ) = FA(Tq,o, R : ) = FA (T~o, R : )

and identifying z(P) with its unique lifting to/~o, that EL(Z(Tr

z(R:)) = Fc%s

z(R~o))

(since, by Lemma 5.5 and Proposition 5.15 in [19], C,/~ ~ can be embedded in a group algebra CL' of a finite central if-extension L' of L). Moreover, by 1.12.1, C , s ~ can be embedded in S o | A~ (since C can be embedded in S o | S) and therefore, again by Proposition 2.14 in [17], we have Fc,s162

), z(R:)) = Fso|

(z(r~o), z( R : ) ) .

Now, 1.16.3 follows from the following general result. Lemma 1.17 3. With the notation above, let D be an interior P-algebra having an (9basis X such that P" X" P = X and [P" x[ = JP] = Ix" P I f or any x ~ X. I f R~ and T o 3 This lemma is not employed in the sequel. Actually, it should have been already in [18], w

Extensions of nilpotent blocks

23

are local pointed groups on D then FS|

Tg,,) = FD(R~, T~,) c~ Fs(gp, T~)

(1.17.1)

(

where p and tr are respectively the unique local points of R and T on S and e' and ~b' the unique local points of R and T on S | D such that and

BrS(p) | Br~(e) c BrS|176

BrS(tr) |

~ Br s| o(~,,)

Proof It suffices to prove that Fs(Rp, T~) contains Fs| e o(R~,, T~,,) (cf. [18], Th. 5.3.). Moreover, if R,, c T,, then R ~ T and therefore Rp ~ T,; hence, we may assume that IRI = ITI (cf. [17], 2.11). The existence of X shows that the canonical image of CR in D~ is isomorphic to OR and a direct summand of D~ as (9(R x R)-modules by left and right multiplication (cf. the proof of 3.4 in [17]); hence CR is a direct summand of S | so a direct summand of (S | D~)~,, as (9(R x R) modules since R• (ResteR) e• (S)) ReshiP(S) | CR ~- Res e• R • R(S) | IndR~)R(c) _~ Ind,(R) where 6(R) is the diagonal subgroup of R x R. In particular, i f j ' e e ' there are b' ~j'(S | D~)Rj' and an (9(R x R)-submodule M ofj'(S | O~)j' such that j'(S | D)j' ~ M (~((~(9(v'b')~ . r \veR /

Let ~b be an element of Fs| T~,,), denote by 6~: R ~ R x T the homomorphism mapping v ~ R upon (v, q~(v)) and choose a' ~ (S | D) • such thatj '~' e ~9' and (v "j')~' =j'~' "tp(v) for any v e R (cf. [17], Prop. 2.12). It is now clear that Ind~R~(C ) is a direct summand ofj'(S | D,)j'a', so a direct summand of S | D, as (9(R• T)-modules; hence, S~174 is a direct summand of SO | S | D always as (9(R • T)-modules. Consequently, since S | S O ~ Ende(S ) as interior P-algebras and S~

R• P• 0 )) Ind~,~g~(C) ~ Ind~AR)(aes~,(gt(S

as (9(R • T)-modules, on one hand S | ~| is a permutation (9(P • P)module and therefore Res~,~R~(S~ is a permutation CR-module; on the other hand we have Rest(Rest(S)) ~ | Rest(S) -~ Endv(aes~R~(S~ C

as interior R-algebras (we identify R with 6~(R)). Finally, since (9 can be embedded in E n d ~ ( R e s ~ ( S ~ and in End~(Res~(,~(S~ it follows that Sp and Rest(S,) can be embedded in Ende,(aes~,~g~(S~ | Rest(S) and therefore Sp ~ Rest(S,) since R has a unique local point on that interior R-algebra; hence (0 e Fs(Rp, T,) (cf. [17], 2.18). 1.18. Another consequence of the isomorphism 1.12.1, together with 1.16. l, is that the (9-algebras A~, Ar and (9,/~ ~ are pairwise Morita equivalent; in particular, we get

24

B. K/ilshamrner and L. Puig

evident isomorphisms Z ( A , ) ~ Z(A~) "~ Z(O,I~ ~

(1.18.1)

which induce bijections between the sets of blocks of these C-algebras. Let a ~ be a block of G lying over b, so that ct~ = {a ~ } is a point of G on A~, and denote by a ~ the corresponding primitive idempotent of Z ( A J and by z(~ ~ = {z(a~ the corresponding point of L on O,/~,~ notice that A~a ~ is still an interior P-algebra and that we have A~a ~ ~- S | (O,/_~,~ ~ (1.18.2) o as interior P-algebras. Now it is quite clear from 1.16.3 that, for any local pointed group R~o on A~, the following conditions are equivalent 1.18.3. R~o is a defect pointed group o f G~o 1.18.4. R~o is a maximal local pointed group on A~a ~ 1.18.5. T(R~o) is a defect pointed group of L~t~% and, in that case, an embedded algebra associated with the unique local point of R on Rest(S) | ((9,/~~ (cf. [19], 2.13) and [18], Th. 5.3) is a source algebra of a~ in particular, as we say above (cf. 1.1), all the invariants currently associated with a ~ can be computed from Rest(S) | (0,/.~,~ and therefore related to the corresponding invariants ofz(a ~ (recall that, by Lemma 5.5 and Proposition 5.15 in [19], (9,L ~ can be embedded in the group algebra of afinite central if-extension L' of L, and therefore z(a ~ is a block of L' too). 1.19. For instance, we will describe the relationship between the full matrices of generalized decomposition numbers of a ~ and z(a~ Assume that Char(O) = 0 and denote by ~r the quotient field of (9, by Irr~c(G, a ~ the set of irreducible J l characters of G associated with a ~ and, for any ~(~ Irr~(G, a~ by X~ the corresponding irreducible W-character of Lo (recall that 1.12.1 and 1.16.1 induce a Morita equivalence between Aa ~ and (O,/~~176 Let R~o be a defect pointed group of G,o such that R c P and m~o ~: 0 (cf. 1.16), and ~ o a set of representatives Ug,o e R~o for the G-conjugacy classes of local pointed elements on Aa ~ Notice that, with evident notation, it follows from 1.16.3 that {z(U~,o)},,oe%o is a set of representatives for the L-conjugacy classes of local pointed elements on (O.L~176 I f z ~ Irr~(G, a ~ and Ur ~ q/~o, we denote by Z(Ur and ~(z(U,o)) the corresponding generalized decomposition numbers; moreover, for any u e P, we denote by co(u) the "sign" of the trace of the image of u in S ~ (cf. [ 18], 1.11.1). Now, as in 1.12 of [18], it is not difficult to prove that: 1.19.1. For any z~Irroc(G, a ~ and any U~,o~ll~o we have

x(U~o) =

co(u);~'(z(U,o)).

1.20. A last remark. Setting G' = NG(Q~), H ' = Q 9CH(Q), A' = OG', B' = OH' and denoting by b' the block of H ' such that Br~(b')Br~(6) = Br~(6) (recall that B(Q) ~- ~r it is clear that b' has a nilpotent local structure (cf. [18], 1.7). Then, with evident notation (cf. 1.6 and 1.7), we have ct' = fl' = {b'}, N ' = G',

Extensions of nilpotent blocks

25

1~' --/V (cf. 1.7.2) and, for any local pointed group R, on B such that Qa c R~, there is a unique local pointed group R e, on B' such that B r g ( g ) = Brg(e) (since Br~(B 'R) ~ ~eCn(Q)R and B(R) _~ xfCn(R)); moreover, it is clear that R~, c P~, if and only if R, ~ Pr (since both inclusions are equivalent to RBr~t~) c PBr~I~))- On the other hand, it follows from 1.8.1 that Q = H c~ P and from 1.13.1 that 6 is the unique local point of Q on B such that Q~ c P~ (cf. [18], th. 5.3); in particular, ifR~ is a local pointed group on B such that Q~ ~ R~ and x an element of N such that ( R y ~ P~, x normalizes Q~. Consequently, identifying N ' with N through 1.7.2 and thus considering N' as an N-permutation group (since N' ~ N), we have

Es,~( R~, P~) = Es,,~(R~,, P~,)

(1.20.1)

for any local pointed group R~ on B such that Q~ c R~ ~ P~, and therefore

1.20.2. the group L and the group homomorphisms n: L ~ N and z: P ~ L fulfil conditions 1.8.1 and 1.8.2 with respect to G', H' and b'. That is, the group N~(Q~) and the block f = b' determine L and, since B(Q~) ~- B'(Q~,) as N~(Q~)-algebras, they determine the central J• L and the interior P-algebra (9.s ~ too (cf. 1.15.2); this shows, for instance, that (cf. 1.12.1 and 1.16.1).

1.20.3. the (f-algebras OGb and CNG(Q~)f are both isomorphic to full matrix algebras over C , s ~ which generalizes the main result of [7]. This paper is divided in eight sections. In w we list notation, terminology and quoted results; although we follow essentially the point of view introduced in [15], it is handy to replace interior G-algebras by slightly more general objects named Galgebra extensions-which not only simplify statements but suggest the adequate equivalence classes of homomorphisms that we may c o n s i d e r - a n d in w we develop the corresponding background. In Sect. 3 we study the so-called semicovering exomorphisms: this class of G-algebra exomorphisms has been introduced by Puig some time ago to understand the relationship between CG and (9(G/Z) when Z is a central p-subgroup of G4; it turns out that the same class is useful to study the relationship between CH and CG when H is a normal subgroup of G such that G/H is a p-group, namely to study p-extensions of blocks; actually, together with the main result of [18], it is the most important tool to prove Theorem 1.12 when G/H is a p-group. It is not difficult to prove that P/Z(Q) is a Sylow p-subgroup of NG(Q~)/Cn(Q) and that the element of H z (P/Z(Q), Z(Q)) determined by P belongs to the image of H 2 (Na(Q,)/Cu(Q), Z(Q)) and therefore determines a group extension L of N by Q (cf. 1.7.2); but such an approach has the disadvantage that L is not necessarily unique up to a unique group exoisomorphism; an important part of this paper is devoted to avoid this difficulty, and in Sect. 4 we expose the general criteria we need 4 In order to prove the existence of a bound for the dimension of a block source algebra in terms of Cartan integers and dimensions of the irreducible source modules. This result has been announced in May 1982, in an Oberwolfach meeting, and will appear in a forthcoming paper

26

B. Kfilshammerand L. Puig

to get unique group exomorphisms between group extensions. In Sect. 5 we discuss to what extent the local category of B, (cf. 2.8) is determined by AT. In w we essentially handle Theorem 1.12 when/V is a p-group. In Sect. 7 we exhibit a wide class of conditions which answer the question on the uniqueness of L raised above: we have chosen conditions 1.8.1 and 1.8.2 as the easiest ones to state. Finally, in Sect. 8 we prove Theorems 1.8 and 1.12.

2. Notation, terminology and quoted results 2.1. We follow essentially the notation and terminology introduced in [15], [17], [18] and [19] except that we often replace groups by permutation 9roups and interior G-algebras by G-aloebra extensions (both terms are defined below). Throughout the paper p is a prime number, ,( an algebraically closed field of characteristic p and C a complete discrete valuation ring with residue field ~ (we allow the case C = ~). For any 2~ (9 we denote by 2 the image of 2 in ~; more generally, if M is a torsion free C-module, we denote by )( the image of any X c M in A | Recall that the canonical group homomorphism C • ~ A• has a unique section (cf. [22], II, Prop. 8); we denote by 2 the canonical lifting of any 2~,( X to C • , 2.2. All the C-algebras we consider are associative with unity and C-free of finite rank as C-modules. An C-algebra isomorphic to a finite direct product of full matrix algebras over (_9is shortly called C-semisimple, and C-simple if there is just one factor. Let A be an C-algebra: all the A-modules we consider are C-free of finite rank; we denote by A ~ the opposite C-algebra, by A • the group of invertible elements of A, by Z(A) the center, by Aut(A) the group of automorphisms, by J(A) the Jacobson radical and by ~(A) the set of A • classes of primitive idempotents shortly called points of A. Whereas, if S is a set, $(S) denotes the set of all subsets of S to avoid confusion. An ideal a of A means always a two-sided ideal, and we set J(a) = a c~ J (A) and denote by ~ ( a ) the set of ~ ~ ~ ( A ) such that c~ c a. For any c~e ~ ( A ) we denote by A (~) the simple quotient of A associated with ~, by s,: A--* A(~) the canonical homomorphism, by A ' ~ ' A the ideal of A generated by ~, and by m, the dimension of any simple A(c0-module. 2.3. A h o m o m o r p h i s m f : A --* A' between C-algebras is not required to be unitary. For any ~ ( A ) and any ~ ' ~ ( A ' ) we denote by m(f)~, (or m~, when no confusion is possible) the dimension of s,,(f(i))" M' where i t c~and M' is a simple A'(~')-module. In non-commutative algebra it is handy to consider homomorphisms up to inner automorphisms: an exomorphism f from A to A' is the orbit ofa homomorphism f : A ~ A' in Horn(A, A') under the evident action of the group A • x A'• Notice that orbits of A • x A'• and A'• on Hom(A, A') coincide (cf. [19], 2.3), and therefore

2.3.1. exomorphisms of C-algebras can be composed. We denote by flom(A, A') the set of these orbits. We say that f ~ ITIom(A, A') is an embeddin9 if K e r ( f ) = {0} and I m ( f ) = f ( 1 ) A ' f ( 1 ) .

Extensions of nilpotent blocks

27

2.4. If G is a group, we denote by Z(G) the center of G, by D(G) the commutator subgroup of G, by Op(G) the subgroup generated by the normal finite p-subgrofips of G, by OP(G) the intersection of the normal subgroups of G such that the quotient is a finite p-group, and by N~(H) and C~(H) the normalizer and the centralizer of a subgroup H ofG. Ift~ is an extension of G by an (abelian) group Z and ~,: H --* G is a group homomorphism, we denote by Re%(~;) the extension of H by Z formed by the group G x H of pairs (.~, y)~(~ x H such that ~b(y) is the image of~ in G and the

group

homomorphisms Z ~ ( ~

x H G

mapping z e Z

upon (z, 1) and

(~ x H ~ H mapping (2, y)e G x H upon y; we write ResH((~ ) instead of Res~,((~) G

G

when the choice of ~ is evident from the context. If q~:Z~(~*• is a group homomorphism (and G is finite), we may consider the twisted group algebra

6)G/

~

(9.(z~ - (p(z)2)

(2.4.1)

zeZ,,~d that we denote by (9.(~ when there is no confusion on p, namely ifZ = ~ • and q~ is the canonical section (cf. 2.1). But in this paper we will systematically consider permutation groups rather than groups: if S is a set, an S-permutation group G is a group still denoted by G - endowed with an action on S (i.e. with a group homomorphism from G to the symmetric group E(S) of S); then any subgroup of G is considered as an S-permutation group with respect to the restricted action. Notice that, as any group can be considered as an ~ - p e r m u t a t i o n group, everything about permutation groups specializes to groups (and then we will omit to mention the empty set). 2.5. Let G and G' be respectively S- and S'-permutation groups for some sets S and S'; a homomorphism from G to G' (or from (G, S) to (G', S') to avoid confusion) is a pair (q~,f) where q~: G ~ G' is a group homomorphism and f : S ~ S' a map such that f ( x ' s ) = ~o(x)'f(s) for any x e G and any seS. If S' = S, we denote by Homs(G, G') the set of S-permutation group homomorphisms from G to G'; similarly, we denote by Auts(G) and Auts(G') the groups of S-permutation group automorphisms of G and G'. The canonical maps G ~ Aut(G) and G' ~ Aut(G'), and the structural homomorphisms G--* IE(S) and G ' ~ Z(S) induce evident group homomorphisms G-~ Auts(G) and G ' ~ Auts(G'); then composition induces an action of G x G' on Homs(G, G') and as above we call S-permutation group exomorphism from G to G' any orbit of G x G' on Horns(G, G') and denote by 17toms(G, G') the set of these orbits. Again G x G' and G' have the same orbits on Horns(G, G'), and therefore

2.5.1. exomorphisms of S-permutation groups can be composed. Moreover we have ( = ~Iom~ (G, G')).

an evident map

from

Homs(G,G' ) to ITtom(G,G')

2.6. Let S be a set and G a finite S-permutation group. A G-algebra B (over (9) is an (9-algebra endowed with a group homomorphism ~0: G ~ Aut(B); we usually write b~ instead of q~(x-~)(b) for any x e G and any b e B. If H is a subgroup of G, BH denotes the unitary subalgebra of H-fixed elements of B and, for any subgroup

28

B. Kiilshammer and L. Puig

K of H, Tr~n: B r ~ B H denotes the relative trace map and B~ its image (cf. [19], 2.8). For any p-subgroup P of G we set B(P) = B P /(~QBQP + J((9)'B P) where Q runs over the set of proper subgroups of P, and we denote by Brp: B P ~ B(P) (or Bre~ to avoid confusion) the canonical homomorphism (see [19], 2.9 for more detail). 2.7. Let B be a G-algebra. A pointed group Hp on B is a pair formed by a subgroup H of G (considered as an S-permutation group!) and a point fl of BH; we say that fl is a point of H on B, set B(Hp) = BH(fi) (cf. 2.2) and denote respectively by Nc;(H~) and Co(H~, S) the stabilizers of fl in No(H ) and in the intersection of Co(H ) with the kernel of the action of G on S; if H = ( x > we say that x~ is a pointed element on B. If K~ is another pointed group on B, we write K~ c Htl (or y~eHp if K = ) and say that K~ is contained in Hr i l K c H and for any i~fl there i s j e ? such that ij = j = ji. A pointed group P~ on B (or a point 7 of P on B) is local if Brp(7') 4: {0 }; we denote by LP~n(P ) the set of local points of P on B and coherently we set ~ 8 ( H ) = ,~'(BH) (cf. 2.2). A defect pointed group P~ of Ha is a maximal local pointed group on B such that Py c Ho or, equivalently, a minimal pointed group on B such that fl c Tren(BP" 7 9B P) (see [19], 2.10 for more detail). Notice that any subgroup and any element of G can be respectively considered as a pointed group and a pointed element on the trivial G-algebra C:J(and then we will omit to mention the point, which is always { 1}). 2.8. Let Ht~ and K~ be pointed groups on B. Any element x of G such that (K~) ~ c H~ (where (K~.)x = (K~)~x) determines an S-permutation group homomorphism from K to H mapping y ~ K on y~ and s ~ S on x - ' 9s (cf. 2.5); we denote by (K~, x, Ho)s the corresponding exomorphism and say that (K~, x, H~)s is a Gexomorphism (or a (G, S)-exomorphism to avoid confusion)from K~ to Hp; that is, (K~, x, H~) s is an element of ITtoms(K, H) (cf. 2.5) and, if y is another element of G such that (K~) ~ c Ho, we have (K~, x, HO)s = (K~,, y, Ho) s if and only ify = zxh for some z~Ca(K~, S) and some h ~ H (see [12], Ch. I, w and [17], Def. 2.1 for similar definitions). We denote by Ea,s(K ~, Ht) the subset of Horns(K, H) of G-exomorphisms from K~ to Ho and set Ea,s(Ho)= E6.s(Ho, He). Notice that the canonical map Horns(K, H ) ~ ITIom(K, H) (cf. 2.5) induces a surjective map (cf. [17], Def. 2.1)

E6,s(K ~, H~) --+ Eo(K~, Hp) ( = EG, ~(K~, H~)).

(2.8.1)

Moreover, as any composition of G-exomorphisms is a G-exomorphism, Eo, s(Hp) is a subgroup of k, uts(H) isomorphic to NG(Hp)/H'Co(H ~, S). We call local category of B the category where objects are the local pointed groups on B and morphisms are the corresponding G-exomorphisms. If P~ is a local pointed group on B, we denote by C~(P~, S) the inverse image of Op(Ea, s(PO) in NG(P~). 2.9. An interior G-algebra A (over O) is an O-algebra endowed with a group homomorphism ~o: G--* A• we usually write x . a . y instead of ~o(x)acp(y) (see [193, 2.11 for more detail). But in this paper it is handy to consider a slightly more general structure: a G-algebra extension A is an interior G-algebra - still denoted by A - e n d o w e d with a G-stable subalgebra B containing the unity of A. Then B has

Extensions of nilpotent blocks

29

an evident G-algebra structure; conversely, as in 2.7 of [18], we denote by BG the interior G-algebra formed by the free B-module over G endowed with the evident product fulfilling

(bx)(cy) = bcx 'xy for any x, y ~ G and b, c e B; except when a different G-stable unitary subalgebra will be explicit, we consider BG endowed with Be, where e is the unit element of G, as a G-algebra extension. If ~9: H ~ G is an S-permutation group homomorphism, we denote by Reso(A ) the H-algebra extension defined by the group homomorphism q~o ~b: H ~ A • and the same subalgebra B; when H is a subgroup of G and 0 the inclusion map (with ids), we set Res0(A ) = ResGn(A). Notice that if B = A then we have nothing more than the interior G-algebra A and therefore everything about Galgebra extensions specializes to interior G-algebras. Similarly, if A = BG then the whole structure depends only on the G-algebra B and therefore everything about G-algebra extensions specializes to G-algebras by restricting to the structural subalgebra; in particular, this remark applies to the exomorphisms defined below. 2.10. Let A and A' be G-algebra extensions and denote respectively by B and B' the structural subalgebras. We denote by A | A' the G-algebra extension formed by the tensor product of A and A' endowed with the subalgebra B | B' and the evident diagonal group homomorphism G ~ A• |

• c(A|

• 6

A homomorphism f from A to A' is an (9-algebra homomorphism such that f ( B ) c B' a n d f ( x ' a ' y ) = x ' f ( a ) ' y for any x, y 6 G and any a~A. We denote by Horn(A, A') the set of G-algebra extension homomorphisms from A to A'. It is clear that conjugation induces group homomorphisms from (B G) • to Aut(A) and from (B 'G) • to Aut(A'), and then composition induces an action of(BG) • • (B m) • on Horn(A, A'): an exomorphism f from A to A' is the orbit of a homomorphism f : A ~ A' under this action; again this orbit coincides with the orbit of f under the action of (B'G) • and therefore 2.10.1. exomorphisms of G-algebra extensions can be composed. We denote by ITtom(A, A') the set of these orbits. We say that.Te I~Iom(A, A') is an embedding if K e r ( f ) = {0},f(A) = f ( 1 ) A ' f ( l ) a n d f ( B ) = f ( 1 ) B ' f ( 1 ) ; notice that

2.10.2. If ~ A ~ A' is an embedding, for any pointed group H~ on B, f(fl) is contained in a unique point fl' of H on B', we have f - l ( f l ') = fl and f induces a ~r algebra embedding f(H~): B(Hp) -* B'(Hp,). Often we denote fl and fl' by the same letter. If ~k: H --* G is anS-permutation group homomorphism and f e H o m ( A , A'), we denote by Res0(f) the exomorphism from Reso(A) to Reso(A') (cf. 2.9) containing .7; as above, when H is a subgroup of G and 0 the inclusion map (with ids), we set Re%,(f) = Resg(f); notice that (as in 2.12.2 of [19])

2.10.3. ifj~ OeFIom(A, A') we have R e s , ( f ) = Reso(0) if and only if f = g.

30

B. K/ilshammer and L. Puig

Indeed, if b' ~(B') • fulfils g(a) = f ( a ) b' for any a t A then the right multiplication by b' induces a B'G-module isomorphism from B'f(1) onto B'g(l) since we have (a')Xb ' = x -1" a'f(1 "x)b' = x -1 "a'b'g(l "x) = (a'b') x (2.10.4) for any a' ~ A'f(1) and any x ~ G; it follows that f(1) and g(l) are (B m) • -conjugate; thus, we may assume that f ( l ) = 9(1) = 1 and then 2.10.4 implies that G fixes b' (see the proof of Lemma 3.7 of [15] for more detail). 2.11. Let A be a G-algebra extension and denote by B the structural subalgebra. An embedded algebra extension (C, O) associated with a pointed group H a on B is a pair formed by an H-algebra extension C and an embedding 0: C ~ Rest(A) such that g(1)~fl; as in 2.13.1 of [19], it is clear that 2.11.1. for any pointed group Hp on B there is an embedded algebra extension ( C, O) associated with Ha; moreover, for any embedded algebra extension ( C', O') associated with H a there is a unique exoisomorphism h: C' ~- C such that O' = 9 '~h.

Indeed, if j ~ fl it suffices to consider the (;'-algebra C = j A j endowed with the subalgebra j B j and the group homomorphism H ~ C • mapping y ~ H upon y-j, and the embedding 0 determined by the inclusion C c A. Moreover, if (C', 0') is another embedded algebra extension associated with H a , we may assume that g'(1) = j ; then 9' induces an H-algebra extension isomorphism C' ~ C and the uniqueness is clear since we may assume that g'(c') = h(c') for any c'~ C'. Usually we denote by ( A p , ~ ) an embedded algebra extension associated with H a chosen once for ever, and by Bp the structural subalgebra. Notice that the argument above, which proves the existence and uniqueness of h, shows that 2.11.2. if H a is a pointed group on B and O: C -* R e s t ( A ) an H-algebra extension exomorphismsuch that g(1)6fl, there is a unique exomorphism h: C ~ Ap such that 0 = fa ~ h

(see [ 18], 2.10.1 for the corresponding result on interior G-algebras). Similarly, it is not difficult to prove that (cf. 2.7) 2.11.3. for any pointed groups H a and K~ on B such that K c H, we have K~ c H a if and only if there is a K-algebra extension exomorphism~P: A e ~ Res~(Ap) such that J~ = Res~(J~)oJ~e, and t h e n # is a uniquely determined embedding

(see [19], 2.13.2 for the corresponding result on interior G-algebras). 2.12. Let us recall the definition of A-fusions (cE [17]) - that we only need in Sect. 1. Here we assume that S = ~ and B = A: that is, G is just a finite group and A an interior G-algebra. Let Hr and Ky be pointed groups on A; an A-fusion Cpfrom K~ to H o is a group exomorphism (5: K --* H such that ~0 is injective and there is an interior K-algebra embedding f~0: Ay --, Re%(Ap) fulfilling Res~(J~) = Resf(J~) o Resf(J~).

(2.12.1)

We denote by FA(K 7, Ha) the set of A-fusions from K~ to H a and set FA(Ha) = FA(Hp, Hp) which is a subgroup of b, ut(H) (cf. [17], 2.6 and 2.7). We call local fusion

Extensions of nilpotent blocks

31

category of A the category where objects are the local pointed groups on A and the morphisms are the corresponding A-fusions (cf. [17], Def. 2.15). Moreover, let Py be a local pointed group on A and denote by N M ( P ) the normalizer in A~ of the structural image of P; the action of NA:(P ) on this image induces a group homomorphism from FA(P~) to NAr (p]/ (?~)x (cf. [17],• 2.13) and, considering NAr + J(A~)) as P. central ,( -extension of N A ~ ( P ) / P ' ( A ~ ) • , we set (cf. [19], 6.7 and 6.14) 7

t~A(Pr) = ResFA(p~)(NAf (P)/P" [l + J(A~)))

(2.12.2)

which does not depend on the choice ofJ~: A~ + Rest(A) (cf. [19], Prop. 6.8). It is not difficult to modify Proposition 6.t2 of [19], replacing G by A • and P by its structural image/5 in A x, to prove that: 2.12.3. I f FA(P,~) ~ k • x FA(P~,), there is a group homomorphism cr: N A~(/SQ--, A(P~) • such that s,+(ab) = ~r(ab) = s~(a)~

for any a ~ ( A p) ~ and any b~NA~(/5v).

By a theorem of Dade (cf. [7], (12) or [ 16]) this situation occurs in the following case: 2.12.4. I f A = End e (M), where M is an ~G-module, and P stabilizes an 6~-basis of A containing the unity then ff'~A(P) = {'/} and ~'A(P~,) ~-- s215x FA(P~.). In particular, if Q~ is a local pointed group on A and for any j e 6 we consider j " M as an C Q-module, an injective group exomorphism (9: Q ~ P is an A-fusion from Q~ to P~ if and only if j " M is a direct summand of Res+(M). Indeed, the existence and uniqueness of 7 follow from Corollary 5.8 of [18] and, together with Proposition 2.18 of [17], they imply the last statement. 2.13. Let H be a subgroup of G and C an H-algebra extension, and denote by D the structural subalgebra of C; the induced G-algebra extension Indg(C) is formed by the tensor product CG | C | CG endowed with the distributive product fulfilling (x|174174174

|174

ifyx'eH

for any x, y, x', y' e G and any c, c' + C, with the subalgebra ~r~a y | D | y - ~ and with the group homomorphism mapping x 9 G on ~y xy | 1 | y - ~ where y runs over a set of representatives for G/H in G. Moreover, we denote by dg(C): C ~ Res~Indg(C)

(2.13.1)

the canonical embedding determined by the H-algebra extension homomorphism mapping c E C on 1 | c | 1 (see [19], 2.14 for the corresponding construction for interior G-algebras). Conversely, 2.13.2. if A is a G-algebra extension and 9: C--* Rest(A) an H-algebra extension embedding such that 1 =Tr~(9(l)), g(l)9(1)~=0 for any x 9 and g(1)

32

B. Kfilshammer and L. Puig

centralizes the structural subalgebra B of A, then there is a unique exoisomorphism I n d , ( C ) ~- A such that ff = Res~(f)od~(C).

Indeed, notice first that, if such an exoisomorphism exists, we may assume that = x ' g ( c ) ' y for any x, y e G and any c ~ C ; conversely, it is clear that there is an C-module homomorphism f: I n d , ( C ) ~ A mapping x | c | y upon x ' g ( c ) ' y and it is easily checked that f is an interior G-algebra isomorphism. Moreover, as g ( D ) = Bg(1) (cf. 2.10) and B = ~x~o Bg(1):', we get B = f ( ~:,~o x | D | x - 1). Finally, if d ~ ( D n) • we have Tr~t(g(d))~(B G) • and (x "g(c)'y) Trf,lgla~) = x ' g ( c ) ' y for any x, y E G and any c ~ C , which proves t h a t f does not depend on the choice of g in g.

f(x|174

2.14. Let A be a G-algebra extension, ~ a point of G on the structural subalgebra B of A, and Pv a defect pointed group of G, (cf. 2.7). The following statement gives only a weak generalization of the corresponding result on interior G-algebras (cf. [15], Prop. 3.6), but it is strong enough to show that the interior G-algebra structure of the G-algebra extension A~ can "almost" be computed from the interior P-algebra structure of the P-algebra extension A t, (the "almost" depending on the so called multiplicity module of 7 as explained in Lemma 9.9 of [19] for interior G-algebras). 2.14.1. I f lip is a pointed group on B such that Hi3 c G~ and ~ c T r ~ ( B n . f l . B n) then there is a unique interior G-algebra embedding ~ : A~ ~ Ind~ such that the following diagram of interior H-algebra embeddings commutes

Rest(A,)

Res,~O~),ResnOlndnO(A#)

J~= T / / / ~

dn~

A# In particular, restriction through f~ induces an equivalence between the categories of A,-and Aa-modules.

Indeed, if i ~ ~ there are j ~/3 and h', b" ~ B n such that (i = j = ji (cf. 2.7) and i = Tr~(b'jb") (since Ci + iJ(BG)i = iB6i = Tr~n(iBnjBni)); now we may assume that A , = i A i and A p = j A j , and that f , , f a and f~ are determined by the inclusions j A j c i A i = A (cf. 2.11); then it is easily checked that the C-module

homomorphism

g: A, ~ Res~Ind~n(Aa)

mapping

a~A~

upon

~ , y x | j b "" x - a . a" y" b'j | y - 1, where x and y run over a set R of representatives for G / H in G, is an interior G-algebra homomorphism 5. Moreover, setting c' = ~ R 1 | x" b 'j | x - 1 and c " = ~ R X | j b " " x - ~ | 1, it is clear that c' c " = 1 | 1 | 1 and g(a) = c"(1 | a | 1)c' for any a~Ap;therefore, it follows from Lemma 3.7 of [15] that d~(A#)= Res~(ff)oJ~" where d~(A#) and J~" denote the

corresponding interior H-algebra embeddings (cf. 2.11.3 and 2.13.1). Finally, the fact that ff is a uniquely determined embedding (of interior G-algebras) follows from 2.10.3 and [19], 2.3.4. 5 This homomorphism, which simplifies the proof of Proposition 3.6 of [15], has been pointed out by C. Picaronny

Extensions of nilpotent blocks

33

2.15. In this paper we are mainly concerned with the following situation: we consider a set S, a finite S-permutation group G, a normal subgroup H of G acting trivially on S, the G-algebra extension A formed by the (P-algebra (PG endowed with the canonical map from G and with the subalgebra B = CH, and a point cc of G on B. For our purposes we have to translate some results in Chap. III of [12] to the local category of 13, (cf. 2.8): we try to be concise rather than complete. First of all we translate some terminology. A G,-biexomorphism (or a (G,, S)biexomorphism to avoid confusion) (~0, ~b) is a pair of elements of EG.s(Qo, P~) (cf. 2.8) where Qa is a local pointed group on B such that Qa c G, and Pr a defect pointed group of G, (cf. 2.7), and we say that IPI/[ Q I is the length I(~o, ql) of(~o, if); if (o = ( Qo, x, P~)s and qJ = ( Qa, y, P~)s, where x, y e G and Pr contains both ( Qa)x and (Q~)r (cf. 2.8), we set (r ~k) = (Q~, x, y, P~)s. Moreover, if qeE~.s(Qa, P~) we write (q~, ~) --- (q~, r/) + (q, q~) (2.15.1) and for any 2eEa,s(Q'a,, Qa) and any #eEG,s(P~., P'~.), where Q~, and P~. are local pointed groups on B, we set 2((p, ~)# = (/~o ~oo 2, #o ~9o 2) ;

(2.15.2)

notice that E~.s(Q'6,, Qa) ~ implies Q,~, c G,, and that E~,s(P~, P'r,) 4 - ~ if and only if P'r, is a defect pointed group of G,. The law of composition 2.15.1. being associative (when defined), we say that (q~, ~,) is a linear combination of a set r of G,-biexomorphisms if there are nonempty ordered families {2~}~ t and {#~}~ of G-exomorphisms and {(~0i, ff~)}i~t of elements of r such that we have

(~o, tp) = ~, 2i(tpi , ~i)l~i

(2.15.3)

i~l

in the obvious sense. Then we say that (~o, ~) is reducible if this G,-biexomorphism is a linear combination of the set of all G,-biexomorphisms of length smaller than l(~0, ~,), and irreducible otherwise (notice that this terminology differs slightly from the terminology introduced in Chap. III of [12]) 2.16. A generator set of the G,-biexomorphisms is a set ~ of G,-biexomorphisms such that any G~-biexomorphism (q~, ~,) is a linear combination o f ~ ; then it is quite clear that (see Lemma 5 of [12], Ch. III).

2.16.1. if fs is a generator set of the G,-biexomorphisms then the set of irreducible elements of ~ is also a generator set. To find minimal generator sets of the G,-biexomorphisms we need the following fact which generalizes Lemmas 3.9 and 3.10 of [18] and can be proved essentially by the same arguments. Notice that for any p-subgroup Q of G we have B(Q) _~ AeCn(Q) (cf. [19], 2.8.4) and therefore any simple ~Cn(Q)-module is associated with a local point of Q on B.

2.16.2. If Qa is a local pointed group on B and K a subgroup of G containing Q " Cn(Q), there is a unique point v of K on B such that Q~ c Kv. Then Q~ is a defect pointed group of K~ if and only if NK(Q~)/Q'CH(Q) is a p'-group and the simple ~e( C n( Q )/( H c~ Z ( Q ) ) )-module associated with 6 is projective, and in that case Br~(6)

34

B. Kfilshammer and L. Puig

is the unique point of B( Q ) f where f is the primitive idempotent of Z ( B( Q ) ) such that f B r ~ ( 6 ) = Br~(6). In particular, if L is a normal subgroup of K containing Cn(Q) and K / L is a p-group, we have K = R" L where R~ is a defect pointed group of K~. Indeed, setting (~ = Q.CH(Q)/Q, we have B ( Q o ) ( ~ ,~ and therefore s~(B K) ~,( which pro_ves the uniqueness of v. On the other hand, setting N = Nr(Q~)/Q, we have B ( Q J [ = IN/C]B(Q~) c and if B(Qo) c ~ ,( then the image f of f in Z(~(?) determines a block of defect zero of C which implies I ~ ( B ( Q ) f ) l = 1. Moreover, setting M = R ' L , we get easily NK(M ) = L'(N~(R~)c~ N~(M)) (since we have R~ ~ M z < Nr(M)u ~ K~), and therefore N r ( M ) / M is both a p-group and a p'group; so, as M is subnormal in K, we get M = K. 2.17. Let P~ be a defect pointed group of G,; as G acts transitively on the set of them (cf. [19], 2.10.2), the set of all the G,-biexomorphisms (Qa, 1, x, P~.)s is a generator set. First of all, we have the following reduction:

2.17.1. The set of all the G,-biexomorphisms ( Q~, 1, n, P~)s such that n 6 N G(Qa) and P~ contains a defect pointed group of NG(Qa) ~, where v is the point fulfilling Q~ ~ NG(Q~),., is a generator set. By 2.16.1 it suffices to prove that any G,-biexomorphism (Qo, 1, x, P~,)s is a linear combination of the union of the set of reducible G~-biexomorphisms and the set above; moreover, we may assume that Q # P and then there are local pointed groups Q~, and Q~,, on B such that (cf. [15], cor. 1.5)

Q~ ~ Q'~, ~ P,~ and

(Q~)~~ (Q~,,)~ ~ P.~.

Let v be the unique point of N~(Q6) on B such that NG(Q6) ~ contains Q6, Q,;, and Q~',, (cf. 2.16.3), R~ a defect pointed group of N~(Q6)~ containing Q~, (cf. 2.7), y an element of G such that ( R y c P~, (cf. [19], 2.10.2) and n an element of NG(Q6 ) such that (Q~,,)" ~ R~ (cf. [19], 2.10.2); now it is clear that

(Q6, 1, x, Py)s = (O6, 1, y, P,)s + (Qa, y, (Qa)Y)s((Q~)', 1, n', P,)s + (Qa, ny, x, P,)s and that the G~-biexomorphisms (Q~, 1, y, P~)s and (Q~, ny, x, Py)s are reducible (since Q;, # Qo :I: Q~,,).

2.17.2. I f (Q~, 1, x, Pr)s is an irreducible G,-biexomorphism then Qo is a defect pointed group of Q'CG(Q~,S) u, where I~ is the unique point such that

Q~ = Q" CG(Q~, S).. Indeed, by 2.17.1 we may assume that xeNG(Q~) and that Py contains a defect pointed group R~ of Q" C~(Qo, S), (notice that Q~ c R,); then we have x = mz where m e N~(R~) and z e CG(Q~, S)(cf. [19], 2.10.2), and therefore (Q~, 1, x, P~,)s = (Qo, 1, R~)s(R~, 1, m, P~.)s which forces Q~ = R~. 2.18. Let Qo be a local pointed group on B and/~ the unique point of Q" CG(Qo, S) on B such that Q6 c Q. C~(Q6, S)u (cf. 2.16.2). Now we assume that Q6 is properly contained in P~ and a defect pointed group of Q " CG( Q6, S)u. First of all, notice that

Extensions of nilpotent blocks

35

2.18.1. we have Cp(Qo, S) c Q. Indeed, as H is a G-stable (5-basis orB, there is a local point e ofQ" Ce(Qo, S) on B such that Q'Cp(Qa, S)~ c P~, and then a local point 6' of Q on B such that Qo, c Q. Cv(Qa, S)~ (cf. [1], Prop. 1.5); but, by 2.16.2 above and Theorem 1.8 o f [ l ] , we have 3' = 6 and therefore Q- Cp(Qa, S)~ c Q. CG(Qa , S)u (eft 2.16.2); consequently, Q = Q'Ce(Qa, S ) (cf. 2.17.2). Recall that EG,s(Qa ) is isomorphic to N~(Qa)/Q. Cc,(Qa, S) {cf. 2.8) and denote by 37 the image of X c N~(Qa) in Ec,s(Qa).

2.18.2. A G~-biexomorphism (Qa, I, x, P~,)s is reducible if and only if there are psubgroups d o . . . . . R. of EG,s(Qa ) such that N p ( Q a ) c Ro, mp. ,(Qa) < R. and Ri__,c~Ri + {I} [1"1 _
and

(R~):' c (R~)~,

and therefore we have (Oa, xi- ~, xi, Py)s = (Qa, 1, R,)s(R ~, zi_ ,xi_ ~, zixi, P~)s ; but Qa 4= R, s i n c e / ~ _ ~ ~ / ~ 4= {I} (cf. 2.I6.2), which completes the proof. Then, denoting by M the minimal subgroup of E~.s(Qo ) which contains Np(Qa ) and fulfils the following condition (cf. [12], Ch. I1, Prop. 1).

2.18.3. for any n~EtLs(Qa) - ~1 the intersection ~1 c~ )~1a is a f-group. It follows from Proposition 2 and Theorem 1 of Chap. II1 of [12], and from 2.18.2 above that (cf. 2.8)

2.18.4. a G~-biexomorphism (_Q~, 1, x, Pr)s such that x e N a ( Q a ) is irreducible if and only if Yc~E~,s(Qa ) - M, and then we have Q = P ~ P ~ and C~(Qa, S) =

Q" c~(o.~, s). 2.19. Finally, assume that H = {1} and that S is endowed with a f-divisible abelian group structure compatible with the action of G; in that case all the points considered

36

B. K f i l s h a m m e r a n d L. Puig

above coincide with {1} and we omit to mention them; so, P is just a Sylow psubgroup of G. For any subgroup K of G we denote by [~"(K, S) the corresponding n-th cohomology group (cf. [4], Ch. XII, w and it is quite clear that any Gexomorphism q~eE~.s(K,L ) induces a group homomorphism H"(~p) from H"(L,S) to H"(K, S) (cf. [4], Ch. XII, w In particular, the inclusion P c G induces an injective group homomorphism H"(G, S) ~ H"(P, S) and its image is the intersection of the kernels [K"((~0, ~)) = Ker(H"(q~) - H"(~9))

(2.19.1)

where (q~, r runs over the set of all the G-biexomorphisms from any subgroup Q of P to P (cf. [4], Ch. XII, Th. 10.1). Now, the argument of Lemma 9 of 1-12] proves that

2.19.2. If X is a subset of G containing N~(P) such that the family { (P ~ P~-', 1, x, P ) s } ~ x is a generator set of the G-biexomorphisms, the inclusion P c G induces an isomorphism H"(G, S) ~ (-] ~"((P c~ px-,, 1, x, P)s). xeX

3. Semicovering exomorphisms 3.1. An important tool to prove Theorem 1.12 when G/H is a p-group is a class of G-algebra exomorphisms that we name semicovering exomorphisms. This class allows us to lift pointed p-groups (see Proposition 3.15 below). Our approach tries to make evident the analogy with the coverin9 exomorphisms introduced in Sect. 4 of [18], the main difference coming from the fact that Theorem 3.13 below fails beyond the set of p-subgroups. 3.2. First of all, it is handy to consider the C-algebra case (i.e. the case where G = { 1}). Let A and A' be C-algebras, a and a' respective ideals of A and A' (cf. 2.2), and f: A ~ A' an C-algebra exomorphism (cf. 2.3) such t h a t f ( a ) c a'. We say t h a t f is a semicovering exomorphism from a to a' if f ( J ( a ) ) c J(a') a n d f ( a ) contains a maximal commutative (9-semisimple subalgebra of a' (cf. 2.2 and [14], w or, equivalently, m,= ~ m,,. (3.2.1) a e ~ l a ) - 2.P(Ker(J'))

~'~,~(a')

Then we say that f i s strict on a if moreover a c~ K e r ( f ) c J ( a ) or, equivalently, ~ ( a ) c~ ,~(Ker(f)) = ~ . Notice that, if f is a semicovering exomorphism from a to a', t h e n f ( a ) contains all the idempotents of a' ~ Z(A'). Moreover, it is clear t h a t f i s a semicovering exomorphism from a to a' if and only if s 1 7 4 @eA ~r | A' is a semicovering exomorphism from fi to fi' (cf. 2.1), and then f is strict on a if and only if s | strict on ft. When a = A and a' = A' we say just t h a t f i s a (strict) semicoverin9 exomorphism and then f(1) = 1.

Proposition 3.3. The foUowin9 conditions on f a r e equivalent: 3.3.1. The exomorphism f: A ~ A' is a semieovering from a to a'.

Extensions of nilpotent blocks

37

3.3.2. We have f ( J ( a ) ) c J(a') and for any ~' e ~ ( a ' ) there is 5g(~') c ~ ( a ) such that m,, = ~ ~,~( ,) m~ and J(~) coL' for any o:e,9~(o~'). 3.3.3. We have f ( J ( a ) ) ~ J(a') and there is a map )7.: ~ ( ~ ( a ' ) ) - - * ~ ( ~ ' ( a ) ) , preserving union and intersection, such that, for any , f ' c ~ ' ( a ' ) , we have ~ , ~,~, m,, = ~'~f*l.~') m, and )T induces an injective unitary exomorphism

H

[I

~e7"(,9~ ~'e.~" In that case, we have ~ ( f - l ( a ' ) ) - ) 7 * ( , ~ ( a ' ) ) = : ~ ( K e r ( f ) ) and, in particular,)Tis strict on a (land only (H~*(~(a')) = ~ ( a ) . Remark 3.4. I f ) 7 is a semicovering exomorphism from a to a', it is clear that )7"(5~') is the disjoint union ~ , ~ , ~ , ,S(~') for any ,9~' c ~ ( A ' ) , and that 5g(~') is the set of ~ . ~ ( A ) such that m(J')~, + 0 (cf. 2.3) for any ~ ' e ~ ( a ' ) (since m,, > ~ e,~(A) m(f)],m~). Proof. If)Tis a semicovering exomorphism from a to a' and T is a maximal commutative (9-semisimple subalgebra of a, t h e n f ( T ) is a maximal commutative (5~-semisimple subalgebra of a' (cf. [14], w and, in particular, for any primitive idempotent i of T such that f ( i ) + 0, f ( i ) is a primitive idempotent of a'; thus, for any ~ ' e ~ ( a ' ) , we have m~, = [~'c~ f ( T ) l = ~ let ~ TI = ~ m~ ~t

where ~ runs over the set of points of a such that f ( ~ ) c ~ ' . Assume now that 3.3.2 holds. To prove 3.3.3 it suffices to consider )7*:~(~(a'))~(;~(a)) mapping ~ ' c ; ) ( a ' ) upon the disjoint union [9 ~'~,'p' '9~(a') 9 Indeed, it is clear that such a map preserves union and intersection; moreover, we have 2 m,,= 2 2 m~= 2 m, (3.3.5) ~'e.~' ~'e ~/" ~e,~/'(~') ~e ~*(,v"') and, in particular, ~ ' ~ : e l , ' ) m,, = ~ f*l.e~)) m, which proves that , ~ ( f - ~(a')) )7*(,~(a')) = ~ ( K e r ( f ) ) (since for any ~ e f * ( ~ ( a ' ) ) there is ~ ' e ~ ( a ' ) such that .f(e) c e ' , whereas if e e , ~ ( f - ~ ( a ' ) ) - f * ( J / ( a ' ) ) , it follows from 3.3.5 that m(f)~, = 0 for any e ' e ~ ( A ' ) ) ; finally, as f ( J ( a ) ) c J(a'), )Tinduces a unitary exomorphism

1]

H

H

and the kernel is clearly equal to H ~ ~ ~lo)-/-*l ~1A(~). If 3.3.3 holds and T and T' are respectively maximal commutative (9-semisimple subalgebras of a and a' such that f ( T ) c T', we have ranke,(T' ) =

~

m,, =

_~

m, < r a n k e ( f ( T ) ) ,

and t h e r e f o r e f ( T ) = T'. Proposition 3.5. Let b and b' be respectively ideals of A and A' such that f ( b ) c b'. 1f f is a semicovering exomorphism from a to a' and from b to b' then f is

38

B. K/ilshammer and L. Puig

also a semicovering exomorphism from a + b to a' + b' and the associated map )7*: ~(,~(a' + b')) ~ ~(,~(a + b)) extends the associated maps 5 ( ~ ( a ' ) ) ~ ~(;~(a))

and

~(?~(b')) ~ ~ ( ~ ( b ) ) .

(3.5.1)

I f moreover f is strict on a and b then f is also strict on a + b. Proof Since J(a + b) = J(a) + J(b), if)Tis a semicovering exomorphism from a to a' and from b to b', we h a v e f ( J ( a + b)) c J(a' + b'). On the other hand, as ~ ( a ' + b ' ) = , ~ ( a ' ) w ~ ( b r ) , it follows from 3.3.2 that for any ~ ' ~ ( a ' + b') there is 5Y(~') c ~ ( a + b) such that m,, = ~ ~,,(~,) m~ and .f(~) c ~' for any ~ 5e(7'). Thus, by Proposition 3.3,fis a semicovering exomorphism from a + b to a ' + b ' ; Then, by Remark 3 . 4 , f * : ~ ( ~ ( a ' + b ' ) ) ~ ( ~ ( a + b ) ) maps {~'} upon ,%~(ct') and therefore extends the associated maps 3.5.1. Finally, the last statement follows from the last statement of Proposition 3.3. Proposition 3.6. Let A" be an C-algebra, a" an ideal of A" and ~: A'--* A" an exomorphism such that g(a') ~ a". 3.6.1. I f f and (t are respectively semicovering exomorphisms from a to ct' and from a' to a" then ~ , , f is a semicovering exomorphism from a to a" and we have (~ of)* = f * o ~*. In that case, if two semicovering exomorphisms among f (1 and ~ o f are respectively strict on a, a' and a, so is the third. 3.6.2. l f ~ o f is a semicovering exomorphism from a to a 'r and g( J (a') ) ~ J (a ") then 0 is a semicovering from a' to a". I f moreover ~ is strict on a' then f is a semicovering from a to a'. Proof If )7 and .~ are respectively semicovering exomorphisms from a to a' and from a' to a", we have g(f(J(a))) ~ g(J(a')) ~ J(a") and, for any maximal commutative C-semisimple subalgebra T of a, f ( T ) is a maximal commutative Csemisimple subalgebra of a', whence g ( f ( T ) ) is a maximal commutative Csemisimple subalgebra of a". Conversely, if g ( f ( T ) ) and T' are respectively maximal commutative C-semisimple subalgebras of a" and a' in such a way that f ( T ) ~ T', it is clear that g ( T ' ) = g ( f ( T ) ) which proves the first statement in 3.6.2. If moreover a ' n K e r ( g ) ~ J(a'), the intersection a'c~ g - l ( J ( a " ) ) has no nonzero idempotents and therefore we g e t f ( J ( a ) ) ~ a'c~ g-~(J(a")) ~ J(a'); on the other hand, as T ' c ~ K e r ( g ) = {0}, the equality g ( T ' ) = g ( f ( T ) ) and the inclusion f ( T ) c T' imply that f ( T ) = T'. Corollary 3.7. Let A and A' be respectively quotient algebras of A and A' such that f induces a homomorphism f: A ~ A', and denote respectively by ~ and fi' the images of a in A and a' in A'. l f f is a semicovering exomorphism Jrom a to a' then ~ is a semicovering exomorphism from ?t to fi'. Proof By 3.6.1, the exomorphism A ~ ,4' induced by f is a semicovering from a to fi' and then, by 3.6.2, the exomorphism f is a semicovering from fi to fi' (since J(fi) is the image of J(a) in A).

Proposition 3.8. Let B and B' be C-algebras, ~ : B ~ A and ~':B'-~ A' 6'-algebra embeddings and (t:B ~ B' a unitary exomorphism such that ~'o~ =fo~. Set

Extensions of nilpotent blocks

39

b = e- 1(a) and b' = e' l (a'). I f f is a semicovering exomorphismfrom o to a' then ~ is a semicovering exomorphismfrom b to b'. I f moreover f is strict on a then ~ is strict on b. Proof Assume that e'~,g =.fo e and choose maximal commutative (-semisimple subalgebras T of a and T' of a' such that T centralizes e(1), T' centralizes e'(1) and f ( T ) ~ T' (which is always possible!); then e - 1 ( T ) and e ' - l ( T ' ) are respectively maximal commutative 6'-semisimple subalgebras of b and b', and we have g(e i (T)) c e'- 1( T'); moreover, it is clear that f ( T ) = T' implies g(e- 1(T)) = e'-l(T'). On the other hand, we have e l(J(a)) = J(b) and e'-l(J(a')) = J(b'). Example 3.9. Let G and G' be finite groups and (p: G -~ G' a group h o m o m o r p h i s m such that Im(q~) is normal in G' and Ker(~o) and Coker(~p) are both p-groups, and denote by.[: 6G--. 6G' the C-algebra homomorphism induced by ~p. T h e n f i s a strict semicovering exomorphism. Indeed, since Ker(~o) is a p-group, we have K e r ( f ) c J(CG) and, as Im(~p) is normal in G', we get f ( J ( ~ G ) ) ~ J((gG'); finally, if P' is a Sylow p-subgroup of G', I m ( f ) is U-stable and we have C G ' = ~,'~e' I m ( f ) u ' which implies that a maximal commutative 6'-semisimple subalgebra of I r a ( f ) is still maximal in 6G' (cf. [14], Th. 2).

3.10. Let G be a finite group: we are ready to discuss the G-algebra case. Let B and B' be G-algebras and ff:B --* B' a G-algebra exomorphism (cf. 2.9 and 2.10). We say that ,q is a semicoverin,q exomorphism if, for any p-subgroup P of G, the induced 6,:algebra exomorphism ,0P: B P --+ B 'e is a semicovering (cf. 3.2), and then we denote by (~P)*: S(~B,(P) ) -~ S(;~(P)) (3.10.1) the associated map (cf. 3.3.3). If moreover ~P is strict or, equivalently, (~P)* ( . ~ ' ( P ) ) = ~B(P) for any p-subgroup P of G, we say that ~ is strict; notice that c)p is strict if and only if we have Ker(g ~') ~ J(BP). Recall that if~ p is a semicovering exomorphism, 7'~ .~B'(P) and ,5/~(7') is the set of ), e ,~B(P) such that m(g)~, 4= 0 then oae induces an injective unitary exomorphism (cf. Prop. 3.3 and Remark 3.4) g(P~.'): 1-[ B(P~,)~ B'(P,,). ;. ~ ~/'(7')

(3.10.2)

First of all, we prove that the relative trace map preserves semicovering exomorphisms: this result is the key point to show in Theorem 3.16 below that semicovering exomorphisms of G-algebras can be recognized locally. Notice that the analogous statement on coverin 9 exomorphisms is true and supplies another approach to prove Theorem 4.22 of [18]. Theorem 3.11. With the notation above, let P and Q be p-subgroups o['G such that Q ~ P, b an ideal oJ'B Q and b' an ideal o r b 'Q such that ~4(b) ~ b'. I f the ~?-algebra exomorphism Oa is a semicovering exomorphismJkom b to b' then ,~P is a semicovering exomorphism from T r y ( b ) t o Tr~(b'). Proof We m a y assume that G = P and P 4= Q. Arguing by induction on IP: QI, if R is a maximal subgroup of P containing Q then ~R is a semicovering exomorphism from Try(b) to Tr~(b'). Consequently, we may assume that IP:Q] = p, so that Q is

40

B. K/ilshammer and L. Puig

normal in P; then B e and B 'e are (P/Q)-algebras and ~Q is a semicovering exomorphism from ~_~,~eb" to ~u~e b'" (cf. Prop. 3.5). Since T r ~ ( ~ , ~ p h") = Try(b) and Tr~(~u~eh'") = Tr~(b'), we may assume that Q = {1} and that P has order p and stabilizes b and b'. Moreover, we may assume that C = ,( (cf. 3.2). Now J(h) and J(b') are respectively P-stable ideals of B and B', and we have g(J(b)) ~ J(b') since the C-algebra exomorphism ~ is a semicovering from b to b' (cf. 3.2). Set /~ = B/J(b) and /~' = B'/J(b') and denote by ~:B--* B' the P-algebra h o m o m o r p h i s m induced by g, and by b and b' the respective images of b in/~ and b' in/~'; by Corollary 3.7, the ~f-algebra exomorphism ~ is a semicovering from b to b'. On the other hand, if the ,~-algebra exomorphism ~e is a semicovering from TrP(b) to Tr~e(b'), the ~-algebra exomorphism from B e to /~'P induced by ~e is a semicovering from Tr le(b) to Tre(b ') (cf. 3.6.1), and therefore it follows from 3.6.2 that ~P is a semicovering exomorphism from Tre(b) to Tr 1e(h') (since J ( T r le(b')) maps onto J(Tr~(b'))). Hence we may assume that J(b) = {0} = J(b'). In that case, b and b' are actually semisimple ,(-algebras and therefore we may assume that b = B and b' = B'. Moreover, the exomorphism defined by the inclusion g(B) ~ B' is clearly a semicovering and, since g(B P) = 9(B) P, we may assume that is injective. In conclusion, we assume that B and B' are semisimple and that ~ is an injective semicovering exomorphism of ,(-algebras, and then it suffices to prove that ~P is a semicovering exomorphism from B1e to B'IP. Let I be the set of primitive idempotents of Z(B); since ~ is an injective semicovering, Bi and 9(i)B'9(i) are simple algebras for any i ~ l and therefore g induces a he-algebra isomorphism Bi ~ g(i)B'g(i). Moreover, by Propositions 3.5 and 3.8, we may assume that P acts transitively on I; then, if [II = 1 or B' is not simple, g induces an isomorphism B ~ B' and therefore an isomorphism B~e ~ B'~. Otherwise, B' is simple and [I[ = p; now, choosing i e I, the trace map T r te induces a ~f-algebra isomorphism Bi "~ B p= B~ and, lifting the action of P on B' to a group h o m o m o r p h i s m P --* (B')• we have B' ~- Ind~(g(i)B'g(i)) (cf. 2.13.2); in particular, J(B P) = {0} and it is easily checked that B 'n' _~ g(i) B' g(i) | heP _~ B p | heP which proves that OP is a semicovering exomorphism.

Corollary 3.12. Let P, Q and R be p-subgroups of G such that R c Q c P. f f ~Q is a semicovering exomorphism from BQRto B'RQ then ~P is a semicovering exomorphism from B~ to B~. Proof. This follows from Theorem 3.11 applied to b = B~ and b' = B~e. Now we generalize Propositions 3.6 and 3.8 to the new context.

Proposition 3.13. With the notation above, let B" be a G-algebra and h:B'--* B" a G-algebra exomorphism. 3.]3.]. l f ~ and h are semicovering exomorphisms, then ho ~ is also a semicovering exomorphism and we have ((/~o ~)P)* = (~e)* o (/~P)* for any ~p-subgroup P of G. In particular, if two semicovering exomorphisms among ~, h and h o ~ are strict, so is the third.

Extensions of nilpotent blocks

41

3.13.2. I f h~ is a semicovering exomorphism and, for any p-subgroup P of G, we have h( J(B'e)) ~ J(B "l') then h is a semicovering exomorphism. I f moreover h is strict then 0 is a semicovering exomorphism too. Proof. This follows easily from Proposition 3.6.

Proposition 3.14. With the notation above, let D and D' be G-algebras, ~: D ~ B and ~' : D' --* B' G~lgebra embeddings and h: D --, D' a unitary G-al~bra exomorphism such that ~' o h = 0 o ~. I f 0 is a semicovering exomorphism then h is a semicovering exomorphism too. I f moreover 0 is strict then h is strict too. Proof. This follows easily from Proposition 3,8.

The following proposition summarizes the lifting features of semicovering exomorphisms. Proposition 3.15. With the notation above, assume that 0 is a semicovering exomorphism. Let Pv and Qo, be respectively pointed p-groups on B and B' fulfilling Q ~ P, and denote by 5'~(6 ') the set of 6e~B(Q) such that g(6) c 6'. 3.15.1. I f Qa ~ P~for some 665P(6'), there is y'e~a,(P) such that g ( y ) ~ 7' and 3.15.2. I f g ( 7 ) ~ 7' for some 7'e~s,(P) then m]', = ~ a esola,)m~. In particv,lar, if Q~, ~ P~, there is 6~ 5~(6 ') such that Q~ ~ P~. 3.15.3. I f 6e,9~(6 ') then Qa is local if and only if Qo, is local and, in particular, Qo is maximal local on B if Qa, is maximal local on B'. Proof Recall that (cf. [18], 2.3.1) m(g)~,m~, ~, m~m(g)~, = m(g)~, = ~, ~ ~' ," e ~BiQ) I" e,~'B(e)

(3.15.4)

on the other hand, by 3.3.2 we have m(g)~, = 1 or 0 according to whether 6 belongs to ,~(6') or not, and therefore if Q~ cc p~ for some 6~5~(6 ') we have m(g)], + 0 (cf. 3.15.4) which implies m(g)~, + 0 # m]', for some 7 ' s ~n,(P); hence, in that case, we have Q~, = P / a n d , again by 3.3.2, g(7) = 7'; moreover, if g(7) = 7' for some )"E~n,(P), 3" is unique such that m(g)~, # 0 and therefore (cf. 3.15.4) m] = m(g)~, = m~: .

(3.15.5)

6 e J(6') I f 6 e ,~(6') is not local, there is a proper subgroup R of Q such that 6 c B~ whence g(6) c B'Q R, and g(6) c 6' forces 6' c B~e; conversely, if 6' c B~e we have ,~(6 )) c 9~(B~) since 0 e is a semicovering exomorphism from Ben to B]e (cf. Cor. 3.12) and (0e) * extends the associated map from S(9~(B~e)) to S(~(B~)) (cf. Prop. 3.5). The following theorem may be considered as the main result of this section. It supplies "local" criteria to recognize semicovering exomorphisms. For any psubgroup P of G, we denote by 0(P): B(P) --* B'(P) the ,(-algebra exomorphism induced by 0.

42

B. K/ilshammer and L. Puig

Theorem 3.16. With the notation above, the following conditions on ~ are equivalent

3.16.1. The G-algebra exomorphism ~: B ~ B' is a semicovering. 3.16.2. For any p-subgroup P of G, the ,~-algebra exomorphism if(P): B( P) ~ B'( P) is a semicoverin9. 3.16.3. For any local pointed group P~, on B' there is 5~(7 ') ~ ~n(P) such that m;,, = ~7~,'~(;")mr and for any ? ~ ( 7 ' ) we have Bre(g(7)) ~ Bre(?' )

and

B r e ( g ( J ( B e ' ? ' B e ) ) ) ~ Bre(J(B'e'7" B'e)) .

3.16.4. For any p-subgroup P of G and any subgroup Q of P, the C-algebra exomorphism ~e: Be__. B,P is a semicovering from B~ to B~. In that case, ~3 is strict if and only if g(P) is strict for any p-subgroup P of G. Proof It is clear that 3.16.4 implies 3.16.1, and the converse follows from Theorem 3.11. On the other hand, it follows from Corollary 3.7 (applied to ~(| ~e:~r @~, B e ~ xr | B 'e for any p-subgroup P of G) that 3.16.1 implies 3.16.2. Moreover, if .~ is a strict semicovering, for any p-subgroup P of G we have (~e)*(~B,(P))=~B(P) (cf. Prop. 3.3) and, by Proposition 3.15, we get (~e)*(cf:~B,(P)) = Lf~n(P) or, equivalently, (t(P)*(~(B'(P))) = ~(B(P)), whence ~(P) is a strict semicovering too (cf. Cot. 3.7); the converse is clear. It follows from Proposition 3.3, applied to ~ ( P ) : B ( P ) ~ B ' ( P ) for any psubgroup P of G, that 3.16.2 implies 3.16.3. Finally, assume that 3.16.3 holds and let P be a p-subgroup of G; we will prove by induction on [P[ that (~e:Be~ B 'e is a semicovering exomorphism. If Q is a proper subgroup of P, it follows from the induction hypothesis and Theorem 3.11 that Oe is a semicovering exomorphism from B~ to B~. So, by Proposition 3.5, ~e is a semicovering exomorphism from Ker(Bren) to Ker(Bre~') (notice that this is trivially true if P = {1 }); in particular, for any 7 ~ 5f?~8(P) and any 7'~ ~B, (P) - 5a~n, (P) we have m(9)~, = 0 (cf. Remark 3.4). Moreover, for any 7'6 5a,~R,(P) there is ,9e(7') c ,~B(P) such that mr, = ~ . e.r and such that for any 7~,~(7') we have Bre(#(7)) ~ Bre(7')

and

Bre(y(J(Be'7"Be))) ~ B r e ( J ( B ' e ' 7 " B ' e ) ) ,

which implies that g(7) c ? ' (since m(9),~,=0 for any 6'~(Ker(Bren'))) and therefore that g ( J ( B e . y . B P ) ) ~ J ( B ' e ' 7 ' . B 'p) (since g ( J ( B e ' 7 " B e ) ) is now contained in B ' e ' 7 " B 'P and maps onto {0} in B'(P~)); hence, as J(~7 ~tT')Be'Y- Be) - ~y ~Ut~')J(BP'7" Be)' Oe is a semicovering exomorphism from ~7 ~J(7')B"- 7' B" to B '~" ~'. B 'e (cf. 3.3.2). Finally, if ? e s - ~ , 5e(?'), where 7' runs over L~n,(P), we have m(g)~, = 0 for any ?' e~n'(P), and therefore 9(7) = {0}; thus fie is also a semicovering exomorphism from B e. 7" Be to {0}. Consequently, as Be =

~

B e ' ? ' B e + Ker(Br~) and B 'e =

the statement follows now from Proposition 3.5.

~

B ' e . y ' . B 'e + Ker(Br~'),

Extensions of nilpotent blocks

43

4. Auxiliary results on group extensions

4.1. Let L, L' and F be groups and p: L ~ F and p': L ' ~ F surjective group homomorphisms; set Z = Ker(p) and Z ' = K e r ( p ' ) . Assume that Z and Z' are if-divisible abelian groups, so that they become Z,p)F-modules through the respective actions of L and L'. Moreover, assume that L and therefore F and Z are finite, denote by P a Sylow p-subgroup of L, so that Z c P, and set p(P) = /5. As usual, we consider respectively L and L' endowed with p and p' as group extensions of F by the Z~ptF-modules Z and Z'. 4.2. It is well-known from homological algebra (cf. [4], Ch. XIV, Th. 3.1 and 4.2) that L and L' determine respectively elements hE H2(F, Z) and h ' e NZ(F, Z') (cf. 2.19), and that there is a group h o m o m o r p h i s m a: L ---, L' lifting idF and extending a Z(v)F-module h o m o m o r p h i s m 0: Z ~ Z ' if and only if the h o m o m o r p h i s m ~ 2 ( F , 0) maps h upon h'. Moreover, as restriction induces an injective map HZ(F, Z ) - ~ 1-t2(/5, Z) (cf. 2.19), there is such a group h o m o m o r p h i s m iT: L ~ L' if and only if there is a group h o m o m o r p h i s m z: P ---,L' lifting the inclusion/5 c F and extending 0. But, in this case, ~ does not necessarily extend z or, more generally, any L'-conjugate of z; indeed, the "difference" between T and the restriction of t7 to P determines an element of H1(/5, Z') (see the proof of Proposition 4.4 below) which is not necessarily the zero element. In this section we discuss a sufficient group-theoretical condition to guarantee that o extends z up to L'-conjugation, which is still useful for non-abelian extensions (see Proposition 4.9 below). 4.3. We consider F endowed with the group h o m o m o r p h i s m F - , Aut(Z') induced by p' as a Z'-permutation group (cf. 2.4). Recall that, considering the trivial F-algebra (5 (and omitting to mention the points), an (F, Z')-biexomorphism ((p, ~) = (/~, 2, )5,/5)z' (cf. 2.15) is a pair of elements of EF~Z'(l~, /5) (cf. 2.8) where/~ is a p-subgroup o f f and 2, )SeF fulfil ,q~ c / S a n d ,q.~ c P; moreover, a generator set of the (F, Z')-biexomorphisms is a set g of (F, Z')-biexomorphisms such that any (F, Z ' ) - b i e x o m o r p h i s m is a linear combination of (g (cf. 2.15.3 and 2.16). Notice that, by 2.16.1, 2.17.1 and 2.18.4, the family { (/5 ~/5~ ,, 1, .~, fi)z' } ~ ~ F is a generator set of the (F, Z')-biexomorphisms. Proposition 4.4. Let X be a subset o f F containing Nv(/5 ) such that the family {(P n P~ ', I, 2,/5)z'}~ ~,~ is a generator set of the (F, Z')-biexomorphisms. Let r :P --* L' be a group homomorphism fulfilling the.following conditions: 4.4.1. We have p'( z(u) ) = p(u) for any u e P, and z(z -~) = z(z)i for any z e Z and any ~eF. 4.4.2. I[ Yc6R there are y e L and y' e L ' such that p ( y ) = z x = p'(y') for some ~ CF(P c~ P~ l, Z') and r(u y) = r(u) y' for any u ~ p - l ( P c~ P~ '). Then there is a unique group exomorphism ~:L--* L' having a representative tr such that p',~cr = p and tr(u)= z(u) for any u ~ P . Remark 4.5. Conditions 4.4.1 and 4.4.2 are obviously necessary.

44

B. Kfilshammer and L. Puig

Remark 4.6. Notice that if q~, ~ ~ Hom(L, L') fulfil the equalities p' o ~o = p' o ~k and ~o(u) = O(u) for any u e P then there is z'eCz,(q~(P)) such that O(x) = ~o(x)=' for any x e L. Indeed, the 1-cocycle mapping x e L upon ~ (x)- ~~o(x) e Z' determines the zero element of H~(L, Resp,,,~o(Z')) since its restriction to P is the trivial homomorphism from P to Z' (cf. 2.19); hence there is z' s Z ' such that ~O(x)-~o(x) = z'-,z,X = z'-~q~(x)-~z'~o(x)

for any x e L , whence tp(x) = tp(x) ='.

Proof. Denoting by O:Z --* Z' the Ztp~F-module h o m o m o r p h i s m induced by r and arguing as in 4.2, we get a group homomorphism ~o: L ~ L' such that p' o q~ = p and q~(z) = z(z) for any z e Z ; in particular, we have p'(q~(u)) = p(u) = p'(z(u)) for any u e P , and therefore r ( u ) = q)(u)~(u) where the map ~: P--*Z' is a l-cocycle; moreover, as ((z) = 1 for any z ~ Z, ( induces 1-cocycle ~: P ~ Z ' which determines an element h of Hi(/5, Z'). Now it suffices to prove that h is the restriction of an element of H I ( F , Z'); indeed, in this case it is quite clear that there is a 1-cocycle ~-:F ~ Z' extending (, and then it is easily checked that the map (r: L ~ L' assigning to x ~ L the prod uct q~(x) ~-(p (x)) ~ L' is a group h o m o m o r p h i s m extending z and fulfilling p' o ~r = p. Moreover, the uniqueness of (i follows from Remark 4.6. To show that h belongs to the image of H i ( F , Z'), it suffices to prove that, any :~ ~)(, the 1-cocycle /5 c~/5~-~ __, Z' mapping ~ e/5 c~ P-~ ~ upon C ( ~ ) - l ( ( ( d ) ~ - ' ~ z ' is a 1-coboundary (cf. 2.19.2). By condition 4.4.2, for any for

there are y e L and y ' ~ L ' fulfilling p ( y ) = z ~ = p ' ( y ' ) for some F~ CF(/Sc~ P~ ', Z') a n d z(u r) = r(u) y' for any u e p - ~ ( P c~ P-~-'); now, setting z' = ~o(y)y'-1, we have 2e.~

, p ( u K ( u ) = ~ ( u ) = T(uy)"-' = q~(u')y'-'~(u')~'-' = ~o(u)Z"~(u') ~'-~

and therefore ((a)-~((a~)~

-~ = ~ ( u ) - ~ ( . , ) ,

'-~ = z,-~z,~

for any ti~/5 c~ P~ - ' , where u ~ P lifts ft. Consequently, the 1-cocycle above is indeed a l-coboundary. In particular, Proposition 4.4 implies the following splitting criterion. Corollary 4.7. With the notation of Proposition 4.4., let ~: fi--*L' be a 9roup

homomorphism fulfillin# the followin O conditions: 4.7.1. We have p'({(ti)) = s for any aeff. 4.7.2. I f 2 e . f a e Pc~ P~-~.

there is x ' e L ' such that p ' ( x ' ) = ~ and ?(ti ~) = {(ti)=' for any

Then there is a unique 9roup exomorphism a: F ~ L' havin# a representative 6 such that p ' o 6 = id~ and 6(if) = ~(~) for any 6z/5. Proof. This follows trivially from Proposition 4.4 setting L = F and p = ida. 4.8. Let E be a finite group and n: L ~ E and n': L ' ~ E surjective group homomorphisms. Assume that n and n' have the same kernel Q and induce the

Extensions of nilpotent blocks

45

same group exomorphism from E to Aut(Q); that is, denoting respectively by q and q' the group homomorphisms from L and L' to Aut(Q) induced by conjugation, and by f/and ~' the corresponding homomorphisms from E to A,ut(Q), we assume that ~ -- ~'. Moreover, assume that Q is a p-group, so that Q c P, and consider L as an ~-permutation group (cf. 2.4) (omitting to mention the empty set). Proposition 4.9. Let X be a subset of L containing NL(P ) such that the family {(P c~ px- ,, 1, x, P) }x~x is a generator set of the L-biexomorphisms. Let z: P --* L' be an injective group homomorphism fulfllling the following conditions: 4.9.1. We have g' ( z( u ) ) = n( u ) for any u ~ P, and O(e) = Ff(e)~ofor any e e E, where z o is the automorphism of Q induced by z. 4.9.2. lf x ~ X there is x' e L' such that z~'( x') = ~(zx )for some z ~ CL(P c~ W - ') and z(u x) = r(u) x' for any u ~ P m px-,. Then there is a unique group exoisomorphism ~: L ~- L' having a representative a such that n ' o a = n and a(u) = v(u),Jbr any u ~ P . Remark 4.10. Conditions 4.9.1 and 4.9.2 are obviously necessary. Remark 4.11. Considering respectively L and L' endowed with n and n', and with left multiplication as E-permutation groups (cf. 2.4), condition 4.9.1 implies that z induces an evident bijection z R : H o m ~ ( R , P ) ~ H o m E ( z ( R ) , z ( P ) ) for any subgroup R of P, and then condition 4.9.2 can be restated as follows: 4.11.1. I f x ~ X and R = P c~ px- ,, there is q~~ EL, E(R, P) lifting (R, x, P) ~ EL(R, P) such that ZR(q9) ~ EL, , ~(z(R), z(P)).

Indeed, if 4.9.2 holds it suffices to set ~o = (R, zx, P)E. Conversely, if ~0 = (R, y, P)~ lifts ( R , x , P ) we have y = z x v for some Z ~ C L ( R ) and v ~ P (cf. 2.8). and if rg(~0)~ EL," ~(r(R), z(P)J there is y ' ~ L' such that rt'(.V') = n ( y ) a n d r(u r) = r(u) r' for any u E P (cf. 2.8), whence n'(y'z(v) -1) = n(zx) and r(u x) = z(u) r~l~- ~ for any u~P. Proof We may assume that F above is the pull-back of t~: E ~ A,ut(Q) and the canonical homomorphism A u t ( Q ) ~ A u t ( Q ) (i.e. the subgroup of pairs (e,O)~E • Aut(Q) such that ~(e)= 0), and that p and p' are respectively the group homomorphisms mapping x ~ L upon (n(x),q(x)) and x ' ~ L ' upon (n' (x'), t/'(x')'~ Then, it is clear that Z = Z(Q) = Z'; moreover, notice that p and p' induce the same group homomorphism F ~ Aut(Z) since t/(x""')= q ~ ) ~ o , for any x 6 L and any x ' ~ L ' such that re(x) = ~'(x'), by condition 4.9.1.

Now, considering F endowed with this group homomorphism as a Z-permutation group (cf. 2.4), it is still clear that p induces a surjective map EL(R, P) ~ Ev, z(P(R), p(P)) for any p-subgroup R of L containing Q; in particular, setting X = p ( X ) and f i = p ( P ) , the family { ( P c ~ P , 1 , ~ , P ) z } ~ . ~ is a generator set of the (F, Z)-biexomorphisms (since any (F, Z)-biexomorphism can be lifted to an L-biexomorphism). Therefore it suffices to prove that conditions 4.4.1 and 4.4.2 (with respect to X and z) follow from conditions 4.9.1 and 4.9.2. Indeed, it follows then from Proposition 4.4 that there is a group homomorphism

46

B. Kiilshammer and L. Puig

a: L ~ L' which extends z and fulfils p' o a = p, whence n ' o a = n; moreover, a is an isomorphism since z is injective and I L i = IL'I, and the uniqueness of ~ follows from Remark 4.12 below (or directly from Proposition 4.4). First of all, we prove that 4.9.1 implies 4.4.1; as (t/'(z(u))~~ = v" = tl(u)(v) for any u e P and any veQ, it is clear that ~t'(r(u)) = 7r(u) implies p'(T(u)) = p(u) for any ueP; moreover, if x s L and x ' e L ' fulfil p ( x ) = p'(x') then r(r/(x)(z))= q'(x')(z(z)), whence r(z ~) = z(z) ~' for any z e Z . Secondly, by condition 4.9.2, if ~ e ) ( and x e X lifts 2 there is x ' e L ' such that rc'(x')=rc(zx) for some Z ~ C L ( P n P ~ ') and r ( u ~ ) = r(u) ~' for any u ~ P c ~ P ~ ', and it suffices to set y = zx and y ' = x' in 4.4.2; indeed, as z(r/(y)(v))= q'(y')(z(v)) for any v~Q (since Q c P c ~ P ~ - ' ) , we have p ' ( y ' ) = p ( y ) = s where s belongs to Cv(Pc~P -~ ' , Z ) (once again since Q c Pc~P~-'), and r ( W ) = r(u) y' for any

u~p~p~-~=

p-~(pc~p-~ ~).

Remark 4.12. Ifq~ and ~ are injective group homomorphisms from L to L' fulfilling the equalities 7r' o r = n ' o O and q~(u) = O(u) for any u~P, there is z~CQ(P) such that ~ ( x ) = q~(x=) for any x ~ L . Indeed, with the notation above where z is the restriction of q~ and ~k to P) it is easily checked that p' o q~ = p' o if, and it suffices to apply Remark 4.6.

5. On the local structure of block extensions 5.1. Let G be a finite group and H a normal subgroup of G, set A = CG and B = CH, and consider A endowed with B as a G-algebra extension (cf. 2.9). Let c~ and fl be respectively points of G and H on B (cf. 2.7) such that H~ ~ G,, set N = No(Ha) a n d / V = N / H , and denote by )f the image of X c N in N. In this section we will discuss to what extent the local category of B, (cf. 2.8 and 2.11) is determined by the quotient N. Actually we are only concerned with the case where the local structure of H a is nilpotent (cf. [18], 1.7) - what we assume from 5.6 on which is completely handled in Theorem 5.8 below. However it is not difficult to discuss the general situation from our results, as we explain in Remark 5.9 below. 5.2. Obviously fl has just one element b and ~ = {Try(b)}; hence fl is still a point of N on B, therefore we get A~ -~ Ind~(Aa) (cf. 2.13.2); in particular, it is easily checked that the inclusion N c G induces a n equivalence between the local categories of B~ and Ba (cf. 2.8). That is, for our purposes it suffices to consider the local pointed groups on B contained in Na, and the next statements show that the set of their images i n / q is the full set of p-subgroups.

Proposition 5.3. A local pointed group P~ on B such that P~ ~ N a is a defect pointed group of G~ if and only if ff is a Sylow p-subgroup of N and there is a defect pointed group Q~ of H a such that Qo ~ P~, and then we have Q = H c~ P.

Remark 5.4. Notice that Proposition 5.3 depends only on the G-algebra structure of B and therefore applies to each subgroup of G containing H.

Proof. Assume first that Pr is a defect pointed group of G~; in particular, Pr is also a defect pointed group of Na, and therefore a defect pointed group Q~ of H a has an

Extensions of nilpotent blocks

47

N-conjugate contained in Pr; so we may assume that Q0 ~ P~. Moreover, as H is a P-stable (?-basis of B, there is e ~ n ( P c ~ H ) such that ( P c ~ H ) ~ P.~ (cf. [1], Prop. 1.5); hence (P c~H)~ ~ Nt~ and therefore (P c~ H), ~ H a (since s~(b) 4: 0); thus there is h~ H such that (P c~ H)~ ~ (Qo)h which forces P c~ H = Q (since Q ~ P c~ H). On the other hand,/3 is still a point of P" H on B and Pr a defect pointed group of P'Ha; therefore, as N ~ ( P r ) ~ N (since b7 = 7), the Frattini argument shows that N~(Py) = N~(fi), so that No(P~)/P. Cn(P) maps surjectively on N s ( P ) / P ; but the quotient N6(P~)/P" Cn(P) is a if-group (cf. 2.16.2); hence/5 is a Sylow p-subgroup of ~7. Conversely, assume that fi is a Sylow p-subgroup of N and that there is a defect pointed group Q0 o f H 0 such that Q~ ~ P~; now, ifR~ is a defect pointed group o f N 0 (and therefore of G,) such that P~ ~ R~, we have f i - - / ~ and, by the argument above, Q = R c~ H so that P c~ H = R c~ H; it follows that P = R, and so P~, = R,.

Proposition 5.5 Let Py and Q~ be respectively defect pointed groups of G, and H o such that Qa c Pr For any subgroup R of P containing Q there is a unique e6 ~LP,~(R ) such that R~ c P~, and then it fulfils the following conditions: 5.5.1. We have Qa c R~, Na(R~) ~ Na(Q~) and NG(R~)c~ Cn(Q) = Z(Q).Cn(R ). 5.5.2. Let L~ be a pointed group on B such that R~ ~ La and Cn(R) ~ L c N~(R~). A local pointed group T~, on B such that R~ c Tg, c Lx is a defect pointed group of Lx if and only if T is a Sylow p-subgroup of [,. Proof First of all, notice that P~ ~ N0; indeed, anyway there is x ~ G such that (P~)X c N 0 (since a defect pointed group o f N 0 is a defect pointed group of G~), and then there is h ~ H such that (Qa)~h = Qo (since (Qa)~ ~ H0) , whence x E N (since Nt~(Q~) ~ N). Moreover, as H is a P-stable (9-basis of B, there is e e L f ~ B ( R ) such that R~ ~ P~, and if e'~5~ there is also ~ 5 ' e ~ e ( Q ) such that Q~, c R~, (cf. [1], Prop. 1.5); now, assuming that R~, ~ P~ we get Q~, c N~ and therefore Qo, = H 0 (since b6' = 6'). Let L~ and T~, be pointed groups on B such that T, is local, R~ ~ T+ ~ L~ and CH(R) ~ L = N~(R~), and set 2 = BrR(2), ~ = Br~(~) and ~, = BrR(c ). As H is an L-stable O-basis of B, it is easily checked that BrR(B L) = B(R) r and BrR(B r) = B(R) r, and therefore 2, tp and ~ are respectively points of L, T a n d R on B(R) fulfilling R r Tq7 c L~; moreover, as B(R)~-,~Cu(R) (cf. [19], 2.9.2) and ~ ~(B(R)), there is a block f of Cu(R) such that 2 = ( f } where f = Br~(f)(since f e = g and L normalizes R~), and therefore 2 is still a point of Cu(R), N~(R~) c~ Cu(Q) and R" Cn(R) on B(R). First of all, it follows from Proposition 5.3 applied to L~, C~(R)~ and T~ that if T~7is a defect pointed group of L~ then 2?is a Sylow p-subgroup of/~ (and it is clear that if T~ is a defect pointed group of Lz then T~ is a defect pointed_group of L~); conversely, again by Proposition 5.3, if T is a Sylow p-subgroup of L then T, is a defect pointed group of L ' H o (cf. Remark 5.4) since Q0,, c T, for some 6 " ~ S a ~ ( Q ) (and then Q~,, is a defect pointed group of H0), and therefore, as L a ~ L" H a, T, is afortiori a defect pointed group o f L a. In particular, R~-is a defect

48

B. Kfilshammer and L. Puig

pointed group of R. Cn(R)~, and therefore the block f of R. CH(R ) has a nilpotent local structure (cf. [18], 1.7); but, asfe = ~ and P~ contains both R, and R~,, we have also f e ' = ~' where ~ ' = BrR(e' ) (cf. [1],Th.l.8); consequently, we have ~ = ~' (cf. [2], Th. 1.2 or [18], Th. 1.6), whence e = e'. It is now clear that 6' = 6 (since Q~, c Pr) and that NG(R~) c NG(Q~) (since Q = R c~ H). Finally, set /~= BrR(fl); as Brg(B/'H) = B(R)NR~HIR) (cf. [1], Lemma 1.126) and R e is a defect pointed group of R- H a (cf. Prop. 5.3 and Remark 5.4), fi is a point of NR.n(R) on B(R) and R~ is a defect pointed group of NR.n(R)~. On the other hand, ~ is also a local point of Z(Q) on B(R) since ~ e ~(B(R)), and setting M = N~(R~)c~ CH(Q) we have Z(Q)~c M~ since f e = ~. Now, as Z ( Q ) ~ R~, M~ ~ NR.H(R)~ and R c~ M = Z(Q), it is quite clear that Z(Q)~ is a (the unique) defect pointed group of M,~, and it follows from Proposition 5.3 (applied to M$, CH(R)$ and Z(Q)e) that M/Z(Q)" CH(R) is a if-group; but for any if-element s of M we have [ s , R ] c R c ~ H = Q and Is, Q ] = 1, whence s centralizes R (cf. [9], K . I, 4.4); so M/Z(Q). CH(R) = {l}. 5.6. From now on we assume that b is a block of H with nilpotent local structure (cf. [18], 1.7). Let P~ and Q~ be respectively defect pointed groups of G~ and Hp such that Q~ ~ Pr, and R~ a local pointed group on B such that Q~ ~ R e ~ P~. As NH(Q~) = Q'Cn(Q) (cf. [18], 1.7.2), on one hand it follows from 5.5.1 that

Nn(g,) = Q'Cn(R).

(5.6.1)

On the other hand, as N = H'No(Q~) by the Frattini argument, the inclusion N~(Q~) ~ N induces a group isomorphism/q ~ No(Q~)/Q 9CH(Q) (cf. 1.7.2); then this isomorphism and the group homomorphism t/: N~(Q~) ~ Aut(Q) induced by conjugation determine a group homomorphism ~/:A7 ~ ~,ut(O)

(5.6.2)

and we set N = AT/Ker(~) and consider ~7, endowed with the action on A~ induced by left multiplication, as an N-permutation group (cf. 2.4); notice that the group exomorphism ~ does not depend on the choice of 6, and therefore N does not depend on the choice of Q~. 5.7. Now consider the trivial/~-algebra (9 (omitting to mention the points) and the full subcategory of the local category of the N-algebra B a (cf. 2.8) formed by the local pointed groups on B which contain a defect pointed group of Ha; the canonical homomorphism N--* N i_nduces an evident functor from this subcategory to the local category of the (N, N)-algebra (9 (cf. 2.6, 2.7 and 2.8); precisely, for any local pointed group T, on B such that Q~ ~ T, r Pr, we have the map

EN(R ~, T,) ~ Eg.~(R, T)

(5.7.1)

assigning to (R~,x, T~,)~Es(R,, T , ) t h e (N, N)-exomorphism (/~,~, fr)g (cf. 2.8). Recall that we denote respectively by C~(R~) and C~,(R, N) the inverse images of 6 Notice that, in Lemma 1.12 of [1], the hypothesis on P is irrelevant

Extensions of nilpotent blocks

49

Ov(EN(R~) ) in NN(R,) and Op(Eff.g(R)) in N~7(/~) (cf. 2.8), and that there is a unique point # of C~(R,) on B such that R~ c Cfv(R~)u (cf. 2.16.2). Theorem 5.8. With the hypothesis and notation above: 5.8.1. We have OP(CN(R~)) = OP(Cg(R, N)) and the maps 5.7.1 are surjective. 5.8.2. The pointed group R~ is a defect pointed 9roup of C~(R~)u if and only if the pgroup R is a Sytow p-subgroup of C~(R, N), and then the maps 5.7.1 are bijective. Remark 5.9. Notice that, without any hypothesis on b, the unique point 2 of Q'CH(Q) on B such that Q~ c Q.CH(Q) ~ (cf. 2.16.2) determines a block of Q'Cn(Q) with nilpotent local structure (cf. [18], 1.7) and therefore Theorem 5.8 always applies to the group NG(Q6 ) and the normal subgroup Q'Cn(Q). Proof. As CN(R~)~ C~(R, N), we have already Ov(CN(R~))~ OP(C~(R, N)). If e N is such that R x ~ Twe have (R" H) x ~ T" H where x ~ N~(Q~) lifts .~; but fl is still a point of both R . H and T ' H on B, and therefore ( R ' H a ) ~ c T'Hp; moreover, it follows from Proposition 5.3 applied to both R ' H and T ' H (cf. Remark 5.4) that R, and T, are respectively defect pointed groups of R ' H p and T.Ha; hence we may choose the element x fulfilling (R~)~ ~ T~,, which proves that the maps 5.7.1 are surjective. Moreover, if ~ is a if-element of C#(/~,/V), we may assume that x is also a p'-element which normalizes R~ (arguing as above and eventually replacing x by a suitable power of x); then, as 2eKer(f/) (cf. 5.6.2), x centralizes Q and, since [x, R] c H c~ R = Q (cf. prop. 5.3), it follows that x centralizes R (cf. [9], K. I, 4.4), whence x ~ OP(CN(R~)). In particular, we have actually proved that NN(R,) maps onto N~7(/~) and that the kernel of the induced group homomorphism EN(R~) ~ Eg,~(R) is a p-group, whence we get C~(R~)= C~(/~,/~); hence, it follows from 5.5.2 that /~ is a Sylow p-subgroup of C~(R, N) if and only if R~ is a defect pointed group of CP(R~),,. In that case, if (R~, x, T~,) and (R,, y, T~,) are two N-exomorphisms such that (/~, Y, T)~7 = (/~, .9, T)~, we have .9 = zxu where u~ T and z ~ R" CN(R~) since /~" C~(R, N) = C~(R, N) = C~(R~) = R" CN(R~) (cf. 5.5.2); in particular, we get R r = ( R ' H ) r c ~ T = ( R ' H ) ~ " r ~ T = R since T ~ H = Q (cf. prop. 5.3); hence, as T~, contains both ( R y and (R~):", the difference yu - 1 x - 1 belongs to N~(R,) c~ R. Cu(R~)" H (cf. Prop. 5.3), and therefore we have (R~, x, Tq,) = (R,, y, T,) by 5.6.1. TM

5.10. If x~N~(Q~) we know by Proposition 5.5 that there is a unique ~, ~ L ~ n ( P ~ P~) such that (P c~ P~)o ~ P~ and we denote the local pointed group (P ~ P~), by Pr c~ (Pr)". Although this notation is not necessarily symmetric, it becomes symmetric if x normalizes Pr c~ (Pr)~ and then this local pointed group is the biggest containing Q~ and contained in both Pr and (pr)x. In the sequel we employ this notation only in the symmetric case. Notice that the family {(p~ ~ (pQx, 1, x, P~)} where x runs over the set of elements of No(Q~) such that

50

B. Kfilshammer and L. Puig

Pvc~(P,Y = (Pv~(P~.)~ff is a generator (cf. 2.16.1, 2.17.1, 2.18.4 and Lemma 5.12).

set

of the

Na-biexomorphisms

Corollary 5.11. With the hypothesis and notation above, let X be a subset qf N~(Qa) such that P;. c~ (P~.)~ = (Pv c~ (P~ff)~for any x~ X.

5.11.1. If x ~ X then (P~.~ (P~)X, 1, x, P~) is an irreducible N~-biexomorphism if and only if (P ~ P~, 1, .~, fi)g is an irreducible (IV, N)-biexomorphism, and then we have 5.11.2. The family is {(P~,c~(P~,)X,l,x,P~)}x~x is a generator set of the Nabiexomorphi_sm~s if and only if the family ~ = {(P c~ px, 1, s ifi)~};r is a generator set of the (N, N)-biexomorphisms. Proof Let x r X and assume that R~ = Pv c~ (Pv)~; it follows from 2. l 7.2, 2.18.4 and 5.8.2 that to prove 5.11.1 we may assume that R~ is a defect pointed group of C~v(R~)~ (cf. 5.7); then, by 5.8.2 we have EN(R~)~-E~,~(/~) and statement 5.11.1 follows from 2.18.4. On the other hand, it follows from 2.16.1 and 5.11.1 that to prove 5.11.2 we may assume that all the elements of ts and Cs are irreducible, and in particular that p ~ px = p c~ P~ for any x ~ X, since

(P c3 P ~, 1, s P)fi = (P c~ P ~, 1, P c~ P~--')~(P c~ P ~- ', 1, ~, fi)~ . As/~ is a Sylow p-subgroup of ~7 (cf. Prop. 5.3) and the maps 5.7.1 are surjective (cf. 5.8.1), it is quite clear that if_ls is a generator set of the Na-biexomorphismsthen ls is a generator set of the (N, N)-biexomorphisms. Conversely, assume that ~ is a generator set; by 2.17.2, 2.18.4 and Lemma 5.12 below, it suffices to show that an No-biexomorphism (R~, 1, x, Pv) is a linear combination of ls (cf. 2.15.3) assurn~g that R, is a defect pointed group of Cw (cf. 5.7); but, by hypothesis, the (N, N)biexomorphism (/~, I, ~,/5)~7 is a linearcombination of ~, and therefore there are ordered families {~i }~, and {~ }~ of (N, N)-exomorphisms and {s } ~ of elements of 3f such that (cf. 2.t5.3)

(t~, 1, s, P),~ = y~ (,(P c~ Px~, 1, s,, p ) ~ , ; i~l

now, choosing liftings xi e X of xi, ~i ~ EN(R~, P~ c~ (P~)~') of ~i, and (~ e EN(Pv, P~.) of ~i (cf. 5.8.1), it follows from 5.8.2 that we have

(R~, 1, x, Pv) = ~ ~i(P~ n (pv)x,, 1, xi, P:,)~i 9 iEl

Lemma 5.12. With the hypothesis and notation above, let 7", be a local pointed group on B such that T~ c P~ and v the point of C~(T~) on B such that T~ ~ C~(T~)v. If T, is a defect pointed group of C~(T~,)~ then Q~ c T~,.

Proof It suffices to prove that N u ( T , ) / C u ( T ) is a p-group; indeed, as Nu(T,) is normal in NN(T~), in that case we have N u ( T ~ ) ~ C~(T~) and a fortiori

Extensions of nilpotent blocks

51

N r . n ( T 0) ~ C~,(T,); thus, if 2 is the point of Nr.H(To) on B such that T~ ~ NT.u(T,) ~ = Cw (cf. 2.16.2), then Tq, is still a defect pointed group of Nr.n(T,)z; but, since Tq, ~ P~ ~ No, we have b~ = ~ and so b)~ = 2, whence N.r.H(Tr ~ ~ T'Hp; consequently, T~ is also a defect pointed group of T ' H ~ (cf. [15], Cor. 1.4), whence Qa C T, by Propositions 5.3 and 5.5. Set U = T ~ H ; as H is a P-stable 6.-basis orB, there is q~6 ~ ' ~ R ( U ) such that U~ ~ T~, (cf. [1], Prop. 1.5); thus, as U,~ = T, ~ P~ = N~, we have U~ = H~ (since b~p = ~p); therefore, E , ( U , ) is a p-group (cf. [18], 1.7.2) and the block of CH(U) associated with q~ is still nilpotent (cf. [2],Th. 1.2), which implies that Nn(Tq,) ~ Nn(U~) (cf. [1], Th. 1.8 and [2], Th. 1.2). So, for any p'-element s of Nn(T,), we have [s, T] = H c~ T = U and s~Cn(U), whence s centralizes T (cf. [9], K . I, 4.4).

6. On p-extensions of nilpotent blocks 6.1. Let G be a finite group and H a normal subgroup of G such that G / H is a pgroup. In this section we will study the unique block of G lying over a nilpotent block of H (cf. [18], 1.7): the main tool to carry out our purpose are the semicoverin 9 exomorphisms between G-algebras introduced in Sect. 3. Precisely, set A = ~G and B = t~;H, and consider first A and B just as G-algebras: the key point is that the inclusion B c A determines a strict semicovering exomorphism, as we show below. Actually, this fact is useful to relate any block b of H with the unique block of G lying over b.

Proposition 6.2. The G-alyebra exomorphism determined by the inclusion B c A is a strict semicovering. In particular, jbr any pointed p-group P~,, on A there is a non empty set ~,~(yo) c ~ n ( p ) such that for any 7~ J~(7 ~ we have 7 c 7~ y is local if and only if?o is local, and then ,90(7~ = { TZ } z ~C,,(p ). Moreover, we have ,~,(P) =

~

51~(7~

(6.2.1)

), e.~'~iP)

Prm~ By Theorem 3.16, to prove the first statement it suffices to show that, for any p-subgroup P of G, the ,4-algebra exomorphism B ( P ) - . A(P) induced by the inclusion B ~ A is a strict semieovering; but A(P) ~ ~CG(P ) and B(P) ~ ~Cn(P) (cf. [19], 2.9.2), and the ~*-algebra exomorphism above is determined by these isomorphisms and by the inclusion Cn(P ) ~ C~(P); so, it suffices to apply Example 3.9. Moreover, the second statement follows from 3.10, 3.3.2 and 3.15.3, and the equality 6.2.1 follows from 3.10 and the last statement of Proposition 3.3. Finally, the transitivity of C~(P) follows from 3.10.2 since A(P.; )c~.w)~ (cf. [19], 2.9.2). The following consequence is more or less well-known. Corollary 6.3. We have ~ A(G) = ~B(G). Moreover, let ~ be a point of G on A, P~ a pointed p-group on B and ~~ the point of P on A such that ~ ~ 7~ then P~ is a defect pointed group of G~ on B if and only if Pr" is a defect pointed group of G~ on A.

52

B. Kiilshammer and L. Puig

Proof. By Proposition 6.2, any idempotent of A has an A • -conjugate contained in B (cf. 3.10 and 3.2); in particular, as A G = Z(A), any idempotent of A G belongs to B e and therefore ~A(G) = ~B(G). Moreover, if Pr is a defect pointed group of G~ on B then ct c Tr~e(Be. ~. B e) and afortiori ct ~ Tr~(A p. 7~ A P) which implies that P~,, is a defect pointed group of G~ on A (cf. 3.15.3 and [ 15], Th. 1.2). Conversely, if P~,ois a defect pointed group of Go on A then P~,ois a maximal local pointed group on A, and therefore P~ is a maximal local pointed group on B (cf. 3.15.3) such that Pr ~ G~. 6.4. From now on, we consider A endowed with B as a G-algebra extension (cf. 2.9). Let ct and fl be respectively points of G and H on B such that H a c G~; by Corollary 6.3, ct is also a point of G on A and the corresponding block of G lies over the block of H determined by ft. Moreover, set N = No(Hp) and let P~ and Q~ be respectively defect pointed groups of G~ and Hp such that Q~ c Pr. c Np (cf. 5.2); by Proposition 5.3 we have Q= Pn H

and

N = P" H;

(6.4.1)

it follows from Corollary 6.3 that the interior G-algebra underlying the G-algebra extension A~ (cf. 2.11) is an embedded algebra associated with G~ on A (cf. [19], 2.13), and that the interior P-algebra underlying the P-algebra extension Ar (cf. 2.11) is a source of the interior G-algebra underlying A~ (cf. [15], Def. 3.2). The next result has already been proved - in an unessentially weaker form in [3]. Proposition 6.5. With the notation above, ct determines a nilpotent block of G if and only if fl determines a nilpotent block of H.

Proof, If ct determines a nilpotent block of G and R~ is a local pointed group on B such that R~ ~ H a, there is a unique e~ L~#A(R ) such that e c e~ (cf. Prop. 6.2) and we have R~o~ G~; hence, E~(R~o) is a p-group (cf. [18], 1.7.2) and Nn(R~) ~ N~(R,o), which implies that En(R~) is still a p-group. Conversely, assume that fl determines a nilpotent block of H; if R~ is a local pointed group on B such that R~ ~ Py and if ~0 ~ ~ # A ( R ) contains e (cf. Prop. 6.2), it suffices to prove that E~(R,o) is a p-group (cf. Prop. 6.2, Cor. 6.3 and [18], Th. 3.8). It follows from 6.4.1 and 5.11.2 that the family {(P~, l, u, P~)}~ ~p is a generator set of the Na-biexomorphisms, and therefore it is easily checked that EN(R~, Pr) = Ep(R~, PT) (cf. 2.15.3); hence, as N~(R~o) = CG(R )" N~(R`) (cf. Prop. 6.2), we get NG(R~o) = C~(R).Ne(R,) (recall that EG(R~) = EN(R~) by 5.2). The next result restates Theorem 1.12 in the present situation. Recall that, ifS is a P-algebra, the interior P-algebra SP is the free S-module over P endowed with the evident multiplication and the canonical map from P (cf. 2.9). Theorem 6.6. With the notation above, assume that fl determines a nilpotent block of H. There is an (P-simple P-algebra S such that, considering SP endowed with SQ as a P-algebra extension, we have a unique P-algebra extension exoisomorphism

Av ~- SP .

(6.6.1)

Extensions of nilpotent blocks

53

Moreover, the P-algebra S is unique up to isomorphism and has a P-stable C-basis which contains the unity as the unique P-fixed element. Proof. By Proposition 6.5, ct determines a nilpotent block of G, and therefore as the interior P-algebra underlying A~ is a source algebra of the block (cf. 6.4), it follows from the main theorem of [18] that there is an C-simple P-algebra S, unique up to isomorphism, such that A~ ~ SP as interior P-algebras and fulfilling all the conditions above. Moreover, we know (cf. [18], 1.8.1) that there is a unique group homomorphism p : P ~ S • lifting the action of P on S such that det(p(u)) = 1 for any u e P , and that SP ~ S @c,,CP considering S endowed with p as an interior Palgebra; hence, since we have a (unique) embedding from the trivial interior Palgebra (9 to S | S O (cf. [18], 5.7), we get an interior P-algebra embedding

(6.6.1)

CP ~ S O| SP

which is an embedded algebra (cf. 2.11 or [18], 2.I0) associated with the unique local point of P on S O@~. SP (cf. [18], 5.3.1). On the other hand, by Propositions 6.2 and 3.14 the inclusion By. ~ A~ determines a strict semicovering exomorphism of P-algebras. Then we claim that 6.6.2. the inclusions S ~ | Bt ~ SO | At and S | S O | termine strict semicovering exomorphisms o f P-algebras.

B~ ~ S |162S O |

At de-

Since B is a direct summand of A as CP-modules, for any subgroup R of P we get canonical inclusions and

(S O | B~.) (R) ~ (S O| A~.) (R) (r,

r

(S | S O| B~.)(R) c (S | S O| A~) (R) r

r

Cc

t~

and, by Theorem 3.16, it suffices to prove that these inclusions determine strict semicovering exomorphisms of s but the inclusion B~(R) ~ A~(R) does it (cf. Th. 3.16); hence, by Corollary 5.8 in [18], the following inclusions do it too S(R) ~ | Bt(R ) ~ S(R) ~ | At(R ) ,f

,r

and

(S | S ~ (R) | Br(R ) c (S | S ~ (R) | At(R )

and now 6.6.2 follows from Proposition 5.6 in [18] (which is still true for Palgebras). Moreover, by 5.3.1 in [18] (which is also true for P-algebras) we know that P has unique local points S O x 7 on S ~ 1 7 4 and S | 1 6 2 ~ x ~ on S | 1 7 6 1 7 4 1 6 2 (where we still denote by ~/the unique point of P on Bt). So, by 6.6.2, it follows from Proposition 3.3 that S ~ 2 175 and S | 1 7 6 ~ are respectively contained in the unique local points of P on S OG e A r and S @e S~ G e A r (cf. 3.15.3 and [18], 5.3.1). In particular, since the interior P-algebra underlying the embedded algebra extension (SO| Ar)s o • ~ ~ S O@e~Ar associated with Ps o • ~ (cf. 2.11) is an embedded algebra associated with the unique local point of P on S O@eAt, and since S O| t ~ S O@eS | as interior P-algebras, it follows from.6.6.1 and the

54

B. Kfilshammer and L. Puig

uniqueness of embedded algebras (cf. 2.11.1 or [18], 2.10.1) that (So | A)s,, x :. -~ (~'P e

(6.6.3)

as interior P-algebras. But such an isomorphism maps necessarily (S o | B,~)so x ~, onto (gQ; indeed, it is quite clear that P c~ B = Q and A = O,~r uB, where T is a left transversal to Q in P, which implies successively that Q" By = B~ and A = G,~r u" Bv that O" (s o | By) = S o | By

and

S~ 1 7 4

|

u.(S ~174

and finally that 9_" (S ~ | B~)so • ;, = (S ~ | By)so x ;, r

r

and

(S~ | A~)sO x ~. = O u'(S~ | By)sO x ;,; 6

ucT

r

so, the image C of (S O | B~,)so x ;. in (5~P through the bijective C-linear map 6.6.3 fulfils Q" C = C and 6'P = GuEr u" C which forces C = ~0Q. Hence, considering (s endowed with 6 Q as a P-algebra extension, the m a p 6.6.3 is an isomorphism of P-algebra extensions, and therefore we have S | (S o | Ay)so x ,/~- SP

(6.6.4)

as P-algebra extensions. O n the other hand, the embeddings (9 ~ S | S~ (as interior P-algebras) and (S o | A~.)so x ,,, ~ So | A~ (as P-algebra extensions) induce the P-algebra extension embeddings S | (S O| Ay)so • y --* S | S O| Ay ~- Ay

(6.6.5)

which are two embedded extension algebras associated with the unique local point S | e S o x 7 of P on S | 1 7 6 1 7 4 (cf. 2.11). Consequently, by 6.6.4, 6.6.5 and 2.11.1 we have Ay ~ SP as P-algebra extensions. N o w it suffices to prove that any P-algebra extension automorphism f of SP is induced by a n element of ((SQ) P) • Set w=

1 + ~ Ker(tr)(1 - p ( v ) - l v ) v~Q

where tr: S ~ (9 denotes the trace map; by Proposition 6.8 of [18] we have W c (SQ) • and there is a unique w ~ W such that f ( S ) = SW; thus P fixes w (since P normalizes W and f ( S ) ) . So we m a y assume that f ( S ) = S, and then there is s ~ S • such that f ( t ) = t ~ for any t ~ S; in particular, the uniqueness of p (cf. [18], 1.8.1) forcesfo p = p and therefore P fixes s. Hence, we may assume that f ( t ) = t for a n y t ~ S, and then f = idse.

7. The projective equivalence between nilpotent block extensions 7.1. Let G be a finite group, H a n o r m a l subgroup of G and b a nilpotent block of H (cf. [18], 1.7). As we say in the introduction, our m a i n purpose is to study the blocks of G lying over b. Clearly we have such a situation whenever H is a p-group and b = 1, a n d roughly speaking, our main results (cf. Th. 1.8 and Th. 1.12) state

Extensions of nilpotent blocks

55

that we can always reduce the general situation to a central p'-extension of such a particular case. In this section we give some characterizations of this reduction. But rather than to discuss just the reduction procedure, we will include it in a full equivalence relation which will allow us to invoke symmetry in all the arguments. This relation will be named projective equivalence to emphasize that central p'-extensions are neglected. 7.2. Take again the notation introduced in 5.1 and 5.6. Similarly, let G' be a finite group, H ' a normal subgroup of G', A' the G'-algebra extension formed by (5'G' endowed with B' = (SJH', b' a nilpotent block of H', fl' the point { b' } of H ' on B', e' the unique point of G' on B' such that H i, c G'~,, N ' the normalizer of H i, in G', P'~, and Q;, respective defect pointed groups of G'~, and H'~, fulfilling Q~, ~ P'~,, ~ N~,, and iV' the quotient N ' / H ' . Recall that N ' _~ N~,(Q'~,)/Q"Cn,(Q' ) (cf. 1.7.2) and that )~' denotes the image o f X ' ~ N' in N'. As in 1.6, we consider respectively N and N', endowed with the actions on the sets N and /V' induced by left multiplication, as /V- and IV'-permutation groups (cf. 2.4). 7.3. Let 6: N ~ AT' and 3: P -~ P' be group isomorphisms. We say that the Gand G'-algebra extensions A, and A'2, are projectively equivalent through 6 and T if the following conditions hold: 7.3.1. We have 6(~) = ~(u)for any u ~ P . 7.3.2. For any local pointed groups R E on B and R'~,on B' such that Qa c R, ~ P,~, Q'~, c R'~, ~ P'~., and r(R) = R', the bijection from Hom~7(R, P) to Homg,(R', P') induced by F and r maps E N, g( R~ , Py) onto E N , ' ~ , ( R '~,, P ';,,).

Then A'~, and A, are projectively equivalent through 6 - ~ and r - ~. We say that A, and A',, are projectively equivalent if there are group isomorphisms N ~ /~' and P g P ' fulfilling conditions 7.3.1 and 7.3.2. This relation is clearly symmetric and transitive, and does not depend on the choice of the defect groups. Moreover, condition 7.3.2 implies that: 7.3.3. We have 6(CN(R~)) = CN,(R'~,).

Indeed, by 7.3.2, for any Z~CN(R,) there is x ' ~ N ' such that (R',,) ~" c u r , , r(u) ~' = z(u) for any u ~ P and ~ ' - 16(r = 6(~-1h) for any h ~ N (cf. 2.5 and 2.8); therefore we have x'~Cu,(R',, ) (cf. Prop. 5.5) and Y ' = 6 ( Y ) , whence 6(~)~CN,(R',,); now We get 7.3.3 by symmetry. 7.4. First of all, we will describe the projective equivalence between A, and A',, in terms of the full subcategories ~2 and ,~' of the local categories of Ba and B~, (cf. 2,8) formed respectively by the local pointed groups on B and B' which contain defect pointed groups of H a and Hi,. Consider the evident functors ~:~ ~ ~

and

f ' : ~ ' ~ 91'

from i~ and E' to the categories 9l and 91' formed respectively by the/q- a n d / ~ ' permutation groups and the permutation group exomorphisms (cf. 2.5). Notice that any bijection from/~ onto N ' induces an equivalence of categories from ~ to 91'.

56

B. K/ilshammer and L. Puig

Proposition 7.5. With the notation above, A~ and A'~, are projectively equivalent if and only if there are a group isomorphism 6: 1V ~- N', an equivalence of categories t: ~ ~ ~ ' and a natural isomorphism ~: ~o~ ~ ~ ' ot, where s is the equivalence from 91 to 9l' induced by 6, such that 7.5.1 for any object R~ of 9~, the iV'-permutation group exomorphism ?(R~) has a representative inducing the identity on N'.

Proof. Assume first that there are 6, t and ~ as above fulfilling condition 7.5.1. Since t(Pr) is a maximal object of ~ ' , we may assume that t(Pr) = P'~,. We claim that, choosing a representative (z~,idg,) of ~(Pr) (cf. 2.5), the group isomorphisms 6: N -~ AT' and zr: P - P ' fulfil conditions 7.3.1 and 7.3.2. By definition, for any u ~ P and any ~ , ~ r , we have u ' ~ ' = z~(u).vi' (cf. 2.5), whence 6(ti)li' = z~(u)~' which proves 7.3.1; in particular, we get z~(Q) = Q' (cf. Prop. 5.3). Moreover, let R, and R'~, be respectively local pointed groups on B and B' such that Q~ ~ R~ ~ P~, Q~, = R'~, = P'~, and rr(R) = R', set t(R~) = R~',, and choose a representative (z,, ida7,) = of ~(R~). For any (N,/V)-exomorphism (R~, x, P~)~7, setting t((R,, x, PT)g) = (R~',,, x", P'~,)~,

and

t((R~, 1, P~)g) = (R",,, y", P'r,)g,,

by the naturality of ? we may assume that (cf. 2.5) r~(u) ~'' = r~(u ~)

and

r~(uy" = z~(u)

for any u e R, and that ~"vi' = x - ~ ' = 6(~)vi'

and

)7"~' = fi'

for any ~ ' e / V ' ; hence we have (R~',,)y ' ' = R'~, (cf. Prop. 5.5) and, setting x ' =

y " - I x " , we get (R',,) x' ~ P'~,, 6(2) = ~' and z~u ~) = z~(u) x' for any u~P, which proves that the bijection from ITIomg(R, P) to H o m g , ( R ' , P') induced by 6 and r~ maps (R~, x, Pr)g upon (R',,, x', P'r,)~7,. So, this bijection maps Es, ~7(R~, Pr) to Es,,g,(R'~,,P'r,), and by symmetry its inverse maps Es,,g,(R',,,P'~,) to EN, ~(Re, P~). Conversely, assume that there are group isomorphisms ~':]q~/V', and r: P ~_ P ' fulfilling conditions 7.3.1 and 7.3.2. F o r any object R~ of ~ there are h e H and x ~ N such that (Q~)* ~ R~ and (R,)~ ~ P~; notice that (Q~)h~ = Q~ by Propositions 5.3. and 5.5. By condition 7.3.2 applied to Q~ and Q~,, the bijection from Hom~7(Q~, Pr) to I~om~7,(Q[,, P'r,) induced by 6 and z maps (Q,~, hx, Pr)g upon some element of E s , ~7,(Q[,, P'r,); that is, there is x ' e N ' such that 6(~) = ~' and r(v *~) = z(v) ~' for any v e Q. Moreover, by Proposition 5.5, there is a unique local pointed g r o u p R'~, on B' such that (R') x' = r ( R ~) and Q'~, ~ (R',,) x' ~ P'~,, and we denote by z,: R --* R' the group isomorphism mapping u e R upon z(uX) x'- '; notice that by condition 7.3.1 we have ~ d u ) = ~(uX) ~ ' - ' = 6 ( a ~ ) ~ ' - ' = 6 ( a )

for any u ~ R, and therefore (r,, 6) is a permutation group isomorphism from (R, N) o n t o ( R ' , / V ' ) (cf. 2.5). Now we consider the functor t: ~ --, i~' mapping R~ upon R',, as above and any morphism (9 ~ E~. g(R~, T~,) upon the unique N ' - p e r m u t a t i o n group exomorphism

Extensions of nilpotent blocks

57

t ( ~ ) from R' to T', where T~,, = i(T,), such that

which belongs to EN,, ~,(R'~,, T~,,) by condition 7.3.2; it is quite clear by symmetry that t is an equivalence of categories. Finally, denoting by ~: ~Ji ---,9~' the equivalence of categories induced by 6, we have an evident natural isomorphism ~o~ ~ - f ' o t mapping any object R~ of ~2 upon the N ' - p e r m u t a t i o n group exomorphism determined by (r~, ida7,). 7.6. Now we will prove that the whole condition 7.3.2 depends only on the behavior under the action of 6 and r of a "small set" of (N, N)-exomorphisms. Recall that for any x~NG(Q~ ) and any x'~N~,(Q'~,) we denote respectively by P~ n (P~)~ and P'~,,c~ (P'~,,y' the unique local pointed groups (P c~ P~)~ on B and ( p , ~ p , x , ) , on B' such that (cf. 5.10) (P c~ px)~ c P~

and

(P' c~ P'~')~, c P~., .

Notice that the family { Pr ~ (PQ~, 1, x, P~)} where x runs over the set of elements of NG(Q~) such that P~ ~ (P~)~ = (P~ ~ (P,y)~ (and so P~ ~ ( P . y = (P;)~ c~ P~,) and P c~P ~ = P c~ P~ is a generator set of the Na-biexomorphisms (cf. 2.16.1, 2.17.1, 2.18.4 and 5.11.1). Theorem 7.7. With the notation above, let 6: IV ~- IV' and r: P ~- P' be group isomorphisms and X a subset of N~(Qa) containing NG(P~) such that P~, c~ (Pr)~ = ( P ~ ( P ~ ) ~ ) ~ and P c ~ P ~ = P n P x for any x r Assume that 6 and z fulfill condition 7.3.1 and that the family {( P~ ~ ( P~)~, 1, x, P~) }xr is a generator set of the Nt~-biexomorphisms. Then A, and A',, are projectively equivalent through 6 and z if and only if the following conditions hold: 7.7.1. We have 6(Cu(Qa)) = Cu,(Q'~,). 7.7.2. For any x 6 X there is x'~NG,(Q'~,) such that P ' ~ P ' ~ ' = P ' n P 'x', P'r, c~ (P'~,)~' = (P'~,c~(P'w)~')~', b(Yr = ~' and z(u ~) = z(u)~' for any u r ~. Proof Conditions 7.7.1 and 7.7.2 are necessary; indeed, 7.3.3 implies 7.7.1; moreover, for any x ~ X , setting R ~ = P r c ~ ( P y and considering the ( N , N ) exomorphism (R~, x, Pr)~7, condition 7.3.2 implies that there is x ' ~ N ' such that a ( x ) = x', r(u ~) = r(u) ~' for any u r and (R'~,)~' c Pr, where R~, is the local pointed group on B' such that R'~, = P'r, and R' = r(R) (cf. Prop. 5.5); in particular, we have !

(R'~,) x' = R'~,

and

!

R' = z(R) = ~(/~) = ~ ( P ~ P~) = / 5 , ~/5,~'

which implies P' ~ P ' ~ ' = P' n f i '~'

and

t

t

P , , c ~ ( P ' , , ) x ' = ( P , , n ( P ' , , ) x ' ) x' .

Conversely, assume that conditions 7.7.1 and 7.7.2 hold. Let R~ and R'~, be respectively local pointed groups on B and B' such that Q~ c R~ c P~,, Q~, c R'~, c P~, and R' = r ( R ) (cf. Prop. 5.5), and/~ the unique point of R" Cu(R~) on B such that R~ c R . C u ( R ~ ) u. We will prove first that:

58

B. Kfilshammer and L. Puig

7.7.3. l f P~ contains a defect pointed group of R" CN( R~),, we have 6( CN( R~) ) ~ CN,( R'~,) . If To is a defect pointed group of R" CN(R~)u, it follows from 5.5.2 that Tis a Sylow p-subgroup ofR" CN(R~), whence Tc~ CN(R~) is a Sylow p-subgroup of CN(R~); but by 5.6.1 we get N r . H ( R ~ ) = T" Cn(R~) and therefore CT.n(R~) = C r ( R ). C n ( R ) (since Cr( R~) = Cr( R ) by Proposition 5.5), whence T c~ C~( R~) = CT( R ). Hence, setting as in 5.6 (notice that Ker(~) = CN(Qa))

= N/CN(R~)

and

/ ~ ' = N'/CN,(R'r)

(7.7.4)

and considering respectively ~r and/V', endowed with the actions on the sets N and N ' induced by left multiplication, as N- and N'-permutation groups (cf. 5.6), it follows from Theorem 5.8 and conditions 7.3.1 and 7.7.1 that if T, ~ P. we have

~(CN(R~)) = ~ ( C T ( R ) ) ' 6 ( O v ( C ~ ( R , S ) ) )

= ~(C~(R)). O"(C~,(~', ~')) ~ C,,,(R'~,). Let (R~, y, P,e)ff be an (N,/V)-exomorphism; now we will prove that the IV'permutation group h o m o m o r p h i s m (cf. 2.5) from (R', AT') to (P', N ' ) mapping z(u) u p o n ,(u y) for any u E R and ~(ff) upon 6(37-1h) for any h e n determines an element of EN, ' ~,(R'~,, P'~.,); that is, we will show that

7.7.5. there is y' E N ' such that (R'~,)Y' c P'v,, ~(37) = 37' and r(u y) = "c(u)Y' for any u~R.

W e may assume that Py contains a defect pointed group of R" CN(R~),; indeed, anyway we have n e N such that P~ contains a defect pointed group of (R- CN(R~),)~; if we assume that 7.7.5 holds for any element ofEN, ~ ( ~R) ~ , P~) and denote by R~',, the local pointed group on B' such that Q~, c R~',, c P~, and R " = ~(R"), then there are firstly n ' e N ' such that (R~',,)~'-' c P~,, 6(t~) = tT and r(u) = z ( u ' ) ~'-' for any u ~ R , whence R[,, -- (R'r)"' (cf. Prop. 5.5), and secondly y ' ~ N ' such that ( R ' r ) r ' = ( R ~ , , ) "' ' " ~ P ' , , , 6 ( ~ - ~ y ) = ~ i ' - ~ ; ' and z ( u r ) = r(u")"'- 'y' for any u e R . Consider the N~-biexomorphism (R,, 1, y, P~); by hypothesis, there are finite ordered families {x~}i~ of elements of X and {(R~,m~,P~c~(P~y')}~t and { (P~, ni, P~)}i~ of N-exomorphisms (cf. 2.8) such that (cf. 2.15.3) (R~, 1, y, P,) = ~ (R~, mi, Py ~ (P-y')(P~. ~ (Py)X', 1, xl, P).)(P:,, ni, P,) 9 tel

Arguing by induction on ]I[, we m a y assume that we have (R~, 1, y, Pr) = (R~, 1, s, P~) + (R~, m, T~,)(Tq,, 1, x, P~)(P~, n, P~) where x r X, T~ = P~ c~ (p~)x, s = mn and y = mxn, and that there is s ' e N' such that (R'c) s' c P'y,, 6(g) = g' and r(u ~) = ~(u) s" for any u e R (cf. 7.7.5). Indeed, i f / i s the last element of/, we certainly m a y assume that y = m~xln~ and, setting s = m~n~, it follows from the induction hypothesis that s = ztv where z z CN(R~), v e P and

Extensions of nilpotent blocks

59

(R~, t, P,,.)~7is an (N, N)-exomorphism fulfilling 7.7.5 (cf. 2.8); thus, there is t ' 9 such that (R'~,)r ~ P'~.,, 6(f) = [' and r(u ~) = r(u)" for any u e R ; but by 7.7.3 we have 6(~) = _~' for some z' 9 ' 9 so, it suffices to set s' = z't'r(v). O n the other hand, by condition 7.7.2 there is x ' ~ N ~ , ( Q ' x ) such that P ,t / ~ (P';)~' = (P'.; c~ ( P ~t , )x ' ) x ' , P c~ P'~' =/5, c~/5';', 8(Y) = .~' and r(u ~) = T(u)~' t x ' , we have for any u~ T; notice that, setting T~,, = P'~.,0 (P>,,)

~( T) = d( P ~ P:') = 5( P c~ P x) = / 5 ' n / 5 ' ~ ' =

f"

and therefore r ( T ) = T' (cf. Prop. 5.3). Similarly, there is n' 9 ) such that 6(if) = if' and z(u") = r(u)" for any u 9 Set m' = s'n '-1 and y' = m ' x ' n ' ; we claim that y ' 9 fulfils 7.7.5. Clearly 6Oh) = rh' and therefore 6()7) = )7'. Moret over, as (R'~,) m' ~ P./ and T~, ~ P'r,, by Proposition 5.5 we have (R'c)"' ~ T~,, (since R ~ = T implies z(R ~) ~ r(T) = T', and so /~'~' = d ( R " i = r ( R " ) = ~?'), and therefore we get z(u ~) = r ( u " 9 " ' = r ( u ~ V ' " ' = r ( u y ' .

Let X' be the set of x' ~ N~,(Q'~,) fulfilling the following conditions:

7.7.6. We have P',/~(Pl~,,)~' =(P';,,~ (P),,) ' ~' )~' and P ' c ~ P 'x' =

P' ~/5"L

7.7.7. There is x ~ X such that P).c~(P},)~=(P.~(P~)~) ~, P ~ P ~ = P ~ P ~ , d(~) = .~,' and r(w ~) = z(u)X' Jbr an),, u 9 l(p, ~p,x'). Now it suffices to prove that X ' contains { ( p ' , ~ ( P ~', ) ~, , l , x ' , P ' ~,')~'~x' ~ is a generator indeed, in that case, we finish the proof invoking fulfils condition 7.7.2 with respect to ~-- 1 and r 7.3.1 and 7.7.1 we have (cf. 7.7.4)

N~,(P'~.,) and that the family . . set of the Na,-b~exomorph~sms symmetry since by 7.7.7 the set X ' 1. By Theorem 5.8 and conditions

Eu(P;,) = EK~, %(/5) = E~, ~,(fi') = E v,(e'),, ) and 6(CN(P~,,)) = d ( Z ( P ) ) ' d ( O P ( C N ( P ~ , ) ) ) = r(Z(P))'OP(Cg,(P',N'))

= CN,(P',

) .

Hence, for any n' 9 ) there is n 9 such that ~(u)" = r(u") for any u e P , and by 7.7.2 there is z' 9 ) such that ~-(ff)= z ' n ' ; since U = 6(~?) for some z 9 Cu(P~), the element x = z - l n 9 fulfils condition 7.7.7 with respect to n' _and therefore n ' 9 Moreover, by Corollary 5.11 the family {(/5~/5x, 1,2,/5)~}x~x is a generator set of the (N,N)-biexomorphisms, and therefore the family {(/5'c~ P'g(;), 1, 6(.~), P-' ) s~' } ~ x is a generator set of the ~ (N', N')-biexomorphisms; but it follows easily from 7.7.2 that 3-()() ~ X- " , consequently, once again by Corollary 5.1 I, the family {(P'y. n (p~,)x,, 1, x ' , P'~')}~'~x" is a generator set of the N~,-biexomorphisms. _

Remark 7.8. Actually, since we may assume that for any x 9 the N~-biexomorphism (P~ c~ (pr)x, 1, x, P~) is irreducible (cf. 2.16.1), a careful reading of our proof shows that in condition 7.7.2 we may replace 6(2) = 2' by ?r(Fc) 9 ~( CNi P ~ c~ (P~)*)),~'.

60

B. Kiilshamrner and L. Puig

8. Source algebra extensions of extensions of niipotent blocks 8.1. In this section we will prove our main results, namely Theorems 1.8 and 1.12. Actually, in the proof of Theorem 1.12, we will give an explicit description of an (9-simple interior P-algebra S such that the P-algebra extension isomorphism 1.12.1 holds. We take once again the notation introduced in 1.6 and 1.7. 8.2. First of all we consider the case when G, H and b fulfil the following additional condition:

8.2.1. There is an indecomposable (gG-module M such that 6" Resin(,( | simple ~eH6-module.

M) is a

Then A maps onto Ende(M ) and, denoting by m the kernel of this map, we set Sa = Aa/fz- 1(m) for any pointed group Ka on B (cf. 2.11) and denote by r~: A~ --* Sz the canonical map. Clearly Sa does not depend on the choice of ]4 in and we consider Sa, endowed with the group homomorphism K ~ S,( determined by ra: A,~ --, $2 and the structural group homomorphism K --* A 2 , as an interior K-algebra, whence as a K-algebra extension (cf. 2.9). 8.3. Let R~ be a local pointed (cf. 8.2.1), the homomorphism section; precisely, considering with Be and S~Q as R-algebra

group on B such that Q~ c R~ c P~. As r#(B#) = S# r~: B~ ~ St is surjective and we claim that it has a ~u~R u'B~ and S,R (cf. 2.9) respectively endowed extensions, we claim that

8.3.1. There is an R-algebra extension isomorphism q~: S~R ~- Eueg u" Be such that r~(q~(s)) = s for any s6S~. Indeed, it is clear that ~,~R u'B~ ~- ( C ( R ' H ) ) , as R-algebra extensions, and it follows from Theorem 6.6 applied to R. H, H and b that there is an C-simple Ralgebra S such that SR ~- ((9(R" H))~ (cf. Prop. 5.3 and Remark 5.4); in particular, r~ induces a surjective R-algebra homomorphism from SQ onto St and therefore an R-algebra isomorphism S -~ S~ (cf. [18], 1.8.1). This proves also that:

8.3.2. The p-group R stabilizes an (9-basis of S~ containing the unity as the unique R-fixed element. 8.4. Consider the R-algebra extension S o | A~; by Theorem 5.3 of [18], R has a unique local point S ~ x e on S~0 | B, and we denote by g~: C~ ~ S o | A~

(8.4.1)

g,

an embedded algebra extension associated with S o x e, and by D~ the structural subalgebra of C~ (i.e. De = h i ~(S ~ | since we have an interior R-algebra embedding d,: (9 ~ S o |162 (cf. [18], 5.7) and an R-algebra extension isomorphism S~R ~- St | (gR (cf. [18], 2.7.1), we have an R-algebra extension embedding (gR --* S o |162S,R and it follows from 8.3.1 and the uniqueness of the local point S o x e that:

8.4.2. There is an R-algebra isomorphism D~ ~- (gQ.

Extensions of nilpotent blocks

61

Another easy consequence of 8.3.1 is that F ~ : B ~ S , and therefore i~l | F,: S O | B~ -~ S O | S, are covering exomorphisms of R-algebras (cf. [18], 4.14 which can be extended to G-algebras) and, in particular, the unique local point of R on S O | B, maps to the unique local point of R on S O | S~; hence, it follows from 2.11.2 applied to the composition of the R-algebra extension exomorphisms/'7~ and id | F~: S O @r; A~ ~ S~174 S, that:

8.4.3. There is an R-algebra extension homomorphism t~: C, ~ C such that d~ o ~ = (i-d | ~)o h-~. 8.5. As R, c Py there is a unique R-algebra extension embedding )~}': A~-+ ResP(A~) such that Res,~(J~)o~'e = ~ (cf. 2.11.3); moreover, since

L- ~(m) (L~)-'(L-'(m)), =

the embedding ~ induces an interior R-algebra embedding ~ : S~ ~ Rest(ST) such that ~ o F~ = Res~(F~)oJ~. Consider the tensor product of these embeddings ~ |

S o | A~ -+ ResP(S ~ | Ay)" r

(8.5.1)

('

it follows from Proposition 5.5 that R has a unique local point on B~ and therefore on S o @~, B~ (cf. [18], Th. 5.3); hence, the composition (9~

f~ ) h~: C~ + ResR(S ~ @ Ay) r

is an embedded algebra extension associated with the second point; but by 8.4.2 we have D~ -~ ResV(D~) and, in particular, ResVR(h'~); Ress(Cy) v ~ ResR(S,r e 0 @eA~) is an embedded algebra extension associated with the same point; therefore, by 2.1 1.1 we get:

8.5.2. There is a unique R-algebra extension exoisomorphism ff~: C, ~- Rese(c~) such that ( ~ | o h~ = Res~(/~) o h'~. In particular t~o h~ = t~. The last equality follows from Rest(aTe,) = (ff~ @f~%') o c7, (cf. [18], 5.7); indeed, we have (cf. 8.4.3 and 8.5.2)

Res~(a-~oty)oh~ = Res~((id | Fr) o #y) o h~ = Res~(icl | Fr) o (g~'r@.~)~~'o ff~ = ( ~ | ~')o(i~l | F~)o ff~ = Res~(d'~) o and we apply 1.3 and 1.5 of [17]. 8.6. F r o m now on we set C r = C, D r = D, S r = S, d~ = d and t~ = t, and choose C, = ResP(C) endowed with the embedding (/~')-~ o/~, denoted again by if, (cf. 2.11 and 8.5.2); moreover, we denote by r: P--* C • the structural group h o m o m o r p h i s m and it follows easily from 8.4 that D = ~ (gv(v)

and

~(P) c N c •

+ Ker(t)).

(8.6.1)

v~Q

In particular . N c , ( v ( Q ) ) normalizes D o = Z(D) and we denote by p: Nc*(~(Q))-+ Nc,(z(Q))/(D~ • the canonical map. Let x be an element of

62

B. Kfilshammer and L. Puig

NG(R~) and denote by ~0 the automorphism of R induced by x (i.e. q~(u) = u x for any u ~ R ) , and by f~: Rest(A) ~ Res,~ResGR(A) (8.6.2)

the R-algebra extension isomorphism mapping a0~ A on a~; as x normalizes R~ and m, and R has a unique local point on S o | B~ (cf. 8.4), it follows from 2.11.1 applied twice that: 8.6.3. There are unique R-algebra extension exoisomorphisms

f ] : A~ ~ Res,AA~), ~,: St ~ Res~o(S~)

and

hx.~-~"Rest(C) =~ ResCRes~(C)

such that

Res,p(J~) oJ'~ =J~ ~,J~, ~ o ~ = Res~o(Y~)ojTf and Res~o(h~)oh~ = (~, |

h~.

But clearly Res~(fx) = i~t as C-algebra exomorphisms, and therefore we get successively Res~(f~)= i~t, Res~(~,)= i'd (a priori true since S~ is C-simple) and Res~(h~) = i~t as C-algebra exomorphisms (cf. [17], 1.5); that is, the maps f~, g~ and h~ are just conjugation by suitable elements of A { , S~ and C • and the following proposition exhibits a particular choice of them.

Proposition 8.7. With the notation and hypothesis above: 8.7.1. There is a ~ A {

such that f~(a)~xB R and f~(ao) = a~ofor an)' a o ~ A ~.

8.7.2. There is c ~ C • such that h~(c)~(rda) -1 | a)(S ~ | for any c o ~ C.

B~) e and h~(co) = C~o

Remark 8.8. In particular aB, = f ~ - ~ ( x B ) and cD = h~-~ (S~o | aBe). Moreover, notice that r(u) ~ = Ux(r(u)) = r(u ~) for any u ~ R . Proof Set f~(1)= i; since i and i~ belong to e there is b'E(Bg) • such that i~b'= i, and therefore there is a unique a ~ A { such that f~(a)= xb'i (since i ( x b ' ) - l x b ' i = i); moreover, denoting by f~: A~ ~A~ the inner C-algebra automorphism induced by a (i.e.fa(ao) = a~ for any ao~A~) we have

f~(f,(ao) ) = ( x b ' ) - ~f~(ao)xb' =f~(f(ao)) b' (since ~(ao)i =fAao)), which~ proves that f, is an R-algebra extension homomorphism and that f, = f ~ (cf. 8.6.3). Similarly, set h~(1) = j ; sincej and (g~ | f j )~( j )9 belong to S~o x (the unique local point of R on S O| B~) and we may assume that (g~ | = J~(~)-' | there is d'e((S~174 • such that j =j(~d~)'| and therefore there is a unique c E C • such that h~(c)= ( r A a ) - l | a)d'j; as above, the inner C-algebra automorphism induced by c is an R-algebra extension homomorphism belonging to h~. Corollary 8.9. With the notation and hypothesis above, assume that R~ = Q6. There is a unique group homomorphism w: N~(Q~) --* Nc~ ('c(Q))/(D Q) • mapping x ~ N ~ ( Q 6) upon p ( c ) and we have Ker(~o)= C ~t( Q ).

(8.9.1)

Extensions of nilpotent blocks

63

Proof. By 8.7.2 we have z(v)C = h ~ ( r ( v ) ) = r(v x) for any v 6 Q and therefore c ~ N c x ( r ( Q ) ). Moreover, if x ' i N g ( Q 6 ) , a ' ~ A ~ and c ' ~ C • is another system fulfilling the conditions above (i.e. f 6 ( a ' ) ~ x ' B Q, f ~ ( a o ) - - a o"' for any ao~A6, h6(c')~(r6(a')-l|176174176176 o for any c o , C ) , then we have = c c' f~(aa')exBex'B O = xx'B Q

and

(f~ ~fx~)(ao) = a"o"'

for any a o 9 A 6, and similarly h6(cc')e(r6(aa')-l|176

| Ba) Q

and

(h~x,~:h~)(Co) = C'o~'

C

for any co e C. But it follows easily from 8.6.3 that .~, of~ =f~x' and ff],o h'~ = ff~x,, ~' , which proves and therefore we may assume thatJ]x,(ao) = ao" ' and h~,(Co) = Co that p(c) does not depend on the choices of a and c (since x' = x ~ forces cc' e Do), and that the map co assigning to x the element p ( c ) s N c • (r(Q))/(D ~ • is a group homomorphism. Moreover, if x ~ Cu(Q) we get necessarily a e B~ and c ~ D o (cf. 8.7.1. and 8.7.2). Conversely, if xf~Cn(Q) either x C H or x~CG(Q); in the first case, we have B c~ xB = ~:~ and we get successively B 6 ca aB 6 = ~ and D ca cD = ~ (cf. Remark 8.8); in the second case, since h~(r(v)) = r(v ~) for any v s Q, we have c r C o (cf. 8.6.1); in both cases p(c) 4= 1. 8.10. Come back to the general situation above (cf. 8.6). Since N~(R~) ~ NG(Qa ) (cf. 5.5.1), we may consider (cf. 8.6.3) f~: A6 --- Res~,(A6),

O~: $6 ~- Res,(S~)

and

ffx~: Rest(C) ~ Res0Res~(C )

where ff denotes the automorphism of Q induced by x, and it follows easily from 2.11.1 and 2.11.3 applied twice that Res~(f~)of~ =f6~ .f;,

ResQ(g~)o 96 = O] ~ 9~

and

= h~

(8.10.1)

where 0]: $6 ~ Res~(S~) is the embedding induced b y ~ ~(cf. 8.2). Now we are able to compare co(x) with p(c) (where a and c have been chosen from Rfl). Corollary 8.11. With the notation and hypothesis above, we have co(x) = p(c). In particular co(u) = p(r(u)) for any u e P. P r o o f ~ B y o u r choices of C~ and Ca (cf. 8.6), and by 2.11.1 we have Res~(ff~) = (O,~| ~ ha and therefore it follows from Remark 8.8 that cD = h? ~(S~ |

=

h ; ~(S~ |

~(xB))

(notice that we may assume ~ =.~ of~ and h~ = (g~ | ,, h6). O n the other hand, by the definition of co and Proposition 8.7, there is c ' e C • such that p ( c ' ) = co(x), c'D = h s 1 7 6 1 7 4 (cf. Remark 8.8) and h~(co) = Co ~' for any coeC. Hence cD = c'D and, since Res~(/~,)= hr~ (cf. 8.10.1), there is d e ( D e ) • such that c'dc -1 e Z ( C ) ; therefore c-~c'd belongs to (De) • and so p ( c ) = p ( c ' ) = co(x). In particular, if x e R we may choose firstly a equal to the structural image of x in A : and secondly c = ~(x), which proves the last equality (setting R~. = P~).

64

B. Kfilshammer and L. Puig

8.12. Set F = Im(co), I = p - I ( F ) n (1 + Ker(t)), J = I c~ (DO) • and Z = Z(Q). By Corollary 8.9 we have F ~- N~(Qa)/Cn(Q); therefore, on one hand co(P) is a Sylow p-subgroup of F (cf. Prop. 5.3) and, on the other hand, by 1.7.2 we have a group homomorphism ~: I ---}hl (8.12.1) mapping c ~ 1 upon 2 for some x~N~(Qo) such that p(c) = co(x) (i.e. co-~(p(c)) = {re(c)}). By 8.6.1 and Corollary 8.11 we have r(P) ~ I

and

J n r(P) = ~r(Z).

(8.12.2)

Moreover, notice that J is a p'-divisible abelian group (cf. 8.6.1). The following statement is the key step in our proof of Theorems 1.8 and 1.12. Theorem 8.13. of I such that:

With the notation and hypothesis above, there is a finite subgroup L

8.13.1. We have ~(P) c L, 1 = J" L and ~(Z) = J n L. 8.13.2. Considering the (f~-algebra CL endowed with the group homomorphism ~: P ~ (CL) • and the subalgebra C(v(Q)) as a P-algebra extension, the inclusion L c C • induces a P-algebra extension isomorphism CL -~ C. 8.13.3. The group L and the group homomorphisms ~: L ~ .~ and r: P - . L fulfil conditions 1.8.1 and 1.8.2. Proof Setting ]-= I/T(Z) and J = J/~(Z), and denoting by /5: I--, F the group homomorphism determined by p, we have the following exact sequence 1--* J ~ l

, F--,1

(8.13.4)

and, since co induces a group isomorphism P / Z ~- ~o(P), ~ induces a group homomorphism ~: c o ( P ) ~ 1 such that ~(~(co(u)))= co(u) for any u ~ P (cf. Cor. 8.11); that is, ~ fulfils condition 4.7.1 and we will prove that ~ fulfils also condition 4.7.2 with respect to the elementsf~ F such that co(P) n ~o(P) s = (co(P) n co(p)s)s. L e t f be such an element and R the subgroup of P containing Q such that ~o(R) = ~o(P) n co(p)s (notice that ~o(Q) is normal in F); by Proposition 5.5, there is a unique s ~ LP~e(R) such that Q~ c R E c P~ and, if 2 is the unique point of R- Cu(Q) on B such that Qa c R ' C n ( Q ) ~ (cf. 2.16.2), R~ is a defect pointed group of R" CH(Q)~ (cf. 5.5.2). Hence, by the Frattini argument, there is x ~ N~(R~) such that co(x) = f and by Corollary 8.11 there is c ~ N c . ( ~ ( Q ) ) such that p ( c ) = co(x) = f and v(u ~) = r(uy for any u ~ R (cf. Remark 8.8); as we may assume that c ~ l (up to replacing e by t(c)- lc), we get t5(~) = f and ~(co(u)) ~ = ~(co(u)s) for any u ~ R, where is the image of c in I. I

It follows now from 2.17.1 and Corollary 4.7 that there is a group homomorphism 6: F ~ / s u c h that ~ o ~ = id v and ~(og(u)) = ~(co(u)) for any u~P. We claim that the inverse image L of Im(6)in I fulfils conditions 8.13.1 to 8. l 3.3 above. Condition 8.13.1 follows easily from the properties of 6. Moreover, we have n(L) = lq ~ N6(Q~)/Q.Cn(Q) (cf. 8.12.1, 8.13.1 and 1.7.2) and, if {x~}~, a n d

Extensions of nilpotent blocks

65

{ y ~ } ~ are respectively sets of representatives for/V in N~(Qa) and in L, it is clear that co(xs) = p(y~) for any fie ,g (cf. 8.12.1), and that Aa = @~ ~ x ~ B a (cf. 1.7); then it follows from Remark 8.8 that

ynD = hs x(S6 |

(xnBp))

for any ~iEN and therefore we get C = ~ y n D , which proves condition 8.13.2. O n the other hand L c~ Ker(r0 = r(Q) (since L c~ J = r(Z)), Ker(r) = {1 } (cf. 8.6.1 a n d Cor. 8.11) and z(P) is a Sylow p-subgroup of L; in particular, in 7.2 we may assume that G' = L, H' = z(Q) and b' = 1, and in 7.3 we may consider the group isomorphisms N -_ LIt(Q) and P _-__r(P) induced respectively by rr and r; then it is quite clear that conditions 7.3.1 and 7.3.2 imply conditions 1.8.1 and 1.8.2. So, it suffices to prove that A~ and (gL are projectively equivalent through the group isomorphisms N ~ L/z(Q) and P ~ z(P) above. Since ~ ( z ( u ) ) = ~ for any u e P (cf. 8.12.1 and Cot. 8.11) and ~(CL(T(Q)) ) = CN(Q~ ) (cf. 8.12.1 and Remark 8.8), conditions 7.3.1 and 7.7.1 hold and, by 7.6 and Theorem 7.7, it suffices to prove that condition 7.7.2 holds with respect to the elements x ~ NG(Q6 ) such that P c~ px = p c~ P~ and P~ c~ (P~)~ = (Pr c~ (p~,)x)~. Let x be such an element and set R~ = P~ c~ (P~)~; by Corollary 8.11 and Remark 8.8 there is ceNc~(z(Q)) such that co(x) = p(c) and z(u ~) = T(uy for any u e R . Hence, since we may assume that c e I (up to replacing c by t(c)-lc) and we have I --- J - L (cf. 8.13.1), we get c = zy for some y e L and some z e J; moreover, as c normalizes r(R) a n d we have J c~ L = z(Z) c r(R), y and therefore z still normalize r(R) and, since N j ( z ( R ) ) = C j ( z ( R ) ) ' z ( N z ( R ) ) by Lemma 8.15 below, we may assume that z centralizes T(R), whence p(z)= 1. In that case, we get g ( y ) - - 2 (since p(y) = p(c) = co(x)) and r(u ~) = z(uy for any u e R , and then it is quite clear that /~ = P ~ P~ implies both z(R) = T(P) ~ z(P)Y and that the image of z(R) in L/~(Q) is the intersection of the images of z(P) and z(P)Y.

Remark 8.14. The existence of 6: F - ~ [ above follows also from a result of Oliver [11]. Indeed, we may assume that ,( is an algebraic closure of the prime field and that (9 is totally unramified. In that case Oliver's result implies that Hx(co(P), J) = {0}. O n the other hand, the existence of g shows already that the exact sequence 8.13.4 splits, and the existence of 6 is equivalent to the existence of a c o m p l e m e n t / 2 of J in I containing f(co(P)). But a complement of J in I contains certainly a complement of J in J" ~(co(P)) and Oliver's result forces the last one to be ~(~o(P)) ~ for some i ~ f . The following temma, quoted above, is a slight generalization of a result by Coleman [5] and our proof employs his argument (see [17] for a different generalization). L e m m a 8.15. Let K be a group, Q a subgroup of K and R a finite p-subgroup of

Nx(Q). I,Ve have NWQ)~(R)=CwQ)~(R).NQ(R ). In particular, if Q c R Nz(cQ)~ (R) = CZWQ)~(R)" Nz(~(R ).

then

Proof. F o r any a ~ NWQ). (R) we consider the action of R on (gQ mapping c e (~Q upon u- ~ca- ~ua for any u s R; clearly R stabilizes Q (since Q = K c~ (gQ in C K ) and

66

B. K/ilshamrner and L. Puig

fixes a through this action; so if a = ~x~o 2xx where 2xE(9 for any x e Q , there is x e Q fixed by R such that 2x e (9 • since the augmentation map (gQ ~ (9 assigns to a an invertible element; that is, we have a - l u a = x - l u x for any u ~ R and therefore a = zx for s o m e 2EC((~Q)~(R). 8.16. Henceforth we prove Theorems 1.8 and 1.12. In order to prove the first one and the existence of S in the second, we may assume that (9 is totally unramified (cf. [18], 2.13.1) and, in particular, that (9 contains a non-trivial root of unity ~ of p-power order only i f p = 2 and ~ = - 1. Let na be the unique ideal of B~ such that B~/n~ is an (?-simple algebra and the determinant of the structural image of any v e Q is one (cf. [18], Th. 1.6 and 1.8.1), and np the unique ideal of B~ such that ( f f ) - l ( n p ) = n~ (cf. 2.14.1). We claim that: 8.16.1. For any x e N, we have (n~)x = n~.

Indeed, by 1.7.2 we may assume that x e NG(Qa ) and, as in 8.6, we denote by f f : R e s t ( A # ) ~ Res~Res~(A~) the (9-algebra extension isomorphism mapping a s A ~ upon a ~, where ~0 is the automorphism of Q induced by x; then, by 2.11.1, there is a Q-algebra extension exoisomorphism f~: A~ ~ Resr such that Res~(J~) o f~ = f ~ o J~; now the uniqueness of n~ forcesff(n~) = rt~, whence we get ff(np) = ng. In particular, it follows easily from 8.16.1 that G stabilizes the isomorphism class of an (gHTr~(b)-module M such that ~f | is a simple ,~Hb-module. 8.17. Thus there is always an extension G' of G by an abelian finite group U' such that H lifts to a normal subgroup of G' - still denoted by H - and such that the triple G', H and b fulfils condition 8.2.1. Moreover, denoting respectively by N' and V' the stabilizers of b in G' and in U', we may assume that (8.17.1)

V' ~ D ( N ' ) n Z ( N ' )

and that V' is isomorphic to a subgroup of ~ • As in 1.6, set A' = (gG', consider A' endowed with B as a G'-algebra extension and denote by N ' the quotient N'/H. Clearly c~is still the unique point of G' on B such that Hp ~ G'~, fl is still a point of N' on B and, denoting by P' the Sylow p-subgroup of the inverse image of P in G', 7 is still a local point of P' on B such that Qo ~ P; c N} c G; and P~ is also a defect pointed group of G;. As in 1.7, denote respectively by J~: A~ ~ A,J~: A~ ~ Res~:(A)

and

J~': A~--* Res~:(A)

embedded algebra extensions associated with G;, N~ and any local pointed group Rs on B such that Qo c R~ ~ P;. Notice that, since A'~ ~- (gN'b (cf. 1.7), A~ has an evident (gV'-algebra structure and then As has a unique (gV'-algebra structure such that the canonical embedding ~'~: A[--* Resgi(A~)(cf. 2.11.3) becomes an embedding of R'-algebra extensions over (9 V'; thus, since Ap ~ (gNb (cf. 1.7), we have N- and therefore R-algebra extension isomorphisms, R being the image of R' in P and ~ the trivial (9 V'-algebra, Aa = s | 6 V'

A~

and

AE = (9 | (~ V'

A[

(8.17.2)

Extensions of nilpotent blocks

67

where (5 | v,A'p and (5 |162v,A'~ are respectively endowed with the subalgebras 1 | Bp and 1 | B~, and with the evident group homomorphisms. 8.18. Since G', H and b fulfil condition 8.2.1, there is an ideal m' of A' such that A'/m' is an r algebra and we have bA'b = Bb + bm'b ,

and we may assume that the restriction U'--* (A'/m') • of the structural homomorphism is injective. As in 8.2, we set S~ = A~/f~ l(m') considered as an interior R'-algebra, whence as an R'-algebra extension (cf. 2.9), and by 8.3.2 R' stabilizes an Cr~'-basisof S~ containing the unity as the unique R'-fixed element. Notice that the 6' V'-algebra structure of A" induces an ~' V'-algebra structure on S~, namely an (5algebra h o m o m o r p h i s m r': CV'--+(5'

(8.18.1)

which does not depend on the choice of R~. 8.19. As in 8.4, we denote by/~': C~ -* S "~ @e A~ an embedded algebra extension associated with the unique local point of R' on S~~ @e B,, by D~ the structural subalgebra of C~ and by t~: C[ ~ ~ the R'-algebra extension h o m o m o r p h i s m 8.4.3. The 6~V'-algebra structures of A~ and S" induce an (5V'-algebra structure on S~~ |162A~ (through the diagonal map V' --* V' x V'); clearly there is a unique 6~'V'algebra structure on C~ such that h~ becomes an embedding of R'-algebra extensions over (r. V' and it is easily checked that t~ is an 6~V'-algebra homomorphism. As in 8.6, we set C~ = C', D~ = D', S I, = S', d~. = d' and t~, = t', and we denote by ~': V " P ' ~ C' • the structural group h o m o m o r p h i s m (both as P'-algebra extension and (9 V'-algebra) and by p': Nc,~ (r'(Q))--+ Nc,~ (r'(Q))/(O 'Q) x the canonical map. 8.20. Now Proposition 8.7 and Corollaries 8.9 and 8.11 apply to our situation, and we denote by co': Nc,,(Qa) --+ Nc,~ (z'(Q))/(D'Q) • the group h o m o m o r p h i s m 8.9.1; notice that: 8.20.1. For any v' ~ V' we have eo(v') = p'(z'(v')).

Moreover, as in 8.12 we set F ' = I m ( o ' ) , I ' = p ' - l ( F ' ) n(1 + K e r ( t ' ) ) and J ' = l'c~ Ker(p'), and denote by n':l'--* N' the group h o m o m o r p h i s m 8.12.1; notice that (cf. 8.20.1 and Cor. 8.11): 8.20.2. We have z' ( V') = I', J ' n ~'(V') = {1} and n' ( ~ ' ( v') ) = ~' for any v' ~ V'. Moreover, the complement of z'( V'c~ P') in z'( V') is the unique complement of J " z ' ( V ' n P ' ) in J " ~ ' ( V ' ) .

Hence, by Theorem 8.13, there is a finite subgroup L' of I ' such that: 8.20.3. We have v'( V'. P') ~ L', I' = J " L' and v'(Z) = J' n L'. 8.20.4. Considering the (5-algebra (SL' endowed with the group homomorphism r': V" P'--* ((5L') • and the subalgebra (5(~'(Q)), the inclusion L' ~ C '• induces an isomorphism (5L' _~ C' both as P'-algebra extensions and as (5 V'-algebras.

68

B. Kflshammer and L. Puig

8.20.5. The group L' and the oroup homomorphisms n' : L' --* IV' and r' : P' ~ L' fulfil conditions 1.8.1 and 1.8.2 with respect to G', H and b. 8.21. We claim that the group L = L'/z'(V') and the group homomorphisms 7t:L ~ N and z: P ~ L induced respectively by ~t' and z' fulfil conditions 1.8.1 and 1.8.2 with respect to G, H and b. Indeed, it is quite clear that condition 1.8.1 with respect to G', H and b implies condition 1.8.1 with respect to G, H and b. Moreover, let R e be a local pointed group on B such that Q6 ~ R, c P~ and denote by R' the inverse image of R in P'. Recall that we consider respectively N and N' as Nand N'-permutation groups (cf. 1.6), and letu ~ s denote by Hom~7,(R', P')v, the set of /q'-permutation group exomorphisms (~o',f') from R' to P' (cf. 2.5) such that ~0'( V' n R') c V' ~ P' and f ' (fi' l?') = f'(fi') I?' for any fi' E/V', which clearly contains EN,~,(r'(R'~, P'~) (cf. 2.8); then, we have an evident map Hom~7,(R', P')v' ~ I7Iom~7(R, P)

(8.21.1)

and it is clear that EN, Iq.(R'~, P'r)_maps onto EN, g(R~, P:.). Similarly, we consider respectively L and L' as N- and N'-permutation groups (cfl 1.8.l), we introduce the set of /V'-permutation group exomorphisms HOmN,('c'(R'), z'(P')) v, containing EL,,Iq,('c'(R'), z'(P')) and we have a map FIom~,(r'(R'), r'(P'))v" --* FIomg,(r (R), z(P))

(8.21.2)

sending EL,,N,(~(R'),z'(P')) onto EL, N(r(R),L(P)). Since the bijections zR: t ~ t r Hom~q(R, P) = Hom~7(r(R), r(P)) and rR' : Hom~7,(R', P')v' ~- Hom~7,(r (R), r'(P'))v' induced respectively by r and z' are compatible with the maps 8.21.1 and 8.21.2, condition 1.8.2 with respect to G, H and b follows from condition 1.8.2 with respect to G', H and b. To finish the proof of Theorem 1.8 it suffices to discuss the uniqueness of L, n and r; this uniqueness follows easily from Proposition 4.9 since conditions 1.8.1 and 1.8.2 imply respectively conditions 4.9.1 and 4.9.2 with respect to any subset of L. 8.22. We will prove now the existence part of Theorem 1.12. Since r' maps injectively V' into (9• and p divides neither ranke(S') (cf. 8.18) nor ]L':z'(P')I (cf. 8.20.5 and Remark 1.9), the composition 0' of the structural homomorphism L' ~ Ind~e,)(S'~') • where S"' is the restriction of S' to r'(P'), with the determinant map maps injectively r ' ( V ' n P') into C• but, as we assume that (9 is totally unramified, 0'(r'( V' n P')) has always a complement ~//in the group of the roots of unity of (9; consequently, L" = 0'-1(o//) is a complement of r ' ( V' n P') in L' and in particular, setting T'(P") = z'(P') n L", P" is a complement of V' n P' in P', so that P" - P. We denote by z": P --* L" the group homomorphism induced by z' and this isomorphism, and by S" the restriction of S' to P. Then, by Remark 1.13, it suffices to prove that we have A~ ~ S" | (9./~o C

as P-algebra extensions.

(8.22.1)

Extensions of nilpotent blocks

69

8.23. First of all, notice that we have the following P-algebra extension embeddings (cf. 8.4 and 8.19)

i~d|174174176174 ~

~|174176174

and C

d,

(8.23.1) C

(9

which are embedded algebra extensions associated with the unique local point of P' on S' |162S '~ | B~ (cf. 8.4.2 and [18], 5.3.1 extended to P-algebras). Moreover, the (9V'-algebra structures of A'~, S' and C' induce successively unique (9V'-algebra structures on S' | S '~ | A'r and S' | C' in such a way that ~ | i~l and i~l | become CV'-algebra embeddings, Consequently, by 2.11.1 and 8.20.4 we have an isomorphism A'~ ~- S' | (gL' both as P'-alqebra extensions and as CV'-algebras, and by 8.17.2 we get a P-algebra extension isomorphism, C being the trivial (9 V'-algebra,

A~.-~(9e@(S'Q(gL' ) .

(8.23.2,

8.24. But denoting by V" the complement of V' r~ P' in V' and considering (gL", endowed with the group homomorphism P --* ((gL") • induced by z" and with the image of (gQ, as a P-algebra extension, it is not difficult to check that we have P-algebra extension isomorphisms

S"~((9"r

)

(8.24.1)

mapping s" | (1 | y") upon 1 | (s" | y") where s"eS", y"eL" and (9" denotes the ring C endowed with the (9 V"-algebra structure mapping any v" e V" upon r'(v")- 1 (cf. 8.18.1). Moreover, as

bA'b/bm'b ~- Bb/(B c~ m')b

and

~r | (Bb/(B c~ m')b) "~ Ba/J(Ba)

(8.24.2)

(

(cf. 8.18), the structural homomorphism N'~(bA'b/bm'b) • induces a group homomorphism ff':/~'-*/V (cfi 1.11) mapping any 17"eI?" upon r'(v")eAr215 (cf. 8.18.1), and therefore a group homomorphism a ' : L ' - * L such that o a' = ~' o n' where/~ = Rest(N) (cf. 2.4) and z~:L --, N is the canonical map; now it can be checked that we have a P-algebra extension isomorphism (9" |

(gL" ~ (9,/_~o

(8.24.3)

~V"

mapping 1 @ y" upon a'(y") -1 for any y"~L". The isomorphism 8.22.1 follows now from the isomorphisms 8.23.2, 8.24.1 and 8.24.3. 8.25, It remains to prove the uniqueness part of Theorem 1.12, without any hypothesis on the ramification of (9. First of all, we claim that

8.25.1. if S is an (9-simple interior P-algebra such that A~ ~-S | ~ then P stabilizes an (9-basis of lnd~e~(S ~) where S ~ is the restriction of S to z( P). Indeed, for any x, y eL, it suffices to prove that z ( R ) = z( P) c~ z( P)* c~ z( P) y stabilizes an 6~-basis of x - a | S' | y, or equivalently, that R stabilizes an (9-basis of

70

B~ K/ilshammer and L. Puig

Res~ ]fR)(S) where 6x y:R ~ P x P maps u ~ R upon (u x- ', (u y-l) 1)(we identify R withX'~,~.(R)); but, if ~ and )) lift respectively x and y to /~, it is clear that ~,~R C3~r(u) ~-1 is a direct s u m m a n d of C , L ~ isomorphic to lnd6~RxxR~y(R)(C)as C(R ~ x RY)-modules and therefore IndfX~ff~(Res~xfR)(S))is a direct su/fimand of A~ as _C(R~x RY)-module; in particular, Res e• ,~RI(S) is a direct s u m m a n d of Res~Xfg)(Ar), so it is a permutation OR-module.' 8.26. Let S and Theorem 1.12. By P-algebras; hence, of the action of P

T be C-simple interior P-algebras fulfilling the conditions of Remark 1.14, we know already that S and T are isomorphic as we have ~ | S ~ ~ | T as interior P-algebras (since the lifting is unique) and, in particular,

~e| Ind~lp~(S ~) ~ ~r | lnd~p)( T ~) (8.26.1) c where S ~ and T ~ are respectively the restrictions of S and T to z(P). But it follows from 8.25.1 and from the fact that p does not divide the dimension of both terms in 8.26.1, that the P-algebras Res~(Ind~te~(S~)) and Res~(Ind~e~ (T~)) fulfil conditions 1.3.2 and 1.3.3 in [20] and therefore they are isomorphic (cf. [20], Cor. 3.7). Moreover, since the determinant maps the images of P in Ind~tp~(S~) • and Ind~te)(T~) • onto {1}, we get Res~(Ind~w~(S~)) -~ Rest(Indite)(T~))

(8.26.2)

as interior P-algebras. As the canonical embeddings S--* Res~(Ind~p~(S~)) and T ~ Rest(IndUe)(T~)) (cf. 2.13.1) are embedded interior P-algebras associated with the unique local point of P on both terms of 8.26.2 (cf. [18], Cor. 5.8), we get finally S -~ T as interior P-algebras.

References 1. Broue, M., Puig, L.: Characters and local structure in G-algebras. J. Algebra 63, 306 317 (1980) 2. Broue, M., Puig, L.: A Frobenius theorem for blocks. Invent. Math. 56, 117-128 (1980) 3. Cabanes, M.: Extensions of p-groups and construction of characters. Comm. Algebra 15, 1297-1322 (1987) 4. Caftan, H., Eilenberg, S.: Homological algebra. Princeton: Princeton University Press 1956 5. Coleman, D.: On the modular group ring of a p-group. Proc. Am. Math. Soc. 15, 511 514 (1964) 6. Dade, E.: Endo-permutation modules over p-groups I-II. Ann. Math. 107, 459-494; 108, 317-346 (1978) 7. Dade, E.: A correspondence of characters. Proc. Syrup. Pure Math. 37, 401-403 (1980) 8. Fong, P.: On the characters of p-solvable groups. Trans. Am. Math. Soc. 98, 263 284 (1961) 9. Huppert, B.: Endliche Gruppen I. Berlin Heidelberg New York: Springer 1967 10. Kfilshammer, B.: Crossed products and blocks with normal defect groups. Comm. Algebra 13, 147-168 (1985) 11. Oliver, R.: Central units in p-adic group rings. K-Theory 1, 507 513 (1987) 12. Puig, L.: Structure locale dans les groupes finis. Bull. Soc. Math. Fr. M~moire 47, 1976 13. Puig, L.: Local block theory in p-solvable groups. Proc. Symp. Pure Math. 37, 385-388 (1980) 14. Puig, L.: Sur un th6or~me de Green. Math. Z. 166, 117-129 (1979)

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71

15. Puig, L.: Pointed groups and construction of characters. Math. Z. 176, 265 292 (1981) 16. Puig, L.: Local extensions in endo-permutation modules split: a proof of Dade's theorem. In: S6minaire sur tes groupes finis III. Paris: Pub. Math. Paris VII 25. 1986 17. Puig, L.: Local fusion in block source algebras. J. Algebra 104, 358 369 (1986) 18. Puig, L.: Nilpotent blocks and their source algebras. Invent. Math. 93, 77 -116 (1988) 19. Puig, L.: Pointed groups and construction of modules. J. Algebra 116, 7 129 (1988) 20. Puig, L.: Affirmative answer to a question of Feit. J. Algebra (1990), to appear 21. Reynolds, W.: Blocks and normal subgroups of finite groups. Nagoya Math. J. 22, 15- 32 (1963) 22. Serre, J-P.: Corps locaux. Paris: Hermann 1968