BOUNDS ON THE GLOBAL DIMENSION OF CERTAIN PIECEWISE HEREDITARY CATEGORIES SEFI LADKANI 8 Abstract. We give boundson theglobal dimension of a finitelength, 0 piecewisehereditarycategoryintermsofquantitativeconnectivityprop- 0 erties of its graph of indecomposables. 2 Weusethistoshowthattheglobaldimensionofafinitedimensional, n piecewise hereditary algebra A cannot exceed 3 if A is an incidence a algebraofafiniteposetormoregenerally,asincerealgebra. Thisbound J is tight. 3 ] A R 1. Introduction . h t Let A be an abelian category and denote by Db(A) its bounded derived a category. A is called piecewise hereditary if there exist an abelian heredi- m tary category H and a triangulated equivalence Db(A) ≃ Db(H). Piecewise [ hereditary categories of modules over finite dimensional algebras have been 3 studied in the past, especially in the context of tilting theory, see [1, 2, 3]. v It is known [2, (1.2)] that if A is a finite length, piecewise hereditary 9 3 category with n non-isomorphic simple objects, then its global dimension 1 satisfies gl.dimA ≤ n. Moreover, this bound is almost sharp, as there are 7 examples [5] where A has n simples and gl.dimA = n−1. 0 6 In this note we show how rather simple arguments can yield effective 0 boundsontheglobaldimensionofsuchacategoryA,intermsofquantitative / h connectivity conditions on the graph of its indecomposables, regardless of t the number of simple objects. a m LetG(A)bethedirectedgraphwhoseverticesaretheisomorphismclasses ′ : of indecomposables of A, where two vertices Q,Q are joined by an edge v ′ ′ Q → Q if HomA(Q,Q)6= 0. i X Let r ≥ 1 and let ε = (ε0,...,εr−1) be a sequence in {+1,−1}r. An r ε-path from Q to Q′ is a sequence of vertices Q = Q,Q ,...,Q = Q′ such a 0 1 r that Q → Q in G(A) if ε = +1 and Q → Q if ε = −1. i i+1 i i+1 i i For an object Q of A, let pd Q = sup{d :Extd(Q,Q′)6= 0 for some Q′} A A and idAQ = sup{d : ExtdA(Q′,Q) 6= 0 for some Q′} be the projective and injective dimensions of Q, so that gl.dimA = sup pd Q. Q A Theorem 1.1. Let A be a finite length, piecewise hereditary category. As- sume that there exist r ≥ 1, ε ∈ {1,−1}r and an indecomposable Q such 0 that for any indecomposable Q there exists an ε-path from Q to Q. 0 Then gl.dimA ≤ r+1 and pdAQ+idAQ ≤ r+2 for any indecomposable Q. We give two applications of this result for finite dimensional algebras. 1 2 SEFILADKANI Let A be a finite dimensional algebra over a field k, and denote by modA the category of finite dimensional right A-modules. Recall that a module M in modA is sincere if all the simple modules occur as composition factors of M. The algebra A is called sincere if there exists a sincere indecomposable module. Corollary 1.2. Let A be a finite dimensional, piecewise hereditary, sincere algebra. Then gl.dimA ≤ 3 and pdQ + idQ ≤ 4 for any indecomposable module Q in modA. Let X be a finite partially ordered set (poset) and let k be a field. The incidence algebra kX is the k-algebra spanned by the elements e for the xy pairs x ≤ y in X, with the multiplication defined by setting exyey′z = exz ′ when y = y and zero otherwise. Corollary 1.3. Let X be a finite poset. If the incidence algebra kX is piecewise hereditary, then gl.dimkX ≤ 3 and pdQ + idQ ≤ 4 for any indecomposable kX-module Q. TheboundsinCorollaries1.2and1.3aresharp,seeExamples3.2and3.3. The paper is organized as follows. In Section 2 we give the proofs of the above results. Examples demonstrating various aspects of these results are given in Section 3. 2. The proofs 2.1. Preliminaries. Let A be an abelian category. If X is an object of A, denote by X[n] the complex in Db(A) with X at position −n and 0 elsewhere. Denote by indA, indDb(A) the sets of isomorphism classes of indecomposable objects of A and Db(A), respectively. The map X 7→ X[0] is a fully faithful functor A → Db(A) which induces an embedding indA ֒→ indDb(A). Assume that there exists a triangulated equivalence F : Db(A) → Db(H) with H hereditary. Then F induces a bijection indDb(A)≃ indDb(H), and we denote by ϕ : indA → indH×Z the composition F indA֒→ indDb(A) −→∼ indDb(H) = indH×Z where the last equality follows from [4, (2.5)]. If Q is an indecomposable of A, write ϕ (Q) = (f (Q),n (Q)) where F F F f (Q) ∈ indH and n (Q) ∈ Z, so that F(Q[0]) ≃ f (Q)[n (Q)] in Db(H). F F F F From now on we fix the equivalence F, and omit the subscript F. Lemma 2.1. The map f :indA → indH is one-to-one. Proof. If Q,Q′ are two indecomposables of A such that f(Q),f(Q′) are isomorphic in H, then Q[n(Q′) − n(Q)] ≃ Q′[0] in Db(A), hence n(Q) = n(Q′), and Q ≃ Q′ in A. (cid:3) As a corollary, note that if A and H are two finite dimensional algebras such that Db(modA) ≃ Db(modH) and H is hereditary, then the represen- tation type of H dominates that of A. We recall the following three results, which were introduced in [1, (IV,1)] when H is the category of representations of a quiver. BOUNDS ON GLOBAL DIMENSION OF PIECEWISE HEREDITARY CATEGORIES 3 Lemma 2.2. Let Q,Q′ be two indecomposables of A, Then Exti (Q,Q′) ≃ Exti+n(Q′)−n(Q)(f(Q),f(Q′)) A H Corollary 2.3. Let Q,Q′ be two indecomposables of A with HomA(Q,Q′) 6= 0. Then n(Q′)−n(Q)∈ {0,1}. Lemma 2.4. Assume that A is of finite length and there exist integers n ,d 0 such that n ≤ n(P)< n +d for every indecomposable P of A. 0 0 If Q is indecomposable, then pdAQ ≤ n(Q)−n0 +1 and idAQ ≤ n0 + d−n(Q). In particular, gl.dimA ≤ d. Proof. See [1, IV, p.158] or [2, (1.2)]. (cid:3) 2.2. Proof of Theorem 1.1. Let r ≥ 1, ε = (ε0,...,εr−1) and Q0 be as in theTheorem. Denotebyr+ thenumberofpositiveεi,andbyr− thenumber of negative ones. Let F : Db(A)→ Db(H)bea triangulated equivalence and write f = f , n= n . F F LetQbeanyindecomposableofA. Byassumption,thereexistsanε-path Q ,Q ,...,Q = Q, so by Corollary 2.3, n(Q )−n(Q ) ∈ {0,ε } for all 0 1 r i+1 i i 0 ≤ i < r. It follows that n(Q)−n(Q ) = r−1α ε for some α ∈ {0,1}, 0 i=0 i i i hence P n(Q0)−r− ≤ n(Q)≤ n(Q0)+r+ and the result follows from Lemma 2.4 with d = r+1 and n0 = n(Q0)−r−. 2.3. Variations and comments. Remark 2.5. The assumption in Theorem 1.1 that any indecomposable Q is the end of an ε-path from Q can replaced by the weaker assumption that 0 any simple object is the end of such a path. Proof. Assume that εr−1 = 1 and let Q be indecomposable. Since Q has finite length, we can find a simple object S with g : S ֒→ Q. Let Q0,Q1,...,Qr−1,S be an ε-path from Q0 to S with fr−1 : Qr−1 ։ S. Replacing S by Q and fr−1 by gfr−1 6= 0 gives an ε-path from Q0 to Q. The case εr−1 = −1 is similar. (cid:3) Remark 2.6. Let G(A) be the undirected graph obtained from G(A) by forgetting the directions of the edges. The distance between two indecom- ′ ′ posables Q and Q, deenoted d(Q,Q), is defined as the length of the shortest path in G(A) between them (or +∞ if there is no such path). ′ ′ The same proof gives that |n(Q) − n(Q)| ≤ d(Q,Q) for any two in- decompoesables Q and Q′. Let d = supQ,Q′d(Q,Q′) be the diameter of G(A). Whend < ∞, inf n(Q)and sup n(Q)arefinite, and byLemma 2.4 Q Q gl.dimA≤ d+1 and pdAQ+idAQ ≤ d+2 for any indecomposable Q. e Remark 2.7. The conclusion of Theorem 1.1 (or Remark 2.6) is still true under the slightly weaker assumption that A is a finite length, piecewise hereditary category and A = ⊕r A is a direct sum of abelian full subcat- i=1 i egories such that each graph G(A ) satisfies the corresponding connectivity i condition. 4 SEFILADKANI 2.4. Proof of Corollary 1.2. Let A be sincere, and let S ,...,S be the 1 n representatives of the isomorphism classes of simple modules in modA. Let P ,...,P be the corresponding indecomposable projectives and finally let 1 n M be an indecomposable, sincere module. Take r = 2 and ε= (−1,+1). Now observe that any simple S is the end i ofanε-pathfromM,aswehaveapathofnonzeromorphismsM ← P ։ S i i since M is sincere. The result now follows by Theorem 1.1 and Remark 2.5. 2.5. Proof of Corollary 1.3. Let X bea poset and k a field. A k-diagram F isthedataconsistingoffinitedimensionalk-vectorspacesF(x)forx∈ X, ′ ′ together with linear transformations rxx′ : F(x) → F(x) for all x ≤ x, ′ ′′ satisfyingtheconditionsrxx = 1F(x) andrxx′′ = rx′x′′rxx′ forallx ≤ x ≤ x . The category of finite dimensional right modules over kX can be iden- tified with the category of k-diagrams over X, see [6]. A complete set of representatives of isomorphism classes of simple modules over kX is given by the diagrams S for x ∈ X, defined by x k if y = x S (y) = x (0 otherwise ′ with ryy′ = 0 for all y < y . A module F is sincere if and only if F(x) 6= 0 for all x ∈X. The poset X is connected if for any x,y ∈ X there exists a sequence x = x ,x ,...,x = y such that for all 0 ≤ i < n either x ≤ x or 0 1 n i i+1 x ≥ x . i i+1 Lemma 2.8. If X is connected then the incidence algebra kX is sincere. Proof. Let k be the diagram defined by k (x) = k for all x ∈ X and X X ′ rxx′ = 1k for all x ≤ x. Obviously kX is sincere. Moreover, kX is inde- ′ composable by a standard connectivity argument; if k = F ⊕ F , write X V = {x∈ X : F(x) 6=0} and assume that V not empty. If x ∈ V and x < y, then y ∈ V, otherwise we would get a zero map k ⊕ 0 → 0 ⊕ k and not an identity map. Similarly, if y < x then y ∈ V. By connectivity, V = X and F =k . (cid:3) X If X is connected, Corollary 1.3 now follows from Corollary 1.2 and Lemma 2.8. For general X, observe that if {X }r are the connected com- i i=1 ponents of X, then the category modkX decomposes as the direct sum of the categories modkX , and the result follows from Remark 2.7. i Corollary 2.9. Let X and Y be posets such that Db(kX) ≃ Db(kY) and gl.dimkY > 3. Then kX is not piecewise hereditary. 3. Examples We give a few examples that demonstrate various aspects of global di- mensions of piecewise hereditary algebras. In these examples, k denotes a field and all posets are represented by their Hasse diagrams. Example 3.1 ([5]). Let n ≥ 2, Q(n) the quiver 0 −α→1 1 −α→2 2−α→3 ... −α−→n n BOUNDS ON GLOBAL DIMENSION OF PIECEWISE HEREDITARY CATEGORIES 5 and I(n) be the ideal (in the path algebra kQ(n)) generated by the paths α α for 1 ≤ i < n. By [1, (IV, 6.7)], the algebra A(n) = kQ(n)/I(n) is i i+1 piecewise hereditary of Dynkin type A . n+1 For a vertex 0 ≤ i ≤ n, let S , P , I be the simple, indecomposable i i i projective and indecomposable injective corresponding to i. Then one has P = S , I = S and for 0 ≤ i < n, P = I with a short exact sequence n n 0 0 i i+1 0 → S → P → S → 0. i+1 i i The graph G(modA(n)) is shown below (ignoring the self loops around each vertex). P0 oo P1 oo P2 ... Pn−2 oo Pn−1 (cid:4)(cid:4)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9) ZZ55555 (cid:4)(cid:4)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9) ZZ55555 (cid:4)(cid:4)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9) __@@@@@@ (cid:127)(cid:127)~~~~~~ ]];;;;;; S0 S1 S2 Sn−1 Sn Regarding dimensions, we have pdS = n−i, idS = i for 0 ≤ i≤ n, and i i pdP = idP = 0 for 0≤ i < n, so that gl.dimA(n) = n and pdQ+idQ ≤ n i i for every indecomposable Q. The diameter of G(modA(n)) is n+1. The following two examples show that the bounds given in Corollary 1.3 e are sharp. Example 3.2. A poset X with kX piecewise hereditary and gl.dimkX = 3. Let X,Y be the two posets: • // • • • ~~~~?? <<<<<(cid:2)(cid:2)(cid:2)(cid:2)@@ @@@@(cid:31)(cid:31) @@@@(cid:31)(cid:31) ~~~~?? • @@@@(cid:31)(cid:31) (cid:2)(cid:2)(cid:2)(cid:2)(cid:2)<(cid:2)<<<<(cid:30)(cid:30) ~~~~?? • ~~~~?? • // • @@@@(cid:31)(cid:31) • // • • • X Y Then Db(kX) ≃ Db(kY), gl.dimkX = 3, gl.dimkY = 1. Example 3.3. A poset X with kX piecewise hereditary and an indecom- posable F such that pd F +id F = 4. kX kX Let X,Y be the following two posets: • • • • (cid:127)(cid:127)(cid:127)(cid:127)?? BBBB ||||>> ????(cid:31)(cid:31) @@@@(cid:31)(cid:31) ~~~~?? •????(cid:31)(cid:31)• ||||>>•xBBBB •(cid:127)(cid:127)(cid:127)(cid:127)??• • ~~~~??• // • // • @@@@(cid:31)(cid:31)• X Y Then Db(kX) ≃ Db(kY), gl.dimkX = 2, gl.dimkY = 1 and for the simple S we have pd S = id S = 2. x kX x kX x We conclude by giving two examples of posets whose incidence algebras are not piecewise hereditary. Example 3.4. A product of two trees whose incidence algebra is not piece- wise hereditary. 6 SEFILADKANI By specifying an orientation ω on the edges of a (finite) tree T, one gets a finite quiver without oriented cycles whose path algebra is isomorphic to the incidence algebra of the poset X defined on the set of vertices of T T,ω by saying that x ≤ y for two vertices x and y if there is an oriented path from x to y. A poset of the form X is called a tree. Equivalently, a poset is a tree if T,ω and only if the underlying graph of its Hasse diagram is a tree. Obviously, gl.dimkX = 1, so that kX is trivially piecewise hereditary. Moreover, T,ω T,ω while the poset X may depend on the orientation ω chosen, its derived T,ω equivalence class depends only on T. Given two posets X and Y, their product, denoted X ×Y, is the poset ′ ′ ′ ′ whose underlying set is X × Y and (x,y) ≤ (x,y ) if x ≤ x and y ≤ y ′ ′ where x,x ∈ X and y,y ∈ Y. It may happen that the incidence algebra of aproductoftwotrees,althoughnotbeinghereditary,ispiecewisehereditary. TwonotableexamplesaretheproductoftheDynkintypesA ×A ,whichis 2 2 piecewise hereditary of type D , and the productA ×A which is piecewise 4 2 3 hereditary of type E . 6 Consider X = A ×A and Y = D with the orientations given below. 2 2 4 • • • (cid:127)(cid:127)~~~~ @@@@(cid:31)(cid:31) @@@@(cid:31)(cid:31) (cid:127)(cid:127)~~~~ • • • @@@@(cid:31)(cid:31) (cid:127)(cid:127)~~~~ (cid:15)(cid:15) • • X Y Thengl.dimkX = 2,gl.dimkY = 1andDb(kX) ≃ Db(kY),henceDb(k(X× X)) ≃ Db(k(Y ×Y)). But gl.dimk(X×X) = 4, so by Corollary 2.9, Y ×Y is a productof two trees of type D whoseincidence algebra is not piecewise 4 hereditary. Example 3.5. The converse to Corollary 1.3 is false. Let X be the poset • // • // • @@ ~?? @@ ~?? @ ~ @ ~ @~ @~ ~@ ~@ ~ @ ~ @ ~ @ ~ @ ~ (cid:31)(cid:31) ~ (cid:31)(cid:31) • // • // • Then gl.dimkX = 2, hence pd F ≤ 2, id F ≤ 2 for any indecompos- kX kX able F, so that X satisfies the conclusion of Corollary 1.3. 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