Math. Z. (2016) 282:973–991 DOI 10.1007/s00209-015-1573-x

Mathematische Zeitschrift

A note on Bridgeland’s Hall algebra of two-periodic complexes Shintarou Yanagida1

Received: 21 August 2013 / Accepted: 12 October 2015 / Published online: 4 November 2015 © Springer-Verlag Berlin Heidelberg 2015

Abstract We show that the Hall algebra of two-periodic complexes, which is recently introduced by T. Bridgeland, coincides with the Drinfeld double of the ordinary Hall bialgebra. Mathematics Subject Classification

16T10 · 17B37

0 Introduction 0.1 Abstract and main result The main object of this paper is the Hall algebra of Z2 (:= Z/2Z)-graded complexes, which was introduced by Bridgeland [1]. Let A be an abelian category over a finite field k := Fq with finite dimensional morphism spaces. Let P ⊂ A be the subcategory of projective objects. Let C (A) ≡ CZ2 (A) be the abelian category of Z2 -graded complexes in A. An object of C (A) is of the form M1 o

f g

/M , 2

f ◦ g = 0, g ◦ f = 0.

Let C (P ) be the subcategory of complexes consisting of projectives, and H(C (P )) be its Hall algebra. One can introduce the twisted Hall algebra Htw (C (P )) as the twisting of H(C (P )) by the Euler form of A. In [1], Bridgeland introduced an algebra DH(A), which is the localization of the twisted Hall algebra Htw (C (P )) by the set of acyclic complexes:   DH(A) := Htw (C (P )) [M• ]−1 | H∗ (M• ) = 0 . The purpose of this note is to show the following theorem, which was stated in [1, Theorem 1.2].

B 1

Shintarou Yanagida [email protected] Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

123

974

S. Yanagida

Theorem Assume that the abelian category A satisfies the conditions • • • • •

essentially small with finite morphism spaces, linear over k, of finite global dimension and having enough projectives, hereditary, nonzero object defines nonzero class in the Grothendieck group.

(A) as an Then the algebra DH(A) is isomorphic to the Drinfeld double of the bialgebra H associative algebra. (A) is the (ordinary) extended Hall bialgebra, which will be recalled in Sects. 1.1.1– Here H 1.1.4. For the review of the Drinfeld double, see Sect. 1.1.5. The proof will be explained in Sect. 2. The organization of this note is as follows. In Sect. 1, we review Bridgeland’s theory [1] and prepare notations and statements which are necessary for the proof of the main theorem. In the Sect. 1.1, we recall the ordinary theory of Hall algebra introduced by Ringel [12]. The next Sect. 1.2 is devoted to the recollection of Bridgeland’s theory. The Sect. 2 is devoted to the proof of the main theorem. We close this note by mentioning some consequences of the theorem in Sect. 3.

0.2 Notations and conventions We indicate several global notations. k := Fq is a fixed finite field unless otherwise stated, and all the categories will be k-linear. √ We choose and fix a square root t := q. For an abelian category A, we denote by Ob(A) the class of objects of A. For an object M  Let K ≥0 (A) ⊂ K (A) of A, the class of M in the Grothendieck group K (A) is denoted by M.  be the subset of K (A) consisting of the classes A ∈ K (A) of A ∈ A (rather than the formal differences of them). For an abelian category A which is essentially small, the set of its isomorphism classes is denoted by Iso(A). di

For a complex M• = (· · · → Mi − → Mi+1 → · · · ) in an abelian category A, its homology is denoted by H∗ (M• ). For a set S, we denote by |S| its cardinality. Finally, assuming the set AutA (M) of automorphisms is finite for an object M of a category A, we set a M := |Aut A (M)|.

1 Hall algebras of complexes 1.1 Hall algebra This subsection gives basic definitions and properties of Hall algebra, following [1, §§2.3– 2.5] and [13, §1]. Let A be an abelian category satisfying the assumptions (a) essentially small with finite morphism spaces, (b) linear over k, (c ) of finite global dimension. Remark 1.1 We will introduce additional conditions (c), (d) and (e) in the following discussion. These symbols except (c ) follow those in [1].

123

A note on Bridgeland’s Hall algebra of two-periodic complexes

975

1.1.1 Definitions Consider a vector space



H(A) :=

C[A]

A∈Iso(A)

linearly spanned by symbols [A] with A running through the set Iso(A) of isomorphism classes of objects in A. Definition/Fact 1.2 (Ringel [12]) The following operation defines on H(A) the structure of a unital associative algebra over C:    Ext1 (A, B)C  A [A]  [B] := · [C]. (1.1) | HomA (A, B)| C∈Iso(A)

Here Ext1A (A, B)C ⊂ Ext 1A (A, B) is the set parametrizing extensions of B by A with the middle term isomorphic to C. The unit is given by [0], where 0 is the zero object of A. This algebra (H(A), , [0]) is called the Hall algebra of A. Below we will denote it by H(A) for simplicity. Remark 1.3 Let us recall the remark on the choice of the structure constant given in [1, §2.3]. We follow the paper [1] to choose | Ext1A (A, B)C |/| HomA (A, B)| for the structure constant of the multiplication. It is proportional to the usual structure constant g CA,B := |{B ⊂ C | B ∼ = B, C/B ∼ = A}| appearing in [12] and [13], namely g CA,B =

| Ext1A (A, B)C | aC . | HomA (A, B)| a A a B

Here we used the notation a A := | Aut A (A)|. With respect to the modified generators [[A]] := the product  can be written as [[A]]  [[B]] =

[A] , aA



g CA,B [[C]],

C

which is the usual definition of the Ringel–Hall algebra.

1.1.2 Euler form and extended Hall algebra Let us recall the notations for Grothendieck group given in Sect. 0.2. For objects A, B ∈ A, the Euler form is defined by  (−1)i dimk ExtiA (A, B), (1.2)

A, B := i∈Z

123

976

S. Yanagida

where the sum is finite by our assumptions on A. As is well known, this form descends to the one on the Grothendieck group K (A) of A, which is denoted by the same symbol as (1.2):

·, · : K (A) × K (A) −→ Z. We will also use the symmetrized Euler form: (·, ·) : K (A) × K (A) −→ Z, (α, β) := α, β + β, α. Definition/Fact 1.4 (1) The twisted Hall algebra Htw (A) is the same vector space as H(A) with the twisted multiplication 

[A] ∗ [B] := t A, B · [A]  [B] √ for A, B ∈ Iso(A). Here t := q is the fixed square root of q. (A) is defined as an extension of Htw (A) by adjoining (2) The extended Hall algebra H symbols K α for classes α ∈ K (A), and imposing relations K α ∗ K β = K α+β ,



K α ∗ [B] = t (α, B) · [B] ∗ K α

(A) has a vector space basis consisting for α, β ∈ K (A) and B ∈ Iso(A). Note that H of the elements K α ∗ [B] for α ∈ K (A) and B ∈ Iso(A). e (A). Remark 1.5 In [1], the extended Hall algebra is denoted by Htw

1.1.3 Green’s coproduct To introduce a coalgebra structure, one should consider a completion of the algebra. Assume that the abelian category A satisfies the conditions (a), (b), (c ) and (e) nonzero object defines nonzero class in the Grothendieck group. The algebra H(A) is naturally graded by the Grothendieck group K (A) of A:   H(A) = H(A)[α], H(A)[α] := C[A].  A=α

α∈K (A)

For α, β ∈ K (A), set  C H(A)[β] := H(A)[α]⊗



C[A] ⊗C C[B],

  A=α, B=β

C H(A) := H(A)⊗



 C H(A)[β]. H(A)[α]⊗

α,β∈K (A)

 H(A) is the space of all formal linear combinations Thus H(A)⊗  c A,B · [A] ⊗ [B]. A,B

 is called a completed tensor product. This tensor product ⊗ Definition/Fact 1.6 (1) (Green [8]) The following maps C H(A),  : H(A) −→ C  : H(A) −→ H(A)⊗

123

A note on Bridgeland’s Hall algebra of two-periodic complexes

977

define a topological coassociative coalgebra structure on H(A):  A ([A]) := t B,C g B,C · [B] ⊗ [C], ([A]) := δ A,0 .

(1.3)

B,C A is introduced in Remark 1.3. Here the structure constant g B,C (2) (Xiao [16]) On the extended algebra, defining the maps

(A),  : H (A)⊗ (A) −→ C (A) −→ H C H :H by ([A]K α ) :=



A t B,C g B,C · ([B]K C+α  ) ⊗ ([C]K α ), ([A]K α ) := δ A,0 ,

B,C

(A). one has a topological coassociative coalgebra structure on H Here the word topological means that everything should be considered in the completed space. For example, the coassociativity in (1) means that the two maps ( ⊗ 1) ⊗  and  H(A)⊗ H(A) coincide. (1 ⊗ ) ⊗  from H(A) to H(A)⊗ Note also that the coproduct on H(A) can be rewritten as ([[A]]) =



t B,C

A a B aC g B,C

B,C

aA

· [[B]] ⊗ [[C]]

in terms of the modified generators [[A]] := [A]/a A explained in Remark 1.3. This formula is the one usually appearing in the literature.

1.1.4 Bialgebra structure and Hopf pairing (A)). In Now we have an algebra structure and a coalgebra structure on H(A) (and on H order that these structures are compatible and give a bialgebra structure, we must impose one more condition on A. Fact 1.7 (Green [8], Xiao [16]) Assume that the abelian category A satisfies the conditions (a), (b), (c ), (e) and (d) hereditary, that is, of global dimension at most 1. Then the tuples (A), ∗, [0], , ) (H(A), , [0], , ), (H  H(A) and are topological bialgebras defined over C. That is,the map  : H(A) → H(A)⊗ (A) → H (A)⊗ (A) are homomorphisms of C-algebras. H :H (A) the bialgebras (H(A), , [0], , ) and Below, we simply denote by H(A) and H  (H(A), ∗, [0], , ) respectively. This bialgebra structure on H(A) has an additional feature, that is, it is self-dual. The self-duality is stated in terms of a natural nondegenerate bilinear form, called Hopf pairing. Recall our notation a A := | Aut A (A)|. Definition/Fact 1.8 (Green [8]) Assume that the abelian category A satisfies the conditions (a), (b), (c ), (d) and (e).

123

978

S. Yanagida

(1) The non-degenerate bilinear form (·, ·) H : H(A) ⊗C H(A) −→ C given by ([A], [B]) H := δ A,B a A is a Hopf pairing on the bialgebra H(A), that is, for any x, y, z ∈ Iso(A), one has (x  y, z) H = (x ⊗ y, z) H .

(1.4)

(2) The non-degenerate bilinear form (A) −→ C (A) ⊗C H (·, ·) H : H given by ([A]K α , [B]K β ) H := δ A,B a A t (α,β)

(1.5)

(A), that is, for any x, y, z ∈ Iso(A), one has is a Hopf pairing on the bialgebra H (x ∗ y, z) H = (x ⊗ y, z) H .

(1.6)

Remark 1.9 (1) In terms of the modified generator [[A]] := [A]/a A explained in Remark 1.3, the Hopf pairing is given by ([[A]], [[B]]) H =

δ A,B , aA

which is the formula usually used. (2) In the right hand sides of (1.4) and (1.6), we used the usual pairing on the product space: (x ⊗ y, z ⊗ w) H := (x, z) H · (y, w) H

1.1.5 Drinfeld double Here we recall the Drinfeld double of the self-dual bialgebra. For the complete treatment of Drinfeld double construction, we refer [10, §3.2] and [13, §5.2]. Fact 1.10 (Drinfeld) Let H be a C-bialgebra with a Hopf pairing (·, ·) H : H ⊗C H → C. Then there is a unique algebra structure ◦ on H ⊗C H satisfying the following conditions (1) The maps H −→ H ⊗C H, a  −→ a ⊗ 1

and H −→ H ⊗C H, a  −→ 1 ⊗ a

are injective homomorphisms of C-algebras. (2) For all elements a, b ∈ H, one has (a ⊗ 1) ◦ (1 ⊗ b) = a ⊗ b. (3) For all elements a, b ∈ H, one has   (a(1) , b(2) ) H · (1 ⊗ b(2) ) ◦ (a(2) ⊗ 1). (a(2) , b(1) ) H · a(1) ⊗ b(2) =

Here we used Sweedler’s notation: (a) = a(1) ⊗ a(2) .

123

(1.7)

A note on Bridgeland’s Hall algebra of two-periodic complexes

979

Remark 1.11 If H is a topological bialgebra, then one should replace the tensor product ⊗ . in the statement by the completed one ⊗

1.2 Hall algebras of complexes We summarize necessary definitions and properties of Hall algebras of Z2 -graded complexes. Most of the materials were introduced or shown in [1]. In this Sect. 1.2, A denotes an abelian category satisfying the following three conditions. (a) essentially small with finite morphism spaces, (b) linear over k, (c) of finite global dimension and having enough projectives.

1.2.1 Categories of two-periodic complexes We shall recall the basic definitions in [1, §3.1]. Let CZ2 (A) be the abelian category of Z2 graded complexes in A. An object M• of this category consists of the following diagram in A: M1 o

/ M , d ◦ d = 0. i+1 i 0

d1 d0

Hereafter indices in the diagram of an object in CZ2 (A) are understood by modulo 2. A morphism s• : M• → N• consists of a diagram M1 o s1

 N1 o

d1 d0 d1

d0

/M 0 s0

/ N 0

with si+1 ◦ di = di ◦ si . Two morphisms s• , t• : M• → N• are said to be homotopic if there are morphisms h i : Mi → Ni+1 such that

ti − si = di+1 ◦ h i + h i+1 ◦ di .

For an object M• ∈ CZ2 (A), we define its class in the K -group by 0 − M 1 ∈ K (A). • := M M Denote by HoZ2 (A) the category obtained from CZ2 (A) by identifying homotopic morphisms. Let us also denote by CZ2 (P ) ⊂ CZ2 (A),

the full subcategories whose objects are complexes of projectives in A. Hereafter we drop the subscript Z2 and just write C (A) := CZ2 (A), C (P ) := CZ2 (P ), Ho(A) := HoZ2 (A).

The shift functor [1] of complexes induces an involution ∗

C (A) ←→ C (A).

(1.8)

123

980

S. Yanagida

This involution shifts the grading and changes the sign of the differential as follows: M• =



M1 o

d1 d0

/M 0



∗

←→ M•∗ =



M0 o

−d0 −d1

/M 1



Now let us recall Fact 1.12 ([1, Lemma 3.3]) For M• , N• ∈ C (P ) we have Ext1C (A) (N• , M• ) ∼ = HomHo(A) (N• , M•∗ ). A complex M• ∈ C (A) is called acyclic if H∗ (M• ) = 0. To each object P ∈ P , we can attach acyclic complexes KP• =

Po

Id 0

/

 P ,

K P •∗ =

Po

0 −Id

/

 P .

Remark 1.13 The complexes K P • , K P ∗• are denoted by K P , K P∗ in [1]. Let us recall the following fact shown in [1]. Fact 1.14 ([1, Lemma 3.2]) For each acyclic complex of projectives M• ∈ C (P ), there are objects P, Q ∈ P , unique up to isomorphism, such that M• ∼ = K P • ⊕ K Q •∗ .

1.2.2 Definition of Hall algebras of complexes Let H(C (A)) be the Hall algebra of the abelian category C (A) defined in Sect. 1.1. As noted in [1, §3.5], this definition makes sense since the spaces Ext1C (A) (N• , M• ) are all finitedimensional by Fact 1.12. Let H(C (P )) ⊂ H(C (A))

be the subspace spanned by complexes of projective objects. Define Htw (C (P )) to be the same vector space as H(C (P )) with the twisted multiplication [M• ] ∗ [N• ] := t M0 ,N0 + M1 ,N1  · [M• ]  [N• ].

(1.9)

Now let us recall the simple relations satisfied by the acyclic complexes K P • : Fact 1.15 ([1, Lemmas 3.4, 3.5]) For any object P ∈ P and any complex M• ∈ C (P ), we have the following relations in Htw (C (P )):  

 

 

 

[K P • ] ∗ [M• ] = t P, M•  · [K P • ⊕ M• ], [M• ] ∗ [K P • ] = t − M• , P · [K P • ⊕ M• ], [K P • ] ∗ [M• ] = t ( P, M• ) · [M• ] ∗ [K P • ], [K P •∗ ] ∗ [M• ] = t −( P, M• ) · [M• ] ∗ [K P •∗ ] In particular, for P, Q ∈ P we have ∗ ], [K P • ] ∗ [K Q • ] = [K P • ⊕ K Q • ], [K P • ] ∗ [K Q ∗• ] = [K P • ⊕ K Q •

[[K P • ], [K Q • ]] = [[K P • ], [K Q •∗ ]] = [[K P ∗• ], [K Q •∗ ]] = 0. In the last line we used the commutator [x, y] := x ∗ y − y ∗ x.

123

(1.10)

A note on Bridgeland’s Hall algebra of two-periodic complexes

981

1.2.3 Bridgeland’s Hall algebra Now we can introduce the main object: Bridgeland’s Hall algebra. We define the localized Hall algebra DH(A) to be the localization of Htw (C (P )) with respect to the elements [M• ] corresponding to acyclic complexes M• :   DH(A) := Htw (C (P )) [M• ]−1 | H∗ (M• ) = 0 . As explained in [1, §3.6], this is the same as localizing by the elements [K P • ] and [K P∗ • ] for all objects P ∈ P . For an element α ∈ K (A), we define K α := [K P • ] ∗ [K Q • ]−1 ,

K α∗ := [K P •∗ ] ∗ [K Q ∗• ]−1 ,

− Q  using the classes of some projectives P, Q ∈ P . This is where we expressed α = P well defined by Fact 1.15 (A) and K α ∈ DH(A) by the Remark 1.16 We will denote two different elements K α ∈ H same symbol, following [1]. By Fact 1.15, we immediately have Corollary 1.17 In the algebra DH(A), we have (1) 



K α ∗ [M• ] = t (α, M• ) · [M• ] ∗ K α , K α∗ ∗ [M• ] = t −(α, M• ) · [M• ] ∗ K α∗ , for arbitrary α ∈ K (A) and M• ∈ C (P ). (2) [K α , K β ] = [K α , K β∗ ] = [K α∗ , K β∗ ] = 0 for arbitrary α, β ∈ K (A).

1.3 Hereditary case In this Sect. 1.3, we assume that A satisfies the conditions (a), (b), (c) and the following additional ones: (d) A is hereditary, that is of global dimension at most 1, (e) nonzero objects in A define nonzero classes in K (A). Then by [1, §4] we have a nice basis for DH(A). To explain that, let us recall the minimal resolution of objects of A.

1.3.1 Minimal resolution and the complex C A • Definition 1.18 [1, §4.1] Assume the conditions (a),(c),(d) on A. Then every object A ∈ A has a projective resolution f

g

0→P− →Q− → A → 0,

(1.11)

and decomposing P and Q into finite direct sums P = ⊕i Pi , Q = ⊕ j Q j , one may write f = ( f i j ) in matrix form with f i j : Pi → Q j . The resolution (1.11) is said to be minimal if none of the morphisms f i j is an isomorphism.

123

982

S. Yanagida

Fact 1.19 ([1, Lemma 4.1]) Any resolution (1.11) is isomorphic to a resolution of the form 1⊕ f

(0,g )

0 → R ⊕ P −−−→ R ⊕ Q −−−→ A → 0 with some object R ∈ P and some minimal projective resolution f

g

→ Q − → A → 0. 0 → P − Definition 1.20 ([1, §4.2]) Given an object A ∈ A, take a minimal projective resolution fA

g

→ A → 0, 0 → PA −→ Q A −

(1.12)

We define a Z2 -graded complex C A • :=



PA o

fA 0

/Q A



∈ C (P ).

Remark 1.21 The complex C A • is denoted as C A in [1]. By Fact 1.19, arbitrary two minimal projective resolutions of A are isomorphic, so the complex C A • is well-defined up to isomorphism. Fact 1.22 ([1, Lemma 4.2]) Every object M• ∈ C(P ) has a direct sum decomposition M• = C A • ⊕ C B •∗ ⊕ K P • ⊕ K Q ∗• . Moreover, the objects A, B ∈ Ob(A) and P, Q ∈ Ob(P ) are unique up to isomorphism.

1.3.2 Triangular decomposition Definition 1.23 ([1, §§4.3–4.4]) Given an object A ∈ A, we define elements E A • , FA • ∈ DH(A) by 

E A := t P, A · K − P ∗ [C A • ],

FA := E ∗A .

Here we used a projective decomposition (1.12) of A and the associated complex C A • in Definition 1.20. Remark 1.24 Even if we replace the complex C A • by another (not necessarily minimal) projective resolution, which can be presented as C A • ⊕ K R with some R ∈ Ob(P ), we get the same element E A . Fact 1.25 ([1, Lemmas 4.6, 4.7]) (1) There is an embedding of algebras (A) −→ DH(A) I+e : H defined by [A]  −→ E A (A ∈ Iso(A)),

K α  −→ K α (α ∈ K (A)).

By composing I+e and the involution ∗, we also have an embedding (A) −→ DH(A) I−e : H defined by [A]  −→ FA (A ∈ Iso(A)),

123

K α  −→ K α∗ (α ∈ K (A)).

A note on Bridgeland’s Hall algebra of two-periodic complexes

983

(2) The multiplication map m : a ⊗ b  −→ I+e (a) ∗ I−e (b) defines an isomorphism of vector spaces ∼ (A) −− (A) ⊗C H → DH(A). m:H

As a corollary, we have Corollary 1.26 DH(A) has a basis consisting of elements E A ∗ K α ∗ K β∗ ∗ FB , A, B ∈ Iso(A), α, β ∈ K (A).

1.4 Main theorem Now we can state our main theorem. Theorem 1.27 Assume that the abelian category A satisfies the conditions (a)–(e). Then the (A) such that the algebra DH(A) is isomorphic to the Drinfeld double of the bialgebra H two natural inclusions H → H ⊗C H is identified with the two embeddings I±e .

2 Proof of the main theorem Because of the description of the basis of DH(A) (Corollary 1.26) and the definition of Drinfeld double (Fact 1.10), the proof of Theorem 1.27 is reduced to check the Eq. (1.7) for (A). the elements consisting of the basis of H Let us write the Eq. (1.7) in the present situation:  ?  (a(2) , b(1) ) H · I+e (a(1) ) ∗ I−e (b(2) ) = (a(1) , b(2) ) H · I−e (b(1) ) ∗ I+e (a(2) ).

(2.1)

(A), it is enough to check this Since {[A]K α | A ∈ Iso(A), α ∈ K (A)} is a linear basis of H equality for a = [A]K α and b = [B]K β with α, β ∈ K (A) and A, B ∈ Iso(A). Let us write  ([A]K α ) = t A1 ,A2  g AA1 ,A2 · ([A1 ]K A2 +α ) ⊗ ([A2 ]K α ), A1 ,A2

([B]K β ) =



B1 ,B2

t B2 ,B1  g BB2 ,B1 · ([B2 ]K  B1 +β ) ⊗ ([B1 ]K β ).

Then the left hand side of (2.1) becomes 

 t A1 ,A2  g AA1 ,A2 t B2 ,B1  g BB2 ,B1 [A2 ]K α , [B2 ]K  LHS of (2.1) = B1 +β H A1 ,A2 ,B1 ,B2

· E A1 ∗ K A2 +α ∗ FB1 ∗ K β∗  = t (α,β) t A1 ,A2 + B2 ,B1  g AA1 ,A2 g BB2 ,B1 ([A2 ], [B2 ]) H A1 ,A2 ,B1 ,B2

· E A1 ∗ K A2 ∗ FB1 ∗ K α ∗ K β∗ .

123

984

S. Yanagida

In the second line we used the Hopf pairing (1.5). Similarly the right hand side becomes  RHS of (2.1) = t (α,β) t A1 ,A2 + B2 ,B1  g AA1 ,A2 g BB2 ,B1 ([A1 ], [B1 ]) H A1 ,A2 ,B1 ,B2

∗ · FB2 ∗ K  ∗ E A2 ∗ K α ∗ K β∗ . B 1

∗ Then by removing the term α ∗ K β from both sides, the Eq. (2.1) reduces to  t A1 ,A2 + B2 ,B1  g AA1 ,A2 g BB2 ,B1 ([A2 ], [B2 ]) H · E A1 ∗ K A2 ∗ FB1

t (α,β) K

A1 ,A2 ,B1 ,B2



?

=

A1 ,A2 ,B1 ,B2

∗ t A1 ,A2 + B2 ,B1  g AA1 ,A2 g BB2 ,B1 ([A1 ], [B1 ]) H · FB2 ∗ K  ∗ E A2 . B 1

(2.2)

By the Hopf pairing (1.5), the left hand side of (2.2) becomes  LHS of (2.2) = t A1 ,A2 + B2 ,B1  g AA1 ,A2 g BB2 ,B1 δ A2 ,B2 a A2 · E A1 ∗ K A2 ∗ FB1 A1 ,A2 ,B1 ,B2



=

A1 ,A2 ,B1

t A1 ,A2 + A2 ,B1  g AA1 ,A2 g AB2 ,B1 a A2 · E A1 ∗ K A2 ∗ FB1 .

Now we note the equality g AA1 ,A2 g AB2 B1 a A2 = |{h ∈ HomA (B, A) | Ker h  B1 , Cok h  A1 , Im h  A2 }|, which follows easily from the definitions of g CB,A and a A . Then we have  g AA1 ,A2 g AB2 B1 a A2 = |{h ∈ HomA (B, A) | Ker h  B1 , Cok h  A1 }| A2 ∈Iso(A)

Thus the formula can be rewritten as  LHS of (2.2) = h∈HomA (B,A)

 

 

t A1 , A2 + A2 , B1  · E A1 ∗ K A2 ∗ FB1 ,

2 := A − A 1 . In the same where in the summation we set A1 := Cok h, B1 := Ker h and A way, we have      ∗ RHS of (2.2) = t B1 , A2 + B2 , B1  · FB2 ∗ K  ∗ E A2 , B h∈HomA (A,B)

1

where in the summation we set A2 := Ker h, B2 := Cok h and  B1 :=  B− B2 . These formulas are rewritten in terms of summation over complexes in the following manner. Lemma 2.1 We have LHS of (2.2) =



P• ∈Iso(C (P ))

      t A1 , A2 + A2 , B1  ExtC (A) (C B ∗• , C A • ) P•  · E A1 ∗ K A2 ∗ FB1

2 := A − A 1 . Similarly, with A1 := H0 (P• ), B1 := H1 (P• ) and A        ∗ t B1 , A2 + B2 , B1  ExtC (A) (C A ∗• , C B • ) Q •  · FB2 ∗ K  ∗ E A2 RHS of (2.2) = B Q • ∈Iso(C (P ))

with B2 := H0 (Q • ), A2 := H1 (Q • ) and  B1 :=  B− B2 .

123

1

A note on Bridgeland’s Hall algebra of two-periodic complexes

985

Proof We will prove the first result only. The result follows from the following equality. |{s ∈ HomA (B, A) | Ker s  B1 , Cok s  A1 }|  

   = ExtC (A) C B ∗• , C A • P•  . P• ∈Iso(C (P )), H0 (P• )A1 , H1 (P• )B1

This equality is a refinement of Fact 1.12, so let us first recall its proof in our situation. Consider the set ExtC (A) (C B •∗ , C A • ). It parametrizes a complex P• sitting in the following commutative diagram. PA o

fA

/Q A

0



/ P 0

P1 o



QB o

/ P B

0 − fB

where both columns give short exact sequences in A. Then, since PA1 and Q A1 are projective, we have P1 ∼ = Q B1 ⊕ PA1 and P0 ∼ = PB1 ⊕ Q A1 . Thus the diagram can be rewritten as follows. fA

PA o

/Q A

0



PA ⊕ Q B o



/ Q ⊕ P A B

f1 f0

/ P B

0

QB o

(2.3)

− fB

Let us call this diagram E( f 1 , f 0 ). Now the commutativity of the diagram restricts the morphisms f 1 , f 0 to the following types:     f A s1 0 s0 f1 = , f0 = (2.4) 0 0 0 − fB with s1 ∈ HomA (Q B , Q A ) and s0 ∈ HomA (PB , PA ) satisfying f A s0 = s1 f B . Therefore from each diagram E( f 1 , f 0 ) one has the following exact commutative diagram. 0

/ PB

fB

s0

0

 / PA

/ QB

/B

/0

/A

/0

(2.5)

s1

fA

 / QA

Let us name this diagram as D(s1 , s0 ). Following the argument in the opposite direction, one finds that every diagram D(s1 , s0 ) arises in this way. Next consider another diagram E( f 1 , f 0 ) giving some D(s1 , s0 ). If the extensions P• and P• associated to E( f 1 , f 0 ) and E( f 1 , f 0 ) are isomorphic, then there is an isomorphism

123

986

S. Yanagida

k• : P• → P• in C (A) commuting with the identity on C A • and C B ∗• . This condition is   1 hi

equivalent to the one that there is a homotopy h • between s• and s• with ki = 0 1 (i = 0, 1). After all we obtained a map ExtC (A) (C B ∗• , C A• ) −→ HomHo(A) (C B • , C A • )  HomA (B, A).

(2.6)

Since any s ∈ HomA (B, A) or the associated s• ∈ HomHo(A) (C B • , C A • ) gives a diagram D(s1 , s0 ) coming from some E( f 1 , f 0 ), the homotopy argument above shows that the obtained map is a bijection. Thus in our situation Fact 1.12 is proved. At this moment we also have the one-to-one correspondence [P• ] ←→ s, where [P• ] is the extension class represented by the complex P• . Next we modify the argument further so that the P• should satisfy H0 (P• )  A1 and H1 (P• )  B1 . But this can  be done  argument  in the projective resolution (2.5).  by a standard f A s1 0 s0 Using the notation f 1 = as in (2.4) of the diagram (2.3) and and f 0 = 0 0 0 − fB denoting s ∈ HomA (B, A) the map corresponding to s• = (s1 , s0 ), we have H1 (P• ) = Ker f 1 / Im f 0

 = {(a1 , b0 ) ∈ PA ⊕ Q B | f A (a1 ) + s1 (b0 ) = 0} {(s0 (b1 ), − f B (b1 )) ∈ PA ⊕ Q B | b1 ∈ PB } ∼

−−→ Ker s, where the last map is given by b0  → b0 + PB ∈ Q B /PB  B. Similarly we have H0 (P• )  Cok s. Thus, restricting the bijection (2.6), we have the desired bijection 

∼

ExtC (A) (C B ∗• , C A• ) P• −−→ {s ∈ HomA (B, A) | Ker s  B1 , Cok s  A1 }.

P• H0 (P• )A1 , H1 (P• )B1

  Then we can further rewrite the formula as follows. We denote by C A,B the set of isomorphism classes of objects in C (P ) of the following form. PA ⊕ Q B o

f g

/Q ⊕P , A B

 f =

   fA ∗ 0 ∗ ,g= . 0 0 0 − fB

Lemma 2.2 We have LHS of (2.2) =

 P• ∈C A,B

 

 

t A1 , A2 + A2 , B1 −2 Q B ,PA  · E A1 ∗ K A2 ∗ FB1

2 := A − A 1 . Similarly we have with A1 := H0 (P• ), B1 := H1 (P• ) and A      ∗ t B1 , A2 + B2 , B1 −2 Q A ,PB  · FB2 ∗ K  ∗ E A2 RHS of (2.2) = B Q • ∈C B,A

with A2 := H1 (Q • ), B2 := H0 (Q • ) and  B1 :=  B− B2 .

123

1

(2.7)

A note on Bridgeland’s Hall algebra of two-periodic complexes

987

Proof Let us denote by C(s0 , s1 )• the element of C A,B with s0 ∈ HomA (PB , PA ) and s1 ∈ HomA (Q B , Q A ). By the proof of Lemma 2.1, there is a surjective map C A,B → ExtC (A) (C B ∗• , C A • ). We also see that for two elements C(s0 , s1 )• and C( s0 , s1 )• in C A,B to be mapped to the same extension class there should be a homotopy between (s0 , s1 ) and ( s0 , s1 ). The set of these homotopies is given by HomA (Q B , PA ) ⊕ HomA (PB , Q A ), but the second factor acts trivially on (s0 , s1 ). Thus the set of homotopies is identified with HomA (Q B , PA ). Now rewriting Lemma 2.1 we have  



LHS of (2.2) =

P• ∈C A,B

 

t A1 , A2 + A2 , B1  · E A1 ∗ K A2 ∗ FB1 |HomA (Q B , PA )|

By the hereditary property of A and the projective property of objects, we have | HomA (Q B , PA )| = q Q B ,PA −0 = t 2 Q B ,PA  . Thus we have the desired equality (2.7). The other one can be shown similarly.   However, the set C A,B is still assymetric, so that we introduce a new set D A,B . It consists of objects in C (P ) of the form     f ∗ / Q ⊕ P , f = fA ∗ , g = ∗ PA ⊕ Q B o . A B ∗ ∗ ∗ − fB g We have a projection map p : D A,B → C A,B which sets morphisms to be 0. We will now show that the formula (2.7) can be rewritten as a summation over D A,B . Fix P• ∈ C A,B and present it as      f 0 s0 f A s1 / Q A ⊕ PB , f = P• = PA ⊕ Q B o ,g= . 0 0 0 − fB g Set I := Im f and K := Ker f . Let us also define R1 •∗ :=



K o

0 g| PB

 / P , R := I o 0• B

ι 0

/Q . A

Here ι is the natural inclusion morphism. Since the hereditary assumption ensures that I and K are projective, R1 • and R0 • are objects in C (P ). Moreover, they are (not necessarily minimal) resolutions of H1 (P• ) and H0 (P• ) respectively. Now we have Lemma 2.3 [R0 • ] ∗ [R1 ∗• ] = t − Q A ,PB − I,K 



[M• ].

(2.8)

M• ∈ p −1 (P• )

Proof By the definitions (1.1) and (1.9) of the products  and ∗ in DH(A), we have 

   ExtC (A) R0 • , R1 •∗ M•   

 · [M• ]. [R0 • ] ∗ [R1 ∗• ] = t Q A ,PB + I,K  HomC (A) R0 • , R1 ∗ 

(2.9)

•

M• ∈Iso(C (P ))

As in the previous argument, the complex M• must be of the form K⊕I o

d1

d0

 / P ⊕ Q , d = 0 B A 1 0

  ∗ g| PB , d0 = 0 ι

 ∗ . 0

123

988

S. Yanagida

Since P1 = PA ⊕Q B ∈ Ob(P ), the isomorphism P1 / Ker f ∼ = K ⊕I . = Im f gives PA ⊕Q B ∼ Using the latter isomorphism, we can rewrite the complex M• as     d1 s0 / Q ⊕ P , d = f A s1 , d = ∗ PA ⊕ Q B o , (2.10) A B 1 0 ∗ ∗ ∗ − fB d0 which implies that M• is a member of p −1 (P• ). Let us prove (2.10) by studying the component f A in d1 . The other components can be shown similarly. We want to know the composition of morphisms PA → PA ⊕ Q B ∼ = d1

→ PB ⊕ Q A  Q A . We are in the following situation. K⊕I − 0

/K

/ PA ⊕ Q B O

f

/ I _

/0

ι

j

 QA

? PA

d1

Here the row line is a split exact sequence. Note also that the composition K ⊕ I − → PB ⊕ Q A  Q A is given by (0, ι). Thus the composition considered is given by ι ◦ f ◦ j in the diagram above, which is nothing but f A . On the other hand, every element in p −1 (P• ) occurs as an extension of R1 ∗• by R0 • . Actually, if an element      d1 f A s1 ∗ s0 / o Q A ⊕ PB , d1 = , d0 = M• = PA ⊕ Q B ∗ ∗ ∗ − fB d0 of p −1 (P• ) is given, then we automatically have d1 | K = 0, since K = Ker f =  Ker ( f A , s1 ) : PA ⊕ Q B → Q A implies Im d1 | K ⊂ PB ⊂ Q A ⊕ PB , but then the conditions d0 ◦ d1 = 0, d0 | PB = t (s0 , − f B ) and the injectivity of f B give Im d1 | K = 0. Similarly we have Im d0 ⊂ K . Therefore the running index in (2.9) can be replaced by M• ∈ p −1 (P• ). Next we calculate ∗ ∼ the coefficient of [M• ]. One can   easily find that HomC (A) (R0 • , R1 • ) = HomA (I, K ). Then we have HomC (A) (R0 • , R1 ∗• ) = |HomA (I, K )| = t 2 I,K  . Finally, since the homotopy of M• is given by HomA (Q A , PB ) as in the proof of Lemma 2.2, it affects the coefficient of [M• ] by t −2 Q A ,PB  . Therefore we get (2.8).   Now recalling Remark 1.24, we use the resolutions R1 • and R0 • of H1 (P• )  B1 and H0 (P• )  A1 to express E A1 and FB1 in (2.7). The result is  ∗

PB ,B1  ∗ E A1 = t I,A1  · K − · K−  ∗ R1 • . I ∗ [R0 • ], FB1 = t P B

Using (2.8) and the commutation relation (Corollary 1.17 (1)), we can proceed the calculation (2.7) as   ∗

∗ t c (P• ) · K − LHS of (2.2) = 2 ∗ K − P  ∗ R1 • I ∗ [R0 • ] ∗ K A P• ∈C A,B

=



P• ∈C A,B

=



P• ∈C A,B

123

B

 ∗

    ∗ t c (P• )−( A2 , R0• )−( PB , R0• ) · K − 2 ∗ K − P  ∗ [R0 • ] ∗ R1 • I ∗ KA B

∗ t c(P• ) · K − PA ∗ K −  ∗ P B



[M• ].

M• ∈ p −1 (P• )

A note on Bridgeland’s Hall algebra of two-periodic complexes

989

0• = Q A −  1 . The coefficients c (P• ) and c(P• ) are given In the last equality we used R I =A by 1 , A 2  + A 2 ,  B1  − 2 Q B , PA  + I, A1  + PB , B1 . c (P• ) := A

2 , R 0• ) − ( P B , R 0• ) − Q A , PB  − I, K . c(P• ) := c (P• ) − ( A 1• = K − P B =  0• = Q A −  1 , we can simplify the coefficient c(P• ) Using R B1 and R I =A as A − P B , A 1 −  B1  − 2 Q A , PB  − 2 Q B , PA . c(P• ) = P

(2.11)

This computation can be done as follows. 1 , A 2  + A 2 ,  c(P• ) = A B1  − 2 Q B , PA  + I, A1  + PB , B1  2 , A 1  − A 1 , A 2  − PB , A1  − A1 , PB  − Q A , PB  −  B +  − A I, P B1  2 ,  1  − 2 Q B , PA  + P B ,  1  +  1 − P B −  1 + Q A , P B  = A B1 − A B1 − A I, A B1  − A B , A 1 −  2 , A 1 −  1 + Q A , P B  = − P B1  − 2 Q B , PA  +  I−A B1  −  I+A B , A 1 −  A , A 1 −  A , P B . = − P B1  − 2 Q B , PA  + P B1  − 2 Q

A − A 1 − A 2 = Q A − A = P A and  1 + Q A = 2 Q A . In the last line we used  I − A2 = Q I+A Since p : D A,B → C A,B is surjective, we finally have  ∗ LHS of (2.2) = t c( p(M• )) · K − PA ∗ K −  ∗ [M• ]. P B

M• ∈D A,B

In the same way, we have



RHS of (2.2) =

N• ∈D B,A

∗ t d( p(N• )) · K − A ∗ [N• ],  ∗ K−P P B

where the coefficient d is given by B − P A ,  2  − 2 Q B , PA  − 2 Q A , PB  B2 − A d(Q • ) := P for Q • ∈ C B,A with A2 := H1 (Q • ) and B2 := H0 (Q • ). Now consider the involution (1.8) on C (A) given by the shift [1] of complex. It induces the involution ∗ on the algebra DH(A). We have  ∗ ∗ (LHS of (2.2))∗ = t c( p(M• )) · K − B ∗ [M• ].  ∗ K−P P M• ∈D A,B

A

On the other hand, we can see (D A,B )∗ = D B,A . Thus, if we denote by the operation of switching A and B after ∗, we have  ∗ ∗ (LHS of (2.2)) = tc( p(M• )) · K − A ∗ [M• ].  ∗ K−P P M•∗ ∈D A,B

B

Here  c(P• ) denotes the quantity c(P• ) with A and B replaced. Now the formula (2.11) means that  c(P• ) = c(P•∗ ). Thus the LHS of (2.2) is invariant under . On the other hand, at the level of the desired Eq. (2.2) [or (2.1)], the operation corresponds to switching the indices 1, 2 of tensor products and switching A and B. That is, corresponds to switching LHS and RHS. Thus the invariance of LHS under means that we have the desired formula.

123

990

S. Yanagida

3 Concluding remarks As mentioned in [1, §1.4], the work of Cramer [3] and our Theorem 1.27 yield the following: Corollary 3.1 For the hereditary abelian category, the algebra DH(A) is functorial with respect to derived invariance. Precisely speaking, let A and B be two abelian categories satisfying conditions (a)–(e). Assume that the bounded derived categories D b (A) and D b (B) of A and B are equivalent by the functor : ∼

: D b (A) −−→ D b (B). Then one can construct an algebra isomorphism ∼

DH : DH(A) −−→ DH(B), DH for all equivalences , . and this construction is functorial: ( 1 ◦ 2 )DH = DH 1 2 1 ◦ 2

See also the recent work of Gorsky [7] for the direct approach to show the invariance of our Hall algebras under derived equivalences avoiding the use of Cramer’s result. Now set T := D b (A) for some hereditary abelian category A satisfying (a)–(e). We also set DH(T ) := DH(A).

This algebra depends only on the triangulated category T by the above corollary. Denote by Auteq(T ) the group of autoequivalences of T . Then, setting AutDH (T ) := { DH | ∈ Auteq(T )}, we have an embedding of groups Aut DH (T ) ⊂ Aut(DH(T )). Let us close this note by mentioning a non-trivial example. For an elliptic curve C defined over k, set A := Coh(C),

the abelian category of coherent sheaves on C. This category satisfies the conditions (a)–(e). Set T := D b (A) as above. Then by the theory of Fourier–Mukai transforms, we have a short exact sequence 0

/ Z ⊕ (C × C) 

/ Auteq(T )

/ SL2 (Z)

/0

 corresponds to the subgroup of Auteq(T ) generated by the of groups. Here Z ⊕ (C × C) shifts [n] of complexes, the pushforward ta∗ by translations on C with a ∈ C, and tensor  := Pic0 (C). The cokernel part SL2 (Z) consists of (non-trivial) products L ⊗ (−) with L ∈ C L

 The generators Fourier–Mukai

E := Rp2∗ (E ⊗ p1∗ (−)) with E ∈ D b (C ⊗ C).  transforms  0 −1 1 −1 S := and T := correspond to the Fourier–Mukai transforms E0 and 1 0 0 1  and L ∈ Pic(C) is a degree one line L ⊗ (−), where E0 is the Poincaré bundle on C ⊗ C, bundle. (See [9, §9.5] for the detailed explanation.)

123

A note on Bridgeland’s Hall algebra of two-periodic complexes

991

Now one can see that the operation D H :  −→ DH yields 0

/ Z ⊕ (C × C) 

0

 / Z/2Z

DH

/ Auteq(T ) 

DH

/ AutDH (T )

/ SL2 (Z) 

/0

DH

/ SL2 (Z)

/0

Here Z/2Z corresponds to the involution ∗ of the algebra DH(T ). Thus SL2 (Z) acts on DH(T ). This action is essentially the same as the SL2 (Z)-automorphisms of the algebra appearing in [2]. The same algebra appeared in the works [4–6,11] and [14]. The SL2 (Z)automorphisms (precisely speaking, the counterpart in the degenerate algebra) play an important role in the argument of [15] in the context of the so-called AGT relation/conjecture. Acknowledgments This work is supported by JSPS Fellowships for Young Scientists (Nos. 21-2241, 244759). This work is also partially supported by the JSPS Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation “Deepening and Evolution of Mathematics and Physics, Building of International Network Hub based on OCAMI”. The author would like to thank Professor Tom Bridgeland for the communication and encouragement. He would also thank Doctor Mikhail Gorsky for his valuable comments. Finally the author would like to thank the referee for valuable suggestions.

References 1. Bridgeland, T.: Quantum groups via Hall algebras of complexes. Ann. Math. (2) 177(2), 739–759 (2013) 2. Burban, I., Schiffmann, O.: On the Hall algebra of an elliptic curve, I. Duke Math. J. 161(7), 1171–1231 (2012) 3. Cramer, T.: Double Hall algebras and derived equivalences. Adv. Math. 224(3), 1097–1120 (2010) 4. Feigin, B., Feigin, E., Jimbo, M., Miwa, T., Mukhin, E.: Quantum continuous gl∞ : tensor products of Fock modules and Wn -characters. Kyoto J. Math. 51(2), 365–392 (2011) 5. Feigin, B., Hashizume, K., Hoshino, A., Shiraishi, J., Yanagida, S.: A commutative algebra on degenerate CP1 and Macdonald polynomials. J. Math. Phys. 50(9), 095215 (2009). 42 pp 6. Feigin, B., Tsymbaliuk, A.: Equivariant K -theory of Hilbert schemes via shuffle algebra. Kyoto J. Math. 51(4), 831–854 (2011) 7. Gorsky, M.: Semi-derived Hall algebras and tilting invariance of Bridgeland–Hall algebras. arXiv:1303.5879v2 8. Green, J.: Hall algebras, hereditary algebras and quantum groups. Invent. Math. 120(2), 361–377 (1995) 9. Huybrechts, D.: Fourier-Mukai Transforms in Algebraic Geometry, Oxford Mathematical Monographs. Oxford University Press, Oxford (2006) 10. Joseph, A.: Quantum groups and their primitive ideals. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 29. Springer, Berlin (1995) 11. Miki, K.: A (q, γ ) algebra. J. Math. Phys. 48(12), 123520 (2007). 35 pp 12. Ringel, C.: Hall algebras and quantum groups. Invent. Math. 101(3), 583–591 (1990) 13. Schiffmann, O.: Lectures on Hall algebras. In: Geometric methods in representation theory. II, pp. 1–141, Sémin. Congr., 24-II, Soc. Math. France, Paris (2012) 14. Schiffmann, O., Vasserot, E.: The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of A2 . Duke Math. J. 162(2), 279–366 (2013) 15. Schiffmann, O., Vasserot, E.: Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A2 . Publ. Math. Inst. Hautes Études Sci. 118, 213–342 (2013) 16. Xiao, J.: Drinfeld double and Ringel–Green theory of Hall algebras. J. Algebra 190(1), 100–144 (1997)

123