Acta Mathematica Sinica, English Series Apr., 2015, Vol. 31, No. 4, pp. 675–694 Published online: March 15, 2015 DOI: 10.1007/s10114-015-3623-z Http://www.ActaMath.com

Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2015

Homological Dimensions of the Extension Algebras of Monomial Algebras Hong Bo SHI Department of Mathematics, Nanjing University of Finance and Economics, Nanjing 210046, P. R. China E-mail : dshi [email protected] Abstract The main objective of this paper is to study the dimension trees and further the homological dimensions of the extension algebras — dual and trivially twisted extensions — with a unified combinatorial approach using the two combinatorial algorithms — Topdown and Bottomup. We first present a more complete and clearer picture of a dimension tree, with which we are then able, on the one hand, to sharpen some results obtained before and furthermore reveal a few more hidden subtle homological phenomenons of or connections between the involved algebras; on the other hand, to provide two more efficient combinatorial algorithms for computing dimension trees, and consequently the homological dimensions as an application. We believe that the more refined complete structural information on dimension trees will be useful to study other homological properties of this class of extension algebras. Keywords

Topdown, bottomup, dimension tree, syzygy, finitistic dimensions

MR(2010) Subject Classification

1

16E05, 16Z05, 16E10, 16G10

Introduction and Notation

Assume that Γ is a quiver, k a field, and I is an ideal generated by a set of linear combinations of paths in the path algebra kΓ. A finite dimensional relations algebra is a finite dimensional quotient algebra over a field k of the form Λ = kΓ/I, for brevity, we call Λ a relations algebra, and further a monomial algebra when I is generated by only paths in the path algebra kΓ. The notion of dimension trees of a monomial algebra Λ and algorithms Topdowm and Bottomup to compute them were first developed and studied by the author in [3–5], leading to immediate homological applications — algorithmically determining projective resolutions of Λ-modules, and homological dimensions of Λ, among others. The whole idea behind the scene is to represent an abstract mathematical object, a projective resolution here, as a concrete combinatorial object, a directed rooted tree or a quiver. This idea is further developed and generalized in [6, 7]. The utility of the idea has been attested by many examples, including an important class of extension algebras — dual extensions and trivially twisted extensions, of monomial algebras, whose finitistic dimensions were investigated successfully using algorithms Topdowm and Bottomup. However, the previous investigations on the extension algebras were Received November 5, 2013, accepted March 4, 2014 Supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD)

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carried out separately and each had its own different foci, of which one is in the determination of the finitistic dimensions, and the other is in the construction of trivially twisted extension algebras of large finitistic dimensions. In order to fulfil then the respective goal of research, we imposed certain restrictions on the trivially twisted extensions so that even dual extensions — their special instances, were excluded from the class, and thus the results obtained were only applicable to a subclass of trivially twisted extensions. This paper takes care of this issue by first dropping the restrictions and then developing a general framework of the dimension trees of a trivially twisted extension algebra using Topdown and Bottomup and then specializing it to the two special cases studied before in [8, 9], offering a unified combinatorial approach. This framework reveals more complete refined structural information on dimension trees that will enable us to improve some results obtained before, and what is more, to discover a few more hidden subtle homological phenomenons of or connections between the involved algebras, and to design two more efficient algorithms for computing dimension trees of any trivially twisted extensions and thus the homological dimensions as applications. Throughout this paper, unless stated otherwise, k is a field, Γ a finite quiver with an arrow set Γ1 and a vertex set Γ0 = {ν1 , ν2 , . . . , νn }, and Λ is a monomial algebra. More precisely, Λ is a finite dimensional algebra over the field k of the form Λ = kΓ/I, where I is an ideal of kΓ generated by a set of paths with length at least 2 in the path algebra kΓ. Given a path p in Γ, we denote by S(p) and E(p) its starting point and end point, respectively. Given paths p, q ∈ Γ with E(p) = S(q), their concatenation is written as qp. A path is called a cycle if it starts and ends at a same vertex. If the canonical image in Λ of a path p of length l is nonzero, we identify this image with p and call it a nonzero path of length l in Λ and we still denote the length by l(p). In particular, we identify the (trivial) paths e1 , . . . , en of length zero corresponding to the n vertices of Γ with the primitive idempotents of Λ. We always denote the sets of all paths in Λ, those of length l and those of length greater than or equal to l, by PΛ or P, PΛ l or Pl and Λ P≥l or P≥l , respectively. We denote by S(Γ) and E(Γ) the set of all the vertices each being the starting point of some arrow of Γ and the set of all the vertices each being the end point of some arrow of Γ, respectively. Note that S(Γ) and E(Γ) do not form a partition of Γ0 in general. Finally, we write gl. dim.(Λ) for the global projective dimension of Λ. Given a left Λmodule M , we write Ωi (M ) for the i-th syzygy of M (i ≥ 0) defined via Ωi+1 (M ) = Ω(Ωi (M )), where Ω0 (M ) = M and Ω1 (M ) is the kernel of a projective cover of M , and we also write l. proj. dim.(M ) for the projective dimension of M . The paper is organized as follows. Sections 1 and 2 contain some background information and well-known facts necessary to understand the topics to be investigated. Section 3 is the main part of the paper, which is divided into two subsections: Sections 3.1 and 3.2. Section 3.1 first depicts a complete picture of a dimension tree, which contains all the structural information, and then describes two more efficient combinatorial algorithms to compute dimension trees, designed based on the thorough understanding of a dimension tree and the two algorithms Topdown and Bottomup. Section 3.2 gives some homological applications of Section 3.1, mainly in global and finitistic dimensions, including a couple of more hidden subtle homological connections or phenomenons discovered and illustrated with examples, and sharpening or reobtaining some

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results obtained before. 2

Preliminaries

For the readers’ convenience, in this section, we collect from [3, 4] some definitions, facts and algorithms to be used later. We proceed by first recalling the following concept that is a key ingredient in the next definition of dimension tree. Definition 2.1 ([3, 4]) Suppose Λ = kΓ/I is a monomial algebra. The minimal left path annihilator of the path p is the set, denoted by LΛ (p) or simply L (p) if there would be no danger of confusion, of all paths α in Λ of minimal non-zero length such that S(α) = E(p) but αp = 0; if there is no such a path, define it to be an empty set ∅. If L (p) = ∅, then p is Λ called left regular. The set of all left regular paths in Λ is to be denoted by Lreg or just Lreg by suppressing the superscript. The concepts of minimal right path annihilator RΛ (p), and being right regular, etc., are defined similarly. Here now comes the concept of dimension tree whose importance lies in its homological connections to the algebras to which it associates (see Remark 2.3). Definition 2.2 ([3, 4]) The dimension tree Δ = ΔΛ (p) (or simply Δ(p) if there would be no confusion!) of p in monomial algebra Λ is a quiver that satisfies the following extra properties : (1) There are two labelling maps Δ0 → P0

and

Δ1 → P,

where as usual Δ0 and Δ1 denote the vertex and arrow sets of the quiver Δ. The image of ν ∈ Δ0 under the first map is denoted as ν¯ ∈ P0 and the image of α ∈ Δ1 under the second map as α. ¯ (2) Δ = Δ(p) has a unique root vertex ν0 and unique root arrow a1 : ν0 → ν1 , and any other vertex is the endpoint of a unique directed path in Δ with first arrow a1 . The vertices ν0 and ν1 and the arrow a1 are sent by the labelling maps to S(p), E(p) and p, respectively. ¯ is a directed path in P from ν¯ to ν¯ . (3) If α is an arrow in Δ1 from ν to ν  , then α (4) For any vertex ν other than the root, there is a unique arrow pν coming into ν. Let Tν be the set of all arrows coming out of ν. Then Tν = L (pν ). (5) The map Δ1 → P in (1) is one-to-one when restricted to Tν for any ν ∈ Δ0 . When there is no risk of confusion, we will identify the arrow α in Δ1 with the directed path α ¯ in P. For any ν ∈ Δ0 , its depth d(ν) is defined as the length of the path from the root vertex ν0 to ν, and the depth or height dΛ (p)(or just d(p) if the context is clear !) of the dimension tree Δ(p) as the maximum depth of any vertex in Δ0 . In particular, the root vertex has depth zero. Both d(ν) and d(p) may be infinite. Finally, we will use d(Λ) to represent the maximum depth or height of all the deepest finite dimension trees of Λ and d(Λ) := 0 if Λ has no any finite dimension tree at all, and use Δ(Λ) and F Δ(Λ) to represent the set of all dimension trees of Λ and the set of all finite dimension trees of Λ, respectively. We often abuse notation, and identify the vertices ν in Δ0 with their images ν¯ in P0 , and arrows in Δ1 with their images in P, when the context is clear such that there is no confusion to be caused.

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Remark 2.3 The dimension tree Δ(p) is actually a combinatorial and graphical description of the minimal projective resolution of the cyclic module Λp generated by the path p. The depth (or height) d(p) of the dimension tree Δ(p), defined above, has a relation with the projective dimension of Λ(p) as follows: l. proj. dim.(Λp) = d(p) − 1. In particular, Λp has an infinite projective dimension if and only if Δ(p) is infinite if and only if Δ(p) contains an infinite path if and only if Δ(p) contains a cycle (after identification!). Furthermore, gl. dim.(Λ) = ∞ if and only if there is some path p such that Δ(p) is infinite. As for relationship in terms of finitistic dimensions, there is . Fin. dim.(Λ) = d(Λ) or d(Λ) + 1. This graphical representation, dimension tree, leads immediately to two algorithms to compute it, Topdown and Bottomup in [3], which have found so far a few applications. For the readers’ convenience, we include a brief summary of them as follows. Topdown constructs a dimension tree Δ(p) recursively from the top down, level by level, beginning with the top (first) level P1 := {p}, consisting of solely the given nonzero path p; Assume the i-th level Pi (i ≥ 1) is constructed and Pi = ∅. Now, construct the (i + 1)-th level  Pi+1 := q∈Pi L (q). Keep it in mind that the level i + 1 constructed in the algorithm determines the i-th syzygy i Ω (Λp), vice versa. Note also that Topdown constructs only one dimension tree each time it is called, and that the program halts once an empty level, consequently a finite dimension tree, is reached, otherwise, it would keep going endlessly, and thus hit an infinite dimension tree. In contrast with Topdown, Bottomup constructs once for all the finite dimension trees from the bottom up beginning with those left regular paths — the “bottoms” of dimension trees, and then building other levels of arrows in a recursive fashion! Clearly, Topdown has its advantage when certain or all dimension trees are concerned, examples of such scenarios include the construction of certain or all minimal projective resolutions, computation of global homological dimensions, etc; while Bottomup is more efficient when only all finite dimension trees are a concern, one important instance of such cases is the computation of finitistic dimensions, among others. Next, we construct the following easy example to illustrate the notion of dimension trees and the related concepts as well as the two algorithms. Example 2.4 Suppose the quiver Γ is as shown below. Let Λ = kΓ/I, where the relation ideal I = ε2 , αε, βε, δ 3 , γα . α 1P -2 q 3 P  βP ε

γ4  δ

First, there are four trivial dimension trees (of depth 0!) determined by the four nonzero paths in Λ of length 0. Such cases will be usually ignored. Then, there are total eight nontrivial dimension trees: Δ(α), Δ(β), Δ(γ), Δ(γδ), Δ(γδ 2 ), Δ(δ), Δ(δ 2 ), and Δ(ε) as α, β, γ, γδ, γδ 2 , δ, δ 2 , ε γ α are the only distinct nonzero paths in Λ. By abusing notation, we may write Δ(α) = 1 → 2 → 3, β γ Δ(β) = 1 → 3 and Δ(γ) = 2 → 4, which are the all finite dimension trees with depth 2, 1, and 1, respectively. The remaining five of them are all infinite, as can be verified easily in a similar manner to the following instance.

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We continue with the preceding Example 2.4 to illustrate the above algorithms. Say we want to use Topdown to construct the dimension tree Δ(δ): P1 := {δ}; P2 := L (δ) = {δ 2 }; P3 := L (δ 2 ) = {δ}; in general, P2k−1 = {δ} and P2k = {δ 2 }(k ∈ N). So, we end up with an infinite dimension tree which can be displayed graphically as follows: •

δ - • δ 2 - • δ -• . . .

To which there corresponds the minimal projective resolution of Λδ: · · · → Λe4 → · · · → Λe4 → Λe4 → Λe4 → Λe4 → Λδ → 0. From this, it’s easy to see that the dimension trees Δ(γδ), Δ(γδ 2 ) and Δ(δ 2 ) are all infinite. As for the dimension tree Δ(ε), we have P2 := L (ε) = {ε, α, β}; since L (α) = {γ} and L (β) = ∅, P3 = P2 ∪ {γ}; since L (γ) = ∅, P4 = P3 ; clearly, Pn = P3 , ∀n ≥ 3. So, we also end up with an infinite dimension tree corresponding to the following minimal projective resolution of Λε: · · · → ⊕41 Λei → · · · → ⊕41 Λei → ⊕41 Λei → ⊕31 Λei → Λe1 → Λε → 0. To construct all the finite dimension trees with Bottomup, we first identify all the left regular paths: β and γ, since L (β) = ∅ and L (γ) = ∅. Then start with them from the bottom up to build easily all the nontrivial finite dimension trees: Δ(α), Δ(β) and Δ(γ) as shown in the example. 3

Trivially Twisted Extensions

In this section, on the one hand, we drop all the restrictions imposed on a trivially twisted extension in [9] and then examine the structure of a general dimension tree of it, which allows the results obtained to be able to be specialized to dual extensions and the trivially twisted extensions studied in [8, 9], on the other hand, two combinatorial algorithms for constructing dimension trees and some homological applications will also be presented. We proceed with the introduction of following construction. Suppose A and B are finite dimensional relations algebras defined over a same field k and ¯ = (Γ ¯ 0, Γ ¯ 1) respectively over a quiver Γ = (Γ0 , Γ1 ) with relations {δi | i ∈ I0 }, and a quiver Γ ¯ 0 . The trivially twisted with relations {τj | j ∈ J0 }. Let S = {s1 , . . . , sm } be a subset of Γ0 ∩ Γ extension Λ of A by B at S is a relations algebra over the field k and the quiver Q = (Q0 := ¯ 1 }. ¯ 0 \ S), Q1 := Γ1 ∪˙ Γ ¯ 1 ) with relations {σi | i ∈ I0 } ∪ {τj | j ∈ J0 } ∪ {αβ | α ∈ Γ1 , β ∈ Γ ˙ Γ Γ0 ∪( ¯ 0 and B is the opposite algebra A∗ of Clearly, Λ is just the dual extension of A if S = Γ0 = Γ A, and Λ is monomial if and only if both A and B are monomial. There is an obvious way to embed A, B in Λ, so we often identify them with their images under the embedding, the subalgebras of Λ, without indication. Correspondingly, the identification is also to be used implicitly in the construction of dimension trees of Λ. From now on, we always assume both A and B above are monomial algebras and Λ is ¯ 0 . Unless stated otherwise, we usually the trivially twisted extension of A by B at S ⊆ Γ0 ∩ Γ exclude the case S = ∅ (trivial as for this paper!) by assuming S = ∅. We denote S0 := {s ∈ ¯ 1 , a ∈ Γ1 } ⊆ S. We remind that the notions of S(Γ), E(Γ) ¯ and S | E(b) = S(a) = s, ∃b ∈ Γ Λ A B ¯ Lreg (resp., Lreg , Lreg ) are given in Sections 1 and 2. We point out that E(Γ) here is different

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¯ that are left regular in from what was given in [9], the set of the end points of the arrows in Γ ¯ B, to be denoted from here on by E ∗ (Γ). We then have the following easy fact. Lemma 3.1 Suppose Λ is as assumed in the above. Then for any nonzero path p ∈ Λ, either p ∈ A or p ∈ B or p = pB pA , where paths pB ∈ B and pA ∈ A. Proof From the defining relations of Λ, it is easy to see that pA pB is always zero for any paths pA ∈ A and pB ∈ B, from which the claim follows immediately as any path in Λ is of one of the following forms: pA ∈ PA ; pB ∈ PB ; pA pB or pB pA with pA ∈ PA , pB ∈ PB .  3.1 Structure of a Dimension Tree of Λ By the definition, a dimension tree ΔΛ (p) of Λ is built with “tree blocks”. A “tree block” is a rooted subtree of ΔΛ (p) with some root label, an arrow (i.e., a path of Λ under identification) q ∈ Δ(p) and a level determined by LΛ (q), and is to be denoted BΛ (q) or simply B(q) if the context is clear. So a “tree block” is either of height 2 if LΛ (q) = ∅ or 1 otherwise (see Figure 1). A “tree block” is called a bottom block or simply bottom if it itself forms a dimension tree of height 1, in other words, its root label is left regular. q

? @

?

··· (a)

Figure 1

q

@

@

R @

(b)

A Tree Block of height 2: (a) or 1: (b)

Therefore, we’ll start with the study of the structure of a tree block in order to understand the structure of a dimension tree. We first look at how a bottom block is formed. 3.1.1 Structure of a Tree Block First, we have the following easy fact. Proposition 3.2 Let A, B, Λ and S be as assumed as before and retain all the notation above. B B B Denote L¯reg := {pB | pB ∈ Lreg , E(pB ) ∈ S0 }; and L2 := {pB pA = 0 | pB ∈ L¯reg ; pA ∈ PA ≥1 }. Then B B B (1) Lreg = L¯reg ∪ {pB ∈ Lreg | E(pB ) ∈ S0 }; Λ A B ¯ (2) Lreg = Lreg ∪ L2 ∪ Lreg . Proof (1) It is obvious. A Λ (2) According to the definition of the relation ideal of Λ, it is easy to see Lreg ⊆ Lreg . Still B by the defining relations of Λ, the conditions: pB ∈ Lreg , E(pB ) ∈ S0 , entail any pB pA ∈ L2 Λ B must be left regular in Λ and thus L2 ⊆ Lreg . Similarly, pB ∈ Lreg and E(pB ) ∈ S0 imply B Λ Λ ¯ pB is left regular in Λ, in other words, Lreg ⊆ Lreg . As a conclusion, we have shown Lreg ⊇ A B Lreg ∪ L2 ∪ L¯reg . On the other hand, from Lemma 3.1, any nonzero path p in Λ is either in A, or B or p = pB pA with pA ∈ PA , pB ∈ PB . Checking case by case in this order according to

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A the defining relations of Λ, we see easily that if p is left regular in Λ, then either p ∈ Lreg , or B p ∈ L¯reg or p ∈ L2 , which shows the other inclusion. 

Remark 3.3 We will see soon from Proposition 3.6 in the following that any path in L2 will never appear in any dimension tree of height 2 or above. So, the part L2 is essentially useless in construction of new dimension trees! The following easy observation will be used frequently in the following when we want to specialize some results to the particular cases — dual extensions studied before in [8] and ¯ ⊆ S(Γ) ∩ S in [9], respectively. trivially twisted extensions with the restriction E ∗ (Γ) Lemma 3.4 Retain all the notation and assumptions above. ¯ ⊆ S0 if E ∗ (Γ) ¯ ⊆ S(Γ) ∩ S; and E(Γ) ¯ ⊆ S0 if S = Γ0 = Γ ¯ 0 and B = A∗ (i.e., Λ (1) E ∗ (Γ) is the dual extension of A). (2) L2 = L¯B = ∅ in the both cases. reg

¯ = ∅ in the case of dual extension, ¯ ⊆ S0 and thus E ∗ (Γ) Proof Since it is easy to see that E(Γ) ∗ ¯ it suffices to show the claims in the other case: E (Γ) ⊆ S(Γ) ∩ S. ¯ ⊆ S(Γ) ∩ S implies that there is some arrow a ∈ Γ1 and some arrow b ∈ Γ ¯1 (1) ν ∈ E ∗ (Γ) ¯ ⊆ S0 , as claimed. left regular in B such that E(b) = S(a) = ν, that is, ν ∈ S0 , and so E ∗ (Γ) B ¯ 1 . Since , we may write pB = aB pB with aB a left regular arrow in Γ (2) For any pB ∈ Lreg ∗ ¯ ∗ ¯ E (Γ) ⊆ S(Γ) ∩ S, we must have E (Γ) ⊆ S0 from (1), and thus E(pB ) = E(aB ) ∈ S0 . So B L2 = L¯reg = ∅. This shows the claim. 

Corresponding to the main cases considered in [8] and [9], there is immediately from Lemma 3.4 the following, which is just Proposition 12(1) and Proposition 3.1(1) therein, respectively. ¯ ⊆ S(Γ) ∩ S or if Corollary 3.5 Retain all the notation and assumptions above. If E ∗ (Γ) Λ A ¯ 0 and B = A∗ (i.e., Λ is the dual extension of A), then Lreg S = Γ0 = Γ = Lreg .

Next, we are going to examine the formation of a tree block of height 2. There might have two types of such tree blocks, one type of which themselves would be also dimension trees, while the other type of which wouldn’t be so. We start with the first type of such tree blocks, and come up with the following result. Proposition 3.6 If a tree block BΛ (p) is also a dimension tree ΔΛ (p) of Λ with height 2, A A B then either LΛ (p) ⊆ Lreg if p ∈ A or LΛ (p) ⊆ (Lreg ∩ Γ1 ) ∪ L¯reg if p ∈ B or p = pB pA , where A B pA ∈ P , pB ∈ P . Graphically, the scenarios may be displayed as in Figure 2. p ∈ B or

p∈A

p = pB pA

? @ Figure 2

···

@ @

A Lreg

⊆ (b1)

? @ R @ ⊆ (L A

reg

···

@

@

R @

B ∩ Γ1 ) ∪ L¯reg (b2)

A Tree Block As Dimension Tree of Height 2

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Proof For any nonzero path p ∈ Λ with l(p) ≥ 1, we may write p = aA pA or aB pΛ with ¯ 1 , and pΛ ∈ PΛ . In the case p = aA pA , the defining relations of Λ aA ∈ Γ1 , pA ∈ PA , aB ∈ Γ Λ A lead to the following: LΛ (p) ⊆ Lreg ⇔ LΛ (p) ⊆ Lreg ; In the other cases p ∈ B or p = aB pΛ , Λ any nonzero path q in Λ, q is in LΛ (p), i.e., q ∈ P≥1 ∩ LΛ (p), if q is the shortest path such that qp = 0. By the defining relations of Λ, q is either in B, or in Γ1 (this case E(p) is in S0 ). Since B the tree block BΛ (p) is just the dimension tree ΔΛ (p) of Λ with height 2, q must be in L¯reg or A A B ¯ Lreg . Therefore, we have LΛ (p) ⊆ (Lreg ∩ Γ1 ) ∪ Lreg , as claimed.  From Proposition 3.6, we know that the part L2 does not intersect LΛ (p), i.e., L2 ∩LΛ (p) = ∅, which confirms our observation in Remark 3.3 that any path in L2 cannot be attached to an arbitrary tree block other than the tree block consisting of the path itself. In other words, paths in L2 form a class of isolated dimension trees of Λ of height 1. As before, we now specialize Proposition 3.6 to the two relevant cases mentioned above: ∗ ¯ E (Γ) ⊆ S(Γ) ∩ S and the dual extension. Corollary 3.7 Let A, B, Λ be as assumed as before. Suppose that a tree block BΛ (p) is also a ¯ ⊆ S(Γ) ∩ S or Λ being dimension tree ΔΛ (p) of Λ with height 2. Then, in the case either E ∗ (Γ) A A a dual extension of A, LΛ (p) ⊆ Lreg if p ∈ A or LΛ (p) ⊆ (Lreg ∩ Γ1 ) if p ∈ B or p = pB pA , A B where pA ∈ P≥1 , pB ∈ P≥1 . Proof It is immediate from Lemma 3.4 that L¯B = ∅, which, together with Proposition 3.6, reg



finishes the proof.

Next, we examine how a general tree block BΛ (p) is formed. Note that tree blocks BΛ (p) A A B with LΛ (p) = ∅ and LΛ (p) ⊆ Lreg or LΛ (p) ⊆ (Lreg ∩Γ1 )∪ L¯reg have been already investigated in Propositions 3.2 and 3.6. Proposition 3.8 Retain all the notation and assume p is any path in Λ with l(p) ≥ 1. Then (1) LΛ (p) = LA (pA ) if p = pA ∈ PA ≥1 . A (2) If p = pB or pB pA with pB ∈ PB ≥1 , pA ∈ P≥1 , denote Γ1 (pB ) := {a ∈ Γ1 | S(a) = E(pB )}. Then ⎧ ⎨ L (p ), E(p) ∈ S0 , B B LΛ (p) = ⎩ Γ1 (pB ) ∪ LB (pB ), E(p) ∈ S0 . Graphically, the tree block B(p) may be displayed as in Figure 3. p ∈ B or

pA

p = pB pA

? @

@

(1)

···

@

⊆ LB (pB )

? @

@

@

@ R @ R ⊆ Γ1 (pB ) ∪ LB (pB@ )

(2)(a): E(p) ∈ S0

Figure 3

p = pB pA

? @

@ R @ ⊆ LA (pA ) ···

p ∈ B or

···

(2)(b): E(p) ∈ S0

A General Tree Block B(p)

Proof Note that p = pA ; pB , or pB pA with pA and pB as prescribed as above. (1) ∀q ∈ LΛ (p), then q is a nonzero path in Λ shortest with respect to qp = qpA = 0 in Λ. Then the defining relations of Λ forces that q ∈ A and is the shortest with respect to qpA = 0

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in A. Therefore, q ∈ LA (pA ), and LΛ (p) ⊆ LA (pA ). The other inclusion is clear. (2) In case p = pB or pB pA , ∀q ∈ LΛ (p), then q is a path in Λ shortest with respect to qp = qpB = 0 or qp = qpB pΛ = 0, respectively. In both cases, the defining relations of Λ give rise to q ∈ LB (pB ) if E(pB ) ∈ S0 , or q ∈ Γ1 (pB ) ∪ LB (pB ) otherwise. So, LΛ (p) ⊆ LB (pB ) or Γ1 (pB ) ∪ LB (pB ). The other inclusions are obvious, and thus complete the proof.  In the case of dual extension of A, the statement (1) of Proposition 3.8 remains the same, ¯ ⊆ S0 by Lemma 3.4. Therefore, we have while statement (2) of it reduces to (2)(b) since E(Γ) the following Corollary 3.9

In case that Λ is a dual extension of A. Let B = A∗ . Then

(1) LΛ (p) = LA (pA ) if p = pA ∈ PA ≥1 .

A (2) If p = pB or pB pA with pB ∈ PB ≥1 and pA ∈ P≥1 , then LΛ (p) = Γ1 (pB ) ∪ LB (pB ), where Γ1 (pB ) := {a ∈ Γ1 | S(a) = E(pB )}.

So far we have had a clear picture of the structure of a tree block, which is the building block of a dimension tree. To construct a dimension tree, we need only know how to attach one to another basing on the algorithms: Topdown or Bottomup. 3.1.2 Fitting Tree Blocks Together Because a tree block BΛ (p) is attachable to another tree block BΛ (q) if p ∈ LΛ (q) which is determined by how the top — path p — is formed. So, in order to get such information, we start out by summing up the various structures of a tree block obtained above: tree blocks Λ A B BΛ (p) of height 1 with p ∈ Lreg — p ∈ Lreg , or Lreg (E(p) ∈ S0 ) or p ∈ L2 (Proposition 3.2); and tree blocks BΛ (p) of height 2 with p = pA , pB or pB pA (Propositions 3.6 and 3.8). Since Remark 3.3 claims that any path in L2 won’t show up in any tree blocks other than the one formed by the path itself, we are concerned only with the remaining tree blocks and are to determine when one can be attached to another. For tree blocks BΛ (p) of types either (b1) or (b2) given in Proposition 3.6, no tree blocks of height 2 can be attached to them, since the level LΛ (p) consists of only left regular paths in Λ. Conversely, in order for them to be able to be attached to blocks BΛ (q) of types (1), (2)(a) or (2)(b) given in Proposition 3.8, then p ∈ LΛ (q), that is, p is a path in PA ≥1 and thus B LΛ (p) = LA (p); or p is a path in P≥1 according to the formation of LΛ (q). The last case to be examined is when tree blocks BΛ (p) of types (1), (2)(a) or (2)(b) given in Proposition 3.8 can be attached to each other. Because the level of BΛ (p) consists of paths in either A or B, it is easy to see that only when p is either in A or B, BΛ (p) may be attachable. The above scenarios may be displayed as in Figure 4 and summarized as follows. p∈A

?

J

J

 ··· J

^ ⊆ LA (p) (¯ 1)

p∈B

?

J

J  ··· J

^

⊆ LB (p) (2)(¯ a): E(p) ∈ S0

Figure 4

p∈B

?

J

J  ··· J

^

⊆ Γ1 (p) ∪ LB (p) (2)(¯ b): E(p) ∈ S0

p∈A

?

J

J

 ··· J

^

A ⊆ LA (p) ∩ Lreg (b1)

Attachable Tree Blocks BΛ (p)

p∈B

? S  S / ··· S  w

A B ⊆ (Lreg ∩ Γ1 ) ∪ Lreg (b2)

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Proposition 3.10

Keep all the notation above. Then

B (1) The path p of all the attachable tree blocks BΛ (p) of height 2 must be in either PA ≥1 or P≥1 . There are five types of such blocks : B(p) with p ∈ A or B and of types (b1), (b2), (1), (2)(a), ¯ respectively. and (2)(b) (see Figure 3.8), to be denoted (b1), (b2), (¯1), (2)(¯ a), and (2)(b),

(2) None of the five types of tree blocks can be attached to type (b1) or (b2) tree blocks, ¯ blocks. This may be indicated as however, they are likely attachable to type (¯ 1), (2)(¯ a) or (2)(b) follows : ¯  {(b1), (b2)}  {(¯1), (2)(¯ ¯ {(b1), (b2), (¯ 1), (2)(¯ a), (2)(b)} a), (2)(b)}. ¯ tree blocks (this case, p ∈ Γ1 (3) Type (¯ 1) tree block may be attachable to itself and type (2)(b) and E(p) ∈ S0 ) but never attachable to type (2)(¯ a) tree blocks ; On the other hand, type (2)(¯ a) ¯ or (2)(b) tree blocks may be attached to themselves or each other but may never be attachable to ¯ but {(2)(¯ ¯  {(2)(¯ ¯  {(¯1)}. type (¯ 1) blocks. That is, {(¯ 1)}  {(¯ 1), (2)(b)} a), (2)(b)} a), (2)(b)} (4) Any tree block BΛ (p) with path p ∈ Λ\(A ∪ B) cannot be attached to any tree blocks. Corollary 3.11 Retain all the notation above. In the case of dual extensions, type (2)(¯ a) (see Figure 4) tree block doesn’t exist, and all the other statements in Proposition 3.10 modified accordingly remain valid. ¯ ⊆ S0 , which always leads to E(p) ∈ S0 for any path p ∈ B with Proof From Lemma 3.4, E(Γ) l(p) ≥ 1. Then the claim is from Proposition 3.10.  We are now in a position to state the structure of a dimension tree. Proposition 3.12

Retain all the notation above. Then

(1) Any non-trivial dimension tree of Λ is of the form : ΔΛ (p) with p ∈ PA ≥1 , or p ∈ B or p = pB pA with pA ∈ PA and p ∈ P ; the (i + 1)-th level P of ΔΛ (p) conB i+1 ≥1 ≥1 sists of paths in LA (pi ) ∪ LB (pi ), pi ∈ Pi ; and furthermore, if p ∈ / A, then for any path · · · am am−1 · · · aj+1 aj · · · a2 a1 in ΔΛ (p) starting with the root a1 = p (under the labeling map !), if there is some as ∈ PA ≥1 then there exists a smallest positive integer t ≥ 1 such that at ∈ Γ1 A and aj ∈ P≥1 for j > t and if there is as ∈ PB ≥1 then there exists a largest positive integer t ≥ s such that aj ∈ PB for j ≤ t. ≥1 PB ≥1 ,

(2) For any nonzero path p ∈ A with l(p) ≥ 1, ΔΛ (p) = ΔA (p). In particular, dA (p) = dΛ (p). (3) For any nonzero path pB ∈ B with l(pB ) ≥ 1 and any path pA ∈ PA ≥1 such that pB pA is nonzero in Λ, the dimension trees ΔB (pB ) may be a subtree of ΔΛ (pB ), but ΔΛ (pB ) and ΔΛ (pB pA ) are the same up to their roots pB and pB pA . In particular, dB (pB ) ≤ dΛ (pB ) = dΛ (pB pA ). (4) A deepest finite dimension tree ΔA (a) of A can be built up into a finite dimension tree ΔΛ (p) of Λ with dΛ (p) = dA (a) + 1 if and only if a ∈ Γ1 and there is some nonzero path q ∈ Λ ¯ 1 such that S(a) = E(aB ) ∈ S0 and p = aB q and ΔΛ (p) is finite. and some arrow aB ∈ Γ (5) Each path in L Λ = L A ∪ L2 ∪ L¯B determines a dimension tree of Λ with height 1. reg

reg

reg

In particular, paths in L2 determine a class of isolated dimension trees of Λ in the sense that none of them is a subtree of other dimension trees. Proof

(1) is clear from Lemma 3.1 and the attaching rules specified in Proposition 3.10 (see

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A Figure 4). From Proposition 3.10 (4), nonzero paths of forms pB pA with pB ∈ PA ≥1 and pA ∈ P≥1 can occur only as roots of dimension trees. By saying two dimension trees are the same we mean the two trees agree level by level starting with their root levels. Thus, based upon the algorithms – Topdown or Bottomup, both (2) and (3) are immediate consequence of Proposition 3.10 (2) and (3). (4) is from Proposition 3.10 (3), (4) and the defining relations of Λ based on algorithms Bottomup or Topdown. (5) is from Proposition 3.2 and the discussion immediately preceding to it. 

From the above and Corollary 3.5, we obtain immediately Proposition 12(1), (3)–(5) in [9] and Proposition 3.1(1), (6)–(8) in [8], respectively. We rephrase two of them as follows. Corollary 3.13

¯ ⊆ S(Γ) ∩ S or Λ is a dual extension of A. In case E ∗ (Γ

(1) Every dimension tree of A is also a dimension tree of Λ. (2) Every finite dimension tree of Λ is built atop finite dimension trees of A. Proof (1) is immediate from Proposition 3.12 (1); and (2) is an immediate consequence of Corollary 3.5 and Proposition 3.10 based upon Bottomup.  3.1.3 Algorithmic Thinking An efficient hybrid algorithm for determining all (finite) dimension trees was supplied in [9], which may be improved further based upon a more exhaustive understanding of the structure of a dimension tree revealed in Propositions 3.10 and 3.12. Towards this end, we provide the following algorithms that take more structural information than ever before into consideration so as to enhance the computational efficiency. In this section, we always assume Λ is a trivially twisted extension of A by B at S, where A and B are monomial algebras. Recall F Δ(A) (resp. F Δ(B)) denotes the set of all finite dimension trees of A (resp. B). Algorithm (A) constructing a dimension tree ΔΛ (p) of Λ (1) If p ∈ A, then ΔΛ (p) = ΔA (p). Call Topdown to compute ΔA (p). Return ΔA (p). Else turn to step (2). (2) If p ∈ B, then follow the subsequent steps: i) Call Topdown to compute ΔB (p); ii) Find the set of all the vertices in ΔB (p) that fall into S0 , and denote it by V0 (p) (= (ΔB (p))0 ∩ S0 , where (ΔB (p))0 is the set of all its vertices (using identification under the labeling maps); iii) Find the set of all the arrows in A each of which starts at some ν ∈ V0 (p), and denote it by A0 ; iv) For any arrow α ∈ A0 , call Topdown to compute ΔA (α)(= ΔΛ (α)); v) For any ν ∈ V0 (p), attach ΔA (α) to ΔB (p) at the vertex ν if S(α) = ν. The dimension tree resulting from this process is the desired one, i.e., ΔΛ (p). Return ΔΛ (p). Else move to Step (3). A (3) If p = pB pA , pB ∈ PB ≥1 , pA ∈ P≥1 , then use procedure (2) to compute ΔΛ (pB ) and then concatenate pA to pB in ΔΛ (pB ) to get ΔΛ (p). Return ΔΛ (p).

Algorithm (B) constructing all the finite dimension trees of Λ

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(1) Call Bottomup to compute all the finite dimension trees ΔB (p) of B and their heights dB (p): {(ΔB (p), dB (p)) | ΔB (p) ∈ F Δ(B), p ∈ PB ≥1 }; (2) Call Bottomup to compute all the finite dimension trees ΔA (p) of A and their heights dA (p) and set P1 := {(ΔA (p), dA (p)) | ΔA (p) ∈ F Δ(A), p ∈ PA ≥1 }; (3) For any ΔB (p) ∈ F Δ(B), if V0 (p) = (ΔB (p))0 ∩ S0 = ∅, then ΔB (p) = ΔΛ (p); else for any ΔA (q) ∈ F Δ(A), attach it to ΔB (p) at the vertex S(q) if S(q) ∈ V0 (p), resulting in a finite dimension tree ΔΛ (p) of Λ, with height dΛ (p). Denote all the found tree-height pairs by P2 ; (4)For any ΔΛ (pB ) ∈ F Δ(Λ) formed in Step (3), extend pB to pB pA by concatenation if possible and end with the last class of dimension trees together with their heights. Denote all such pairs of tree and height by P3 ; (5) Return P1 ∪ P2 ∪ P3 , and anything of interest. 3.2 Global and Finitistic Dimensions Having well understood the structure of dimension trees of Λ, we are now able to describe the homological dimensions of Λ in terms of the homological dimensions or the heights d(A) and d(B) of A and B. Recall from Remark 2.3 the height-dimension relationship that the height d(Λ) and the finitistic dimensions of Λ can differ by at most 1, so determination of the height of Λ will be a key issue in order to determine its homological dimensions. In the following, we’ll focus on the relationships of the heights of Λ, A and B, while the corresponding relationships of their homological dimensions are either given naturally as a consequence of the height-dimension relationship, or are simply omitted. B B Denote H(A) := {p ∈ PA ≥1 | dA (p) = d(A)}; H(B) := {y ∈ P≥1 | dB (y) = d(B)} ⊆ P≥1 . We first have the following result, which generalizes Theorem 14 in [9] and further Theorem 3.4 in [8] in the sense that the restrictions there have been discarded here. Proposition 3.14 Suppose Λ is the trivially twisted extension of A by B at S. Then (i) If S0 = ∅, then d(Λ) = max{d(A), d(B)} and d(Λ) = 0 ⇔ d(A) = 0 = d(B). (ii) d(A) = 0 = d(B) ⇒ d(Λ) = 0 ⇒ d(A) = 0, and d(B) = 0 ⇒ d(Λ) = d(A). But not conversely. (iii) d(Λ) ≤ d(A) + d(B). (iv) d(Λ) = d(A) + d(B), d(A) > 0, d(B) > 0 ⇔ there exists a deepest finite dimension tree ΔB (pB ) with pB ∈ PB ≥1 of B such that ΔΛ (pB ) is finite and the bottom level of ΔB (pB ) contains a path q ∈ B with E(q) = S(a) ∈ S0 for some arrow a ∈ H(A). Proof (i), (ii) d(Λ) = 0 ⇔ L Λ = ∅. From Proposition 3.2, L Λ = L A ∪ L2 ∪ L¯B = reg

reg

reg

reg

A B B Lreg ∪ L2 ∪ Lreg since S0 = ∅. Note that L2 = ∅ ⇔ Lreg = ∅ from their definitions. Note also that S0 = ∅ implies any dimension tree of Λ is either a dimension tree of A or a dimension tree of B. Now, the claims come as a consequence of the above. (iii) Any finite dimension tree of Λ can be constructed with Bottomup starting with the Λ tree bottoms given by Lreg in Proposition 3.2 and following the attaching rules described in Proposition 3.10. The resulting dimension tree is formed like capping certain dimension trees of A with a dimension tree of B (up to its root!), from which the claim follows easily. (iv) (⇐) On the one hand, the assumption entails the existence of a finite dimension tree ΔΛ (pB ) of Λ such that dΛ (pB ) = d(A) + d(B). On the other hand, d(Λ) ≤ d(A) + d(B) from

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(iii). So, d(Λ) = d(A) + d(B). Conversely, the assumptions, together with Proposition 3.12(3), there is a deepest finite dimension tree ΔΛ (pB ) of Λ with pB ∈ PB ≥1 such that d(Λ) = dΛ (pB ) = d(A) + d(B). Equivalently, there is a longest path in ΔΛ (pB ) whose length is d(A) + d(B). By Proposition 3.12(1), this path is of the form au · · · a2 a1 bv · · · b2 b1 with b1 = p, u = d(A) and v = d(B), and furthermore, the path bv · · · b2 b1 is in the finite dimension tree ΔB (pB ) of B obtained by truncating all the finite dimension trees of A from ΔΛ (pB ) while the path au · · · a2 a1 is in ΔA (a1 ) with a1 ∈ Γ1 – one of the truncated finite dimension trees of A. Now the proof is complete.  ¯ ⊆ S(Γ) ∩ S We specialize the above to the cases of trivially twisted extensions with E ∗ (Γ) and dual extensions, and have the following. ¯ ⊆ S(Γ) ∩ S or a dual Corollary 3.15 (i) If Λ is a trivially twisted extension with E ∗ (Γ) extension of A, then d(Λ) = 0 ⇔ d(A) = 0; d(B) = 0 ⇒ d(Λ) = d(A). (ii) (c.f. [8, Theorem 3.4]) If Λ is a dual extension of A, then d(Λ) = d(A)+d(A∗ ), d(A) > 0, ∗ ∗ d(A∗ ) > 0 ⇔ there exists a deepest finite dimension tree ΔA∗ (pA∗ ) with pA∗ ∈ PA ≥1 of A ∗ such that ΔΛ (pA∗ ) is finite and the bottom level of ΔA∗ (pA∗ ) contains a path q ∈ A with E(q) = S(a) ∈ S0 for some arrow a ∈ H(A). Λ A Proof (i) Under either of the assumptions, Lreg = Lreg by Corollary 3.5, from which the claim follows. (ii) is just a version of Proposition 3.14 (iv) in the case of dual extension. 

It’s tempting to conjecture that d(A) = d(A∗ ), and that d(Λ) = d(A) implies d(B) = 0, and furthermore that d(A) = d(A∗ ) > 0 implies d(Λ) = d(A) + d(A∗ ), and vice versa. However, the following example shows it is not the case, even in the case of dual extensions. Example 3.16 Suppose Λ is a dual extension of A, where A is a monomial algebra defined over certain quiver with relation ideal as specified as follows, respectively. (1) A is defined over the lower part (labeled with a, b, c, e!) of the following quiver with the relation ideal generated by all paths of length 2, while its opposite algebra A∗ is defined over the upper part of the quiver. Then it is easy to check with either Bottomup or Topdown following the rules described in Proposition 3.10 and Corollary 3.11 that d(A) = d(Λ) = 1 = d(A∗ ) = 2 = 0(note if switching A and A∗ , one then has d(A) = d(Λ) = 2 > d(A∗ ) = 1!). Thus d(A) = d(A∗ ) and d(Λ) = d(A) + d(A∗ ) in general even in such an extremal case — dual extension. ∗ c∗ a∗ -e b∗• • • • a e c b

Figure 5

Example 3.16 (1)

(2) If one modifies the above quiver by truncating the last two arrows labeled by c and c∗ from the original one and retaining all the other assumptions and notation in (1), then one may verify easily that d(A) = d(A∗ ) = d(Λ) = 1, but d(Λ) = d(A) + d(A∗ ). (3) If one redefines the above quiver by truncating the four arrows labeled by b, b∗ , c, c∗ , but retaining all the other assumptions and notation in (1), then it is easy to see d(A) =

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1 = d(A∗ ) = 0 and d(Λ) = d(A) + d(A∗ ), which means d(Λ) = d(A) + d(A∗ ) doesn’t imply d(A) = d(A∗ ) in general. In order to disclose more hidden relationships among A, A∗ and Λ, we need the following lemma, which extends and refines Proposition 3.1(4) in [8]. Lemma 3.17

Let Λ be a dual extension of A. Then

(1) Any sequence of paths in A: pt := p, pt−1 , . . . , p2 , p1 such that pi ∈ LA (pi−1 ), t ≥ i ≥ 2, induces a sequence of paths in A∗ : q1 , q2 , . . . , qt−1 , qt := p∗ such that qi = pi,2 pi+1,1 ∈ LA∗ (qi+1 ), and p∗i = pi,1 pi,2 , l(pi,2 ) ≥ 1, 1 ≤ i < t.

∗ ), qt−1 , . . . , q2 , q1 such that qi ∈ LA∗ (qi−1 ), (2) Any sequence of paths in A∗ : qt (= b∗ qt,1 t ≥ i ≥ 2, induces a sequence of paths in A: p1 , p2 , . . . , pt−1 , pt such that p1 = b ∈ Γ1 , ∗ pk = qt−k+1,2 qt−k+2,1 ∈ LA (pk−1 ) and qt−k+1 = qt−k+1,1 qt−k+1,2 with l(qt−k+1,2 ) ≥ 1, where 1 < k ≤ t and qt,2 = b.

Proof (1) Taking the dual of the sequence, we end with a sequence of paths in A∗ : p∗1 , p∗2 , . . ., p∗t−1 , p∗t = p∗ such that p∗i ∈ RA∗ (p∗i−1 ), 1 < i ≤ t. We’ll construct the desired sequence from the induced one as follows. Set qt = p∗t = p∗ . Since p∗t ∈ RA∗ (p∗t−1 ) implies p∗t−1 p∗t = 0, there are nonzero paths pt−1,1 , pt−1,2 ∈ A∗ such that p∗t−1 = pt−1,1 pt−1,2 , l(pt−1,2 ) ≥ 1 and pt−1,2 ∈ LA∗ (p∗t ). Set qt−1 := pt−1,2 . Similarly, p∗t−1 ∈ RA∗ (p∗t−2 ) implies p∗t−2 p∗t−1 = (p∗t−2 pt−1,1 )pt−1,2 = 0. But p∗t−2 pt−1,1 = 0 unless l(pt−1,2 ) = 0, a contradiction. Consequently, there are nonzero paths pt−2,1 , pt−2,2 ∈ A∗ such that p∗t−2 = pt−2,1 pt−2,2 , l(pt−2,2 ) ≥ 1 and pt−2,2 pt−1,1 ∈ LA∗ (qt−1 ). Set qt−2 := pt−2,2 pt−1,1 . In general, if qt−k = pt−k,2 pt−k+1,1 has been constructed, where l(pt−k,2 ) ≥ 1, pt−k,1 pt−k,2 = p∗t−k and pt−k+1,1 pt−k+1,2 = p∗t−k+1 . Then qt−k−1 (t > k − 1) can be constructed similarly as follows: p∗t−k ∈ RA∗ (p∗t−k−1 ) implies p∗t−k−1 p∗t−k = (p∗t−k−1 pt−k,1 )pt−k,2 = 0. As before, p∗t−k−1 pt−k,1 = 0 unless l(pt−k,2 ) = 0. So, p∗t−k−1 pt−k,1 = 0 and thus there exist nonzero paths pt−k−1,1 , pt−k−1,2 ∈ A∗ such that l(pt−k−1 ) ≥ 1, p∗t−k−1 = pt−k−1,1 pt−k−1,2 and pt−k−1,2 pt−k,1 ∈ LA∗ (qt−k ). Set qt−k−1 = pt−k−1,2 pt−k,1 . This shows the claim. (2) We will provide an argument for this because the procedure in the proof produces a particular sequence that will be used later, although it does go in a similar fashion to the above. ∗ ∗ First, we write qt = qt,2 qt,1 , where qt,2 = b for some arrow b ∈ A and qt,1 ∈ PA . Then, take ∗ ∗ the dual of qt qt−1 · · · q1 and end up with q1∗ q2∗ · · · qt−1 qt∗ with qt∗ = qt,1 qt,2 and qi∗ ∈ RA (qi−1 ).  ∗ ∗ ∗ ∗ ∗ Set p1 := b. From qt−1 qt = (qt−1 qt,1 )b = 0 and qt−1 qt,1 = 0, we see that there would be qt−1 = ∗ ∗ qt−1,1 qt−1,2 such that l(pt−1,2 ) ≥ 1, p2 := qt−1,2 qt,1 ∈ LA (p1 ). Similarly, from qt−2 qt−1 = ∗  ∗ ∗ ∗ 0 and thus (qt−2 qt−1,1 )p2 = (qt−2 qt−1 )qt,1 = 0, and qt−2 qt−1,1 = 0 unless l(qt−1,2 ) = 0, a ∗ contradiction. We see that there would be qt−2 = qt−2,1 qt−2,2 such that l(pt−2,2 ) ≥ 1, p3 :=   qt−2,2 qt−1,1 ∈ LA (p2 ). Inductively, assume pk = qt−k+1,2 qt−k+2,1 (t > k ≥ 3) ∈ LA (pk−1 ) have ∗ ∗ ∗ been constructed such that qt−k+1 = qt−k+1,1 qt−k+1,2 , l(pt−k+1,2 ) ≥ 1. Clearly, qt−k qt−k+1 =0 ∗  ∗ implies (qt−k qt−k+1,1 )pk = 0. On the other hand, l(pt−k+1,2 ) ≥ 1 forces qt−k qt−k+1,1 = 0. So, ∗ there are qt−k = qt−k,1 qt−k,2 and pk+1 := qt−k,2 qt−k+1,1 such that pk+1 ∈ LA (pk ) as desired. 

This lemma tells us more or less some sort of “homological symmetry” between A and A∗ . It, together with the structural information of a dimension tree, forms a powerful tool to study

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the homological properties of the dual extension of A. Next result is a typical example of such applications. Although Example 3.16 shows many seemingly true statements are actually false, we do have the following results that reveal more subtle homological relations among A, A∗ and Λ. Theorem 3.18 Suppose Λ is a dual extension of A. If d(Λ) = d(A) + d(A∗ ), d(A) > 0, d(A∗ ) > 0, then d(A) ≥ d(A∗ ). In particular, d(Λ) ≤ 2d(A). Proof By the assumptions, there exists certain finite deepest dimension tree ΔΛ (p∗ ) of Λ with p ∈ A, which, according to Proposition 3.12 (1) and (4), is built atop some finite deepest dimension trees ΔA (ai ) (ai ∈ Γ1 , i ∈ I) of A and thus contains a longest path ps ps−1 · · · p1 qt qt−1 · · · q1 , where s = d(A), t = d(A∗ ); pi ∈ LA (pi−1 ) (s ≥ i > 1), p1 = aj , j ∈ I; E(qt ) = S(p1 ) ∈ S0 ; qj ∈ LA∗ (qj−1 ) (t ≥ j > 1), q1 = p∗ and qt = b∗ qt,2 for some paths b ∈ Γ1 and qt,2 ∈ A∗ (after identification under the labeling maps in the definition of a dimension tree!). First, the claim is true if s = t = 1. Next, we suppose s ≥ 2 and t ≥ 2. Note that the dimension tree ΔA (b) is a subtree of ΔΛ (p∗ ) since b ∈ LΛ (qt ) and thus is finite. Now, applying Lemma 3.17(2) to the sequence of paths in A∗ : qt qt−1 · · · q1 gives rise to a sequence of paths in A: p1 , p2 , · · · , pt−1 , pt such that p1 = b ∈ Γ1 and pi ∈ LA (pi−1 ), where 1 < i ≤ t. Clearly, this sequence is also in ΔA (b). Therefore, t = d(A∗ ) ≤ d(b) ≤ s = d(A), and particularly, d(Λ) = d(A) + d(A∗ ) ≤ 2d(A), as claimed.  Corollary 3.19 Suppose Λ is a dual extension of A. If d(Λ) = d(A) + d(A∗ ), d(A) > 0, d(A∗ ) > 0, then . Fin. dim.(Λ) ≤ 2d(A) + 1. Proof

This is due to Theorem 3.18 and the fact that . Fin. dim.(Λ) = d(Λ) or d(Λ) + 1.



Next, we give an example to demonstrate the scenario described in Theorem 3.18. Example 3.20 Suppose Λ is a dual extension of a monomial algebra A defined over the subquiver labeled by a, b, c, e of the following quiver with relations cb = be = ba = ε2 = 0. Then it is easy to see d(A) = 3, d(A∗ ) = 1 and d(Λ) = 4 = d(A) + d(A∗ ). Thus . Fin. dim.(Λ) ≤ 5. e c - b • • • • a

a∗

∗ b∗ e

c∗

In the proof of Theorem 3.18, it’s enticing to attempt to apply Lemma 3.17(1) to the sequences of paths in A in order to create a desired sequence of paths in A∗ . Indeed, it will result in a sequence for sure to do so, however, it might be of infinite length and thus useless in our case. Otherwise, we would have d(A) = d(A∗ ) under the assumptions, to which there is a counterexample as shown in Example 3.20. The forgoing phenomenon also demonstrates that finitistic dimensions are more evasive than global dimensions and harder to compute. The quest for nice homological relationships among A, A∗ and Λ leads to an interesting result, Theorem 3.4 in [8]. We paraphrase it and supply a clearer proof and record it here for completeness, contrast with other results in this line and also for the reader’s convenience. Theorem 3.21 ([8, Theorem 3.4]) Suppose Λ is a dual extension of A. Then d(Λ) = 2d(A) > 0 ⇔ any deepest finite dimension tree of Λ is determined by some nonzero path p∗ α such that A p ∈ Lreg , α ∈ PΛ and ΔΛ (p∗ ) contains a longest path beginning with p∗ , ending with p but

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containing an arrow a ∈ A in between with d(a) = d(A). Proof (⇐) Since ΔΛ (p∗ α) and ΔΛ (p∗ ) may differ only by their root labels by Proposition 3.12 (3). Let ΔΛ (p∗ α), and thus ΔΛ (p∗ ), be a deepest finite dimension tree containing a path of the prescribed property: pt (= p)pt−1 · · · p2 p1 (= a)qs (= b∗ qs,2 )qs−1 · · · q2 q1 (= p∗ ) with b ∈ Γ1 and t = d(a) = d(A) ≥ 1. Applying Lemma 3.17(1) to the sequence of paths in A: pt (= p)pt−1 · · · p2 p1 (= a) gives rise to a sequence of paths in A∗ : q1 , q2 , . . . , qt−1 , qt = p∗ such that qi ∈ LA∗ (qi+1 ). So, this sequence of paths is in ΔA∗ (p∗ ), which shows d(A) = t ≤ s and d(Λ) = d(p∗ ) ≥ s + t ≥ 2d(A). Now, applying Lemma 3.17(2) to the sequence of paths in A∗ : qs (= b∗ qs,2 )qs−1 · · · q2 q1 (= p∗ ) leads to a sequence of paths in A: p1 , p2 , . . . , ps−1 , ps such that p1 = b ∈ Γ1 and pi ∈ LA (pi−1 ), where 1 < i ≤ s. Thus the sequence is also in ΔA (b), a subtree of ΔΛ (p∗ ), and thus finite. This shows s ≤ t = d(A), and thus d(Λ) = d(p∗ ) = s + t ≤ 2d(A). As a consequence of the above, d(Λ) = 2d(A). (⇒) Suppose ΔΛ (q) is a deepest finite dimension tree. Without loss of generality, we may assume q ∈ A∗ by Proposition 3.12(3). Now, d(Λ) = 2d(A) > 0, along with Proposition 3.12(1), ensures that ΔΛ (q) contains a longest sequence of paths: pt pt−1 · · · p2 p1 (= a)qs (= b∗ qs,2 )qs−1 · · · q2 q1 (= q) with a, b ∈ Γ1 and s + t = 2d(A). We first show s = t = d(A). Suppose t < d(A). Applying Lemma 3.17 (2) to the sequence of paths in A∗ : qs (= b∗ qs,2 ), qs−1 · · · q2 , q1 (= q) gives rise to a sequence of paths in A: p1 , p2 , . . . , ps−1 , ps such that p1 = b ∈ Γ1 , pi ∈ LA (pi−1 ), where 1 < i ≤ t, which is also in the subtree ΔA (b) of ΔΛ (q). This shows s ≤ t < d(A), and thus d(Λ) = s + t < 2d(A), a contradiction. So, t = d(A) and thus s = t = d(A). Now, among the induced sequence, by the choice of pi , pt = q1,2 q2,1 and q ∗ = q1∗ = q1,1 q1,2 with l(q1,2 ) ≥ 1. But t = d(b) = d(A) forces that pt is left regular in A, and so is q1,2 in A. So, ΔΛ (q) has the claimed properties. Now, the proof is complete.  This result and its proof imply the following Theorem 3.22

If Λ is a dual extension of A and d(Λ) = 2d(A) > 0, then

(i) Any deepest finite dimension tree of Λ contains a deepest finite dimension tree of A determined by an arrow in A. (ii) d(A) ≤ d(A∗ ). Proof

(i) This is immediate from Theorem 3.21.

(ii) d(Λ) = 2d(A) > 0 implies there is a deepest finite dimension tree ΔΛ (q) containing a longest path: pt pt−1 · · · p2 p1 (= a)qs (= b∗ qs,2 )qs−1 · · · q2 q1 (= q) with a, b ∈ Γ1 , q ∈ A∗ and s + t = 2d(A). From the proof for (⇒) of Theorem 3.21 above we know s = t = d(A). But this means the finite dimension tree ΔA∗ (q) contains a sequence of d(A) paths, and thus d(A) ≤ d(A∗ ), as claimed.  We will see in the following that Example 3.25 illustrates this scenario. In addition, Theorem 3.21 has also an immediate consequence on finitistic dimensions, Theorem 3.5 in [8], which may be regarded as a finitistic analogue of Corollary 3.33. We record it here to be used in the following examples and for completeness as well. Theorem 3.23 ([8, Theorem 3.5]) Suppose Λ is a dual extension A. Then . Fin. dim.(Λ) = . fin. dim.(Λ) = 2d(A) if d(Λ) = 2d(A) > 0.

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Now, we give an example to demonstrate the scenario described in Theorem 3.23. Example 3.24 Suppose Λ is a dual extension of a monomial algebra A defined over the subquiver labeled by b, c, e of the following quiver with relations given by bc = e2 = 0. Then it is easy to see that gl. dim.(A) = ∞, d(A) = d(A∗ ) = 2 and d(Λ) = 4 = 2d(A). Thus . Fin. dim.(Λ) = 4 by Theorem 3.23. ∗ e b∗c∗ • • •  c b e

Figure 6

Example 3.24

Does d(Λ) = 2d(A) > 0 imply that d(A) = d(A∗ )? The following example answers the question negatively, and also illustrates Theorem 3.22 (ii). Example 3.25 Suppose Λ is a dual extension of a monomial algebra A defined over the subquiver labeled by a, b, c, e of the following quiver with relations given by bc = eb = e2 = 0. Then it is easy to see that gl. dim.(A) = ∞, d(A) = 1, d(A∗ ) = 2 and d(Λ) = 2 = 2d(A). Thus . Fin. dim.(Λ) = 2 by Theorem 3.23. But d(A) = d(A∗ ). ∗ e a∗ c∗ b∗-a 6 • •? •   c b e

Figure 7

Example 3.25

Now, we may ask a similar question about d(A∗ ): if d(Λ) = 2d(A∗ ), do we have the analogue of Theorem 3.21 — any deepest finite dimension tree of Λ contains a deepest finite dimension tree of A∗ , and furthermore, d(A) = d(A∗ )? The following example offers a negative answer. Example 3.26 Suppose Λ is a dual extension of a monomial algebra A defined over the subquiver labeled by a, b, c, e of the following quiver with relations given by cb = be = e2 = 0. Then it is easy to see that gl. dim.(A) = ∞, d(A) = 2, d(A∗ ) = 1 and d(Λ) = 2 = 2d(A∗ ), which answers the last part of the question. On the other hand, the deepest dimension tree ΔA (b) = ΔΛ (b) is also a deepest dimension tree of Λ with height 2. But there is nothing atop it! which answers the first part of the question. ∗ ∗ ∗ ∗ a a 6 e b ? c • -• -•  c b e

Figure 8

Example 3.26

Nevertheless, we still have the following, the partial analogue of Theorem 3.22. Theorem 3.27

Suppose Λ is dual extension of A. If d(Λ) = 2d(A∗ ), then d(A∗ ) ≤ d(A).

Proof As d(Λ) = 0 if and only if d(A) = 0, so it is clear in this case. Now, assume d(Λ) > 0, and thus d(A∗ ) > 0. If there is no any finite dimension tree of A∗ that is extendable as a finite

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dimension tree of Λ according to the attaching rules in Propositions 3.10 and 3.12 based upon Bottomup or Topdown, then d(Λ) = d(A). Thus 2d(A∗ ) = d(A) > 0 implies d(A∗ ) ≤ d(A). Otherwise, from Proposition 3.12, Λ has a deepest finite dimension tree ΔΛ (p∗ ) (p∗ ∈ A∗ ) that contains a longest sequence of paths: pt · · · , p2 aqs · · · q2 q1 where pi ∈ A, a ∈ Γ1 , qj ∈ A∗ and qs = b∗ qs,1 for some b ∈ Γ1 such that d(A∗ ) ≥ s ≥ 1, d(A) ≥ t ≥ 1 and s + t = d(Λ). If s < d(A∗ ), then 2d(A∗ ) < d(A∗ ) + t, which shows t > d(A∗ ). Thus d(A) > d(A∗ ). If s = d(A∗ ), as before, applying Lemma 3.17 to the sequence of paths: qs . . . q2 q1 in ΔA∗ (p∗ ) gives rise to a sequence of no less than s paths that are in the dimension tree Δ(b). But Δ(b) is a subtree of ΔΛ (p∗ ), and thus is finite. Therefore, s ≤ d(b) and thus s ≤ d(A), i.e., d(A∗ ) ≤ d(A). Now, the proof is complete.  The following is immediate from Theorem 3.27 and the hight-dimension relationship: Corollary 3.28 d(Λ) = 2d(A∗ ).

. Fin. dim.(Λ) ≤ 2d(A∗ ) + 1 ≤ 2d(A) + 1 if Λ is dual extension of A and

So far, our interest has been restricted in finite dimension trees of the involved algebras, since finitistic dimensions are our main concern, and generally are more elusive than global dimensions. Nevertheless, many results obtained above may be adapted or used directly or together with Lemma 3.29 to get global dimension analogues. Next, we’ll give a few results to illustrate the above ideas. In particular, in the case of dual extensions, we sharpen a main result ([8, Theorem 3.7]) and reobtain the main result in [10] in monomial case. We refer to [8, 9] for other related results. In addition, we refer those who are interested in the finitistic dimension conjecture to [1, 14] among others, for past well-known works, and [2, 11–13] among others, for recent developments on the issue. We start with the following observation. Lemma 3.29 Suppose C is a monomial algebra over a field k and whose quiver is given by Γ. Then gl. dim.(C) = d(a) for some arrow a ∈ Γ1 , where d(a) is the depth of the highest dimension tree Δ(a) of C. In particular, gl. dim.(C) = d(C) if gl. dim.(C) is finite. Proof Note that the Jacobson radical J := J(C) of C is generated by arrows ai of Γ1 ,  and note also that C is monomial, it is now easy to see J = i∈I Cai , and thus Jej =   Ca , where I and I are finite index sets, and e is the idempotent at vertex j. Now, i j j∈I  ⊆I it follows easily that for the vertex simple Sej , its projective dimension l. proj. dim.(Sej ) =  1 + l. proj. dim.(Jej ) = 1 + j∈I  ⊆I Cai . As a conclusion, gl. dim.(C) = max{Sei | i ∈ I} = 1 + max{l. proj. dim.(Cai ) | i ∈ I} = 1 + l. proj. dim.(Ca) = d(a) for some a ∈ Γ1 .  Proposition 3.30

Suppose Λ is a trivially twisted extension of A by B at S. Then

(i) gl. dim.(Λ) ≥ max{gl. dim.(A), gl. dim.(B)}. In particular, the equality holds if S0 = ∅ or one of them is infinite. (ii) gl. dim.(Λ) < ∞ ⇔ gl. dim.(A) < ∞ and gl. dim.(B) < ∞. (iii) gl. dim.(Λ) ≤ gl. dim.(A) + gl. dim.(B). (iv) If gl. dim.(Λ) < ∞, then gl. dim.(A) + gl. dim.(B) = gl. dim.(Λ) ⇔ there exists some y ∈ H(B) such that the bottom level of ΔB (y) contains some path b ∈ PB ≥1 such that E(b) = S(a) for some a ∈ H(A).

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Proof (i) From Proposition 3.12(2), gl. dim.(A) ≤ gl. dim.(Λ). On the other hand, any dimension tree ΔB (p) with p ∈ B can be expanded according to Proposition 3.10 and algorithms Topdown or Bottomup into a dimension tree ΔΛ (p) with dΛ (p) ≥ dB (p) by Proposition 3.12(3), which shows the first part. If S0 = ∅, then any dimension tree ΔΛ (p) with p ∈ Λ is either a dimension tree of A if p ∈ A or a dimension tree of B if p ∈ B, thus the equality holds. The remaining case is still from Proposition 3.12(1)–(3). (ii) Implied in the argument for (i). (iii) From Proposition 3.14 (iii) and Lemma 3.29. (iv) gl. dim.(Λ) < ∞ implies that all dimension trees of Λ, and consequently, all dimension trees of A and B, are finite. From Proposition 3.12, there is some dimension tree ΔΛ (pB pA ) of Λ such that d(Λ) = dΛ (pB pA ) = dΛ (pB ). Then the claim is from Proposition 3.14(iv) and Lemma 3.29.



Specializing the above to the dual extensions and meanwhile noticing the fact that gl. dim.(A) = gl. dim.(A∗ ) which is an immediate consequence of Lemma 3.17 and Proposition 3.1(5) in [8] (Note: this provides an alternative combinatorial argument in the case of monomial algebras to the standard result: gl. dim.(R) = gl. dim.(R∗ ) if R is a left and right noetherian ring!), we have immediately the following Corollary 3.31

Suppose Λ is a dual extension of A. Then

(i) gl. dim.(Λ) ≥ gl. dim.(A) = gl. dim.(A∗ ). In particular, the equality holds if one of them is infinite. (ii) gl. dim.(Λ) < ∞ ⇔ gl. dim.(A) < ∞ or gl. dim.(A∗ ) < ∞. (iii) gl. dim.(Λ) ≤ gl. dim.(A) + gl. dim.(A∗ ) = 2 gl. dim.(A). (iv) If gl. dim.(Λ) < ∞, then gl. dim.(A) + gl. dim.(A∗ ) (= 2 gl. dim.(A)) = gl. dim.(Λ) ⇔ ∗ there exists some y ∈ H(A∗ ) such that the bottom level of ΔA∗ (y) contains some path b ∈ PA ≥1 such that E(b) = S(a) for some a ∈ H(A). From the preceding discussions, we have seen from examples many abnormal behaviors of finitistic dimensions of dual extensions. However, the situation will become much simpler (for example, d(A) = d(A∗ ) = gl. dim.(A) = gl. dim.(A∗ ), etc.) if we assume one of the algebras A, A∗ or Λ has finite global dimension, a case studied in [8]. Recall from [8, Lemma 3.6(2)] which states that if gl. dim.(A) < ∞, then d(Λ) = 2d(A) if d(A) = d(a) for some arrow a ∈ A. This, together with Lemma 3.29, sharpens a main result ([8, Theorem 3.7]) as follows. Theorem 3.32 If gl. dim.(A) < ∞, then . Fin. dim.(Λ) = . fin. dim.(Λ) = 2 gl. dim.(A) = 2 . Fin. dim.(A) = 2 . Fin. dim.(A∗ ). From this, we reobtain the main result in [10] in monomial case. Corollary 3.33 ([10]) of A.

gl. dim.(Λ) = 2 gl. dim.(A) = 2 gl. dim.(A∗ ) if Λ is a dual extension

Acknowledgements The author would like to thank the anonymous referees for helpful comments and suggestions.

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