STABLE EQUIVALENCES OF GRADED ALGEBRAS

STABLE EQUIVALENCES OF GRADED ALGEBRAS ALEX S. DUGAS 1 AND ROBERTO MARTÍNEZ-VILLA2 Abstract. We extend the notion o stable equivalence to the class o locally inite graded algebras. For such an algebra Λ, we ocus on the Krull-Schmidt category gr Λ o initely generated Z-graded Λ-modules with degree 0 maps, and the stable category gr Λ obtained by actoring out those maps that actor through a graded projective module. We say that Λ and Γ are graded stably equivalent i there is an equivalence α : gr Λ grγ that commutes with the grading shit. Adapting arguments o Auslander and Reiten involving unctor categories, we show that a graded stable equivalence α commutes with the syzygy operator (where deined and preserves initely presented modules. As a result, we see that i Λ is right noetherian (resp. right graded coherent, then so is any graded stably equivalent algebra. Furthermore, i Λ is right noetherian or k is artinian, we use almost split sequences to show that a graded stable equivalence preserves inite length modules. O particular interest in the nonartinian case, we prove that any graded stable equivalence involving an algebra Λ with soc Λ = 0 must be a graded Morita equivalence. Understanding where and how stable equivalences arise between algebras poses an important, albeit diicult, problem in the represention theory o artin algebras and inite groups. To address this problem it is natural to look or clues by considering stable module categories that arise in other contexts where they admit alternative descriptions. In the most notable examples, the stable categories that appear are quotients o categories o graded modules. Such categories appear, or instance, in the work o Bernstein, Gel and, and Gel and on the bounded derived category o coherent sheaves on projective n-space [8], and in the work o Orlov generalizing these results to certain noncommutative projective varieties [24]. Stable categories o graded modules are also important in the theory o Koszul duality or sel-injective Koszul algebras [22], and in the study o derived equivalences or inite-dimensional algebras o inite global dimension [13]. It thus appears worthwhile to generalize the classical theory o stable equivalences o inite-dimensional algebras to categories o graded modules over graded algebras. In this paper we extend the unctor category methods o Auslander and Reiten to study stable equivalences between categories o graded modules over a wide class o graded algebras which are not necessarily noetherian. Throughout, we ix a commutative semilocal noetherian ring k that is complete with respect to its Jacobson radical m, and ocus on nonnegatively graded, locally inite k-algebras, by which we mean those graded k-algebras Λ = i 0 Λ i where each Λ i is a initely generated k-module. Notice that i Λ = Λ 0 we recover precisely the class o noetherian algebras arising, or example, in the integral representation theory o inite groups. Alternatively, i k is artinian, we obtain the class o locally artinian graded algebras studied in [20], which includes graded quotients o path algebras o quivers. In particular, our theory applies to the preprojective algebras as well as quantum polynomial rings arising in noncommutative alegebraic geometry. We shall work primarily with the category gr Λ o initely generated Z-graded right Λ-modules and degree 0 morphisms. As usual, the graded stable category o Λ is deined as the quotient category gr Λ obtained rom gr Λ by actoring out all maps that actor through a graded projective module. This category inherits a sel-equivalence given by the grading shit, and we shall say that two algebras Λ and Γ are graded stably equivalent i there is an equivalence α : gr Λ grγ that commutes with the grading shit. While the module categories we consider are typically not dualizing R-varieties in the sense o [5] (since the duality does not preserve initely presented modules, they are still Krull-Schmidt categories. As a consequence, our proos 2000 Mathematics Subject Classiication. Primary 16W50, 16D90, 16G10; Secondary 16S38, 18A25. 1 Department o Mathematics, University o Caliornia, Santa Barbara, CA 93106, USA. 2 Instituto de Matemáticas, Universidad Nacional Autónoma de México, Unidad Morelia, Apartado Postal 6-13, Morelia, Michoacán, 58089, México. This research began while the irst author was visiting the Instituto de Matemáticas in Morelia. He wishes to thank the second author or his hospitality and inancial support. The second author thanks the program PAPIIT rom Universidad Nacional Autónoma de México or unding this research project. 1

oten avoid use o the duality and easily generalize to stable equivalences between abelian Krull-Schmidt categories with enough projectives. The irst examples o nontrivial graded stable equivalence or nonartinian algebras can be ound in [21], where it is observed that the processes o constructing and separating nodes work just as nicely or ininitedimensional graded actors o path algebras o quivers. In developing the theory o invariants o graded stable equivalences below, one o our main motivations is to determine the nature o other examples. In this direction, the main result o this paper is essentially a non-existence result, stating that there are no nontrivial graded stable equivalences involving an algebra Λ with zero socle. The proo o this is given in Section 8, ater much preparation. Along the way we ind many interesting eatures o graded algebras that are determined by the stable category. In Section 3, or instance, we show that the property o being right noetherian can be detected in gr Λ. The proo requires a close study o the eect o a stable equivalence on a class o extensions. We will also see that the subcategories o initely presented modules correspond, and that on initely presented modules, a stable equivalence commutes with the syzygy operator. Using this result, we extend our analysis o short exact sequences and show that a graded stable equivalence induces an isomorphism o stable Grothendieck groups. We also interpret these results in terms o Ext-groups, establishing isomorphisms between them in certain cases. We conclude with another example o graded stable equivalence, obtained by modiying a construction o Liu and Xi. However, it is still largely open as to what types o more interesting examples may arise. 1. Notation and preliminaries All algebras we will consider in this article are assumed to be positively graded, locally inite k-algebras or a commutative semilocal noetherian ring k, complete with respect to its Jacobson radical m, unless noted otherwise. Recall that a nonnegatively graded k-algebra Λ = i 0 Λ i is locally inite i each Λ i is initely generated over k and k Z(Λ 0. As mentioned above, Λ is said to be a noetherian algebra i Λ = Λ 0, or a locally artinian algebra i k is artinian. Furthermore, we assume throughout that our algebras have no semisimple blocks. In this section, we shall review some basic acts about these algebras, including the relevant results rom [20]. When k is a ield, a large class o examples o locally artinian graded k-algebras is given by path algebras o quivers with relations. I Λ 0 = k n and Λ is generated as a k-algebra by Λ 0 Λ 1, then Λ is realizable as the path algebra o some inite quiver Q, modulo an ideal I o relations that is homogeneous with respect to the natural path-length grading o the path algebra kq. Other examples o locally artinian graded algebras can be obtained rom quivers by assigning arbitrary nonnegative integer degrees (or lengths to the arrows and actoring out an ideal that is homogeneous with respect to the induced grading on the path algebra, so long as the resulting degree-0 part Λ 0 has inite dimension over k. It is interesting to note that generalized graded quiver algebras can be deined and are still locally artinian or quivers Q with ininitely many arrows between initely many vertices, provided that only a inite number o arrows have degree n or each positive integer n. In addition, the Yoneda algebras Ext A(M, M o initely generated modules M over an artin algebra A provide many other examples o locally artinian graded algebras. We let Gr Λ denote the category o Z-graded right Λ-modules and degree-0 morphisms. By deault, we will work with right modules, and all modules (over graded rings are assumed to be Z-graded unless explicitly stated to the contrary. For a graded module M = i M i, we write M[n] or the n th shit o M, deined by M[n] i = M i n or all i Z. We let S Λ denote the grading shit unctor S : M M[1], which is a sel-equivalence o Gr Λ. Furthermore, common homological unctors such as Hom Λ (,, Ext Λ(,, etc. will always reer to Gr Λ, and thus are to be computed using degree-0 morphisms only. We will denote the set o morphisms o degree d by Hom Λ (X, Y d = HomΛ (X[d], Y, and similarly or extensions o degree d. We reer the reader to [20] or a detailed treatment o basic homological results in this setting. For the most part, we will be working inside the ull subcategory gr Λ o initely generated modules. We will also write.p.gr Λ and.l.gr Λ or the ull subcategories o initely presented modules and inite length modules, respectively. Notice that gr Λ is a subcategory o l..gr Λ, which consists o the locally inite graded modules, meaning those modules M = i M i with each M i initely generated over k. As k is noetherian, it is easy to see that this is an abelian category, whereas gr Λ and.p.gr Λ may not be. Furthermore, we append the superscripts +,, b to the names o these categories to speciy the corresponding ull subcategories o graded modules that are respectively bounded below, above, or both. 2

We deine stable categories o graded modules as usual and denote them by underlining. Thus gr Λ is the quotient category o gr Λ obtained by actoring out those morphisms that actor through a projective. I X is a.g. Λ-module we will sometimes write X to indicate that we are viewing X as an object o the stable category. We will also use this notation to denote the largest direct summand o X with no projective summands, which is unique up to isomorphism (see below. I Hom Λ (X, Y, we write Hom Λ (X, Y or its image in the stable category. Furthermore, we will oten abbreviate these Hom-groups as (X, Y and (X, Y respectively. We shall say that two graded algebras Λ and Γ are graded stably equivalent i there is an equivalence α : gr Λ grγ that commutes with the grading shits in the sense that there is an isomorphism o unctors = η : α S Λ S Γ α. In this case it automatically ollows that the inverse o α also commutes with the grading shits. While this restriction appears rather strong, it is reasonable in light o the deinition o graded (Morita equivalence in [10]. It also allows us to recover classical results or artin algebras, and new results or noetherian algebras, as a special case since we may regard such an algebra as a locally inite graded k-algebra that is concentrated in degree 0. In act, or this trivial grading we have an equivalence gr Λ mod(λ (Z rom which it ollows that the notions o graded and ordinary stable equivalence coincide or algebras concentrated in degree 0. We also point out that, despite the requirement that a graded stable equivalence commutes with the grading shit, several o our results remain true under the weaker hypothesis that α(m[1] = (αm[1] or all M Λ, and it would be an interesting problem to determine whether this condition is even truly necessary. We now state some elementary acts about locally inite graded algebras that ollow easily rom analogous results or noetherian algebras as ound in [15]. First, notice that i M = i Z M i is.g., then M i = 0 or i suiciently small, and each M i is.g. over k. Furthermore, or any.g. graded Λ-modules M and N, the k-module Hom Λ (M, N o degree-0 morphisms is.g. To see this, suppose M is generated in degrees d 1,..., d n, and observe that the restrictions to the degree-d i components induce an injection Hom Λ (M, N n i=1 Hom k(m di, N di. The latter is clearly.g. over k since each M di and N di is, and thus so is the ormer as k is noetherian. Proposition 1.1. Let Λ be a locally inite k-algebra, and M a.g. graded Λ-module. Then (a M has a decomposition M = n j=1 M j into indecomposable modules M j. (b End Λ (M is a noetherian algebra, and hence is local i M is indecomposable. Consequently, gr Λ is a Krull-Schmidt category. We let J Λ = rad Λ 0 Λ 1 denote the graded Jacobson radical o Λ. We say that an epimorphism π : P M with P graded projective is a (graded projective cover i ker π P J. As we will be working primarily with graded modules, we shall oten omit the word graded here. Proposition 1.2. A locally inite graded algebra Λ is graded semiperect, meaning that all.g. graded Λ- modules have graded projective covers. Consequently, all indecomposable graded projective Λ-modules are initely generated. Lemma 1.3 (Nakayama s lemma. Let Λ be a locally inite graded algebra, and M a graded Λ-module in l..gr + Λ. I MJ = M, then M = 0. In particular, i N + MJ = M, then N = M. Since k is assumed to be m-adically complete, so is any initely generated k-module. Using this act, we now extend Proposition 2.2 rom [20] to show that initely generated graded modules over a locally inite algebra are J-adically complete. Lemma 1.4. Let Λ be a locally inite graded algebra, and let M be a module in l..gr + Λ. Then the natural map M lim M/MJ n is an isomorphism (where the inverse limit is computed in Gr Λ. Let E = E(k/m be the injective envelope o k/m over k. Then i R is a noetherian k-algebra, the unctor Hom k (, E gives a duality between the categories o noetherian right R-modules and artinian let R-modules. We can extend this duality to the categories l..gr Λ o locally inite (equivalently, locally noetherian graded right Λ-modules and l.a.gr Λ op o locally artinian graded let Λ-modules, by which we mean those graded modules with each homogeneous component artinian over k. This duality is given by D(M i = Hom k (M i, E or each i Z. In addition, D restricts to dualities l..gr + Λ l.a.gr Λ op, 3

gr Λ.cg.Gr Λ op,.p.gr Λ.cp.Gr Λ op, and inally.l.gr Λ.l.gr Λ op, where.cg.gr Λ op and.cp.gr Λ op denote the categories o initely cogenerated modules and o initely copresented modules, respectively. Notice that a.g. Λ-module M has each M i.g., and thus noetherian, over k, while or a.cg. Λ-module N, each N i is.cg., and thus artinian, over k. As a consequence o this duality, we see that the locally inite indecomposable injective modules are the duals o the locally inite indecomposable projectives. As the latter are necessarily.g. (as Λ is semiperect, all projectives are direct sums o.g. indecomposable projectives, the ormer must be initely cogenerated. In particular, as long as Λ is initely generated, any.g. injective module will have inite length. The Auslander-Reiten transpose also extends to this context [20], yielding another duality between stable categories Tr :.p.gr Λ.p.gr Λ op. Furthermore, with only minor modiications, the proo o existence o almost split sequences given in [20] or locally artinian algebras, extends to this setting to show that Gr Λ has almost split sequences in the ollowing sense. I M.p.gr Λ is nonprojective and indecomposable, then there exists an almost split sequence 0 DTrM E M 0 in Gr Λ [2, 20]. Here, DTrM is initely copresented, while E is usually not initely generated. However, E is necessarily.g. i DTrM has inite length, and in this case the sequence is also almost split in the smaller category gr Λ. Dually, there exists an almost split sequence beginning in any initely copresented, noninjective indecomposable module. In case Λ = Λ 0, we may in act replace Gr Λ and gr Λ with Mod(Λ and mod(λ, respectively. The act that gr Λ is a skeletally small, exact Krull-Schmidt category is essentially all that is needed to generalize results on unctor categories that Auslander and Reiten used to study stable equivalences between artin algebras in [3, 4]. We will now review the elements o their methods that carry over to this more general setting. Thus let A be a skeletally small, exact Krull-Schmidt category with enough projectives, and let mod(a denote the category o initely presented contravariant additive unctors rom A to the category o abelian groups. Notice that mod(a is an exact subcategory o the abelian category Mod(A o all contravariant additive unctors rom A to the category o abelian groups. Furthermore, let mod(a denote the ull subcategory o mod(a consisting o those unctors F that vanish on all projective objects o A. This category can be naturally identiied with the category mod(a o initely presented contravariant additive unctors on the stable category A, and it ollows that the projective objects o mod(a are the representable unctors (, C or objects C o A. Futhermore, the inclusion unctor mod(a mod A has a let adjoint, which we express as F F [3]. O course, i F mod(a then F = F. The ollowing theorem summarizes the results on minimal projective presentations that we shall use. The proo is virtually identical to the one given in [3], and so we omit it. Theorem 1.5. Any F mod(a has a minimal projective presentation (, B (, (, C F 0 in mod(a. Furthermore, the induced exact sequence (, B (, (, C F 0 is a minimal projective presentation or F in mod(a. Thus, i F mod(a, then F has a minimal projective presentation (, B (, (, C F 0 in mod(a with : B C an epimorphism in A, and this induces a minimal projective presentation (, B (, (, C F 0 in mod(a. I has a kernel A in A, and A, B and C each have partial projective resolutions o length n in A, then as in [3], a long exact sequence o homology groups rom [9] provides a partial projective resolution or F in mod(a: (, Ω n A (, Ω n B (, Ω n C (, ΩC (, A (, B (, C F 0. While this resolution is not necessarily minimal beyond the irst two terms, its minimality at the third term in certain cases serves as a key ingredient in some o our proos. 2. Separation o nodes For path algebras o quivers with relations, a node corresponds to a vertex v o the quiver such that all paths that pass through v are contained in the ideal o relations. This can be thought o as a local radical-square-zero condition at v. I one separates such a vertex v into two new vertices, with one a sink 4

and the other a source, the resulting path algebra with relations is stably equivalent to the original one. While this has long been known or artin algebras, these ideas have recently been extended to categories o graded modules over arbitrary graded quiver algebras [21]. The same proos easily adapt to the case o a locally inite graded algebra. In this section, we review the relevant deinitions and results, phrasing them in this generality. Even though we will requently take advantage o these constructions to assume that our algebras have no nodes, we remark that they so ar provide one o the more interesting classes o examples o nontrivial graded stable equivalences between nonartinian graded algebras. Deinition 2.1. A node o Λ is a nonprojective, noninjective simple module S Λ such that every morphism : S M either actors through a projective module or is a split monomorphism. Lemma 2.2. I S Λ is a node and α is a stable equivalence, then αs is either a node or a simple injective. Proo. For any Y Γ that does not contain αs as a direct summand, we have Hom Γ (αs, Y = Hom Λ (S, α 1 Y = 0. Since epimorphisms between indecomposable nonprojective modules do not actor through projectives, we can conclude that αs must be simple. Since αs is not projective, it will be a node unless it is injective. As in the artinian case [17], we have the ollowing equivalent characterizations o nodes. Proposition 2.3 (c. [17]. Let S Λ be a simple module with projective cover Q. Then the ollowing are equivalent. (i S is projective, injective or a node. (ii I : Q P is a nonisomorphism with P indecomposable projective, then (Q soc P. (iii The image o any map : Q[i] J Λ is contained in soc Λ (this is analogous to saying that S does not occur as a composition actor o J Λ /soc Λ. (iv For all nonisomorphisms : P 1 Q and g : Q P 2 with P 1, P 2 indecomposable projectives, we have g = 0. I S is initely copresented, then the above conditions are also equivalent to (v Either S is injective or there is a let almost split morphism ϕ : S P with P projective. We now briely review the process o separation o nodes. Let S = S 1 S n be a sum o nonisomorphic nodes (each concentrated in degree 0, let a = τ Λ (S be the trace ideal o S in Λ, i.e., the ideal generated by the images o all homomorphisms S[i] Λ, and let b = ann r (a be the right annihilator o a in Λ. Notice that both o these are homogeneous ideals. Deine Γ to be the triangular matrix ring with the given grading ( Λ/a a Γ = = ( (Λ/ai a i. 0 Λ/b 0 (Λ/b i i 0 As k is noetherian and a i Λ i, each a i is.g. over k, and thus Γ is also locally inite. Right Γ-modules can be identiied with triples (A, B, where A is a Λ/a -module, B is a Λ/b -module, and : A Λ/a a B is a morphism o graded Λ/b-modules. We have a unctor F : gr Λ deined on Λ-modules X by F (X = (X/Xa, Xa, µ where µ : X/Xa Λ/a a Xa is induced by multiplication, and deined on morphisms in the obvious manner. Also notice that F preserves the lengths o inite length modules. The ollowing theorem summarizes the necessary results rom [17, 21]. Theorem 2.4 (c. [17, 21]. Let Λ, S Λ and Γ be as above. Then F : gr Λ commutes with the grading shit and induces an equivalence gr Λ grγ. Furthermore, the nodes o Γ are precisely the Γ-modules o the orm (T, 0, 0 where T is a node o Λ not isomorphic to any S i. 3. Right noetherian algebras We now begin our analysis o stable equivalence between graded algebras by showing that i Λ is right noetherian, then so is any graded stably equivalent algebra. We remark that to show a graded algebra is right noetherian, it suices to check the ascending chain condition on homogeneous right ideals [23]. This, in turn, is equivalent to the condition that any graded submodule o a.g. projective is also.g., or in other words that the syzygy o any.g. graded module is also.g. Our proo rests on a careul study o the eect o a stable equivalence on a special class o extensions which we now deine. 5

Deinition 3.1. We say that an extension ξ : 0 A B g C 0 in Ext 1 Λ(C, A is stable i 0 and unstable i = 0. It is easy to see that these notions depend only on the equivalence class o ξ in Ext 1 Λ(C, A. Lemma 3.2. Let ξ denote an extension 0 A B C 0 in gr Λ where A has no projective summands, and let u : ΩC A denote the connecting morphism. Then ξ is unstable i and only i u is a split epimorphism. (π C,0 Proo. We may realize ξ as the pushout o the short exact sequence 0 ΩC P A P C P A C 0 along the epimorphism (u, π A : ΩC P A A, where we write π A : P A A and π C : P C C or the projective covers o A and C respectively. We thus have a commutative diagram with exact rows. 0 ΩC P A (u,π A 0 A i 0 0 1 g P C P A B (π C,0 (p,π A g C 0 C 0 i 0 0 1 I uv = 1 A, we have = uv = piv, which shows = 0. Conversely i = 0, then actors through the epimorphism ( (p, ( π A, say via a map h : A P C P A. Then π C h = g(p, π A h = g = 0 implies that h = i 0 v 0 1 q or : A ΩC P A, and it ollows that = (uv + π A q. Since is a ( v q monomorphism, we can cancel it to get 1 A = uv + π A q. As A has no projective summands, π A q is contained in the radical o End Λ (A, and uv = 1 A π A q is an automorphism o A. It ollows that u splits. Proposition 3.3. The ollowing are equivalent or a initely generated nonprojective graded module C over a graded algebra Λ. (1 ΩC is initely generated. (2 For all initely generated modules A Λ, Ext 1 Λ(C, A[n] = 0 or n >> 0. (3 For all initely generated nonprojective modules A Λ, Ext 1 Λ(C, A[n] contains no nonzero stable extensions or n >> 0. In particular, Λ is right noetherian i and only i one o these equivalent conditions holds or all.g. nonprojective C Λ. Proo. I ΩC is initely generated, then or any initely generated A Λ, Hom Λ (ΩC, A[n] = 0 or n >> 0. Since Ext 1 Λ(C, A[n] is a quotient o Hom Λ (ΩC, A[n], we have (1 (2, and (2 (3 is trivial. In order to prove (3 (1, assume that ΩC is not initely generated. We may o course assume that C has no projective summands. Since ΩC is a submodule o a initely generated projective, it is locally inite and bounded below, and we have (ΩCJ Λ ΩC. By Nakayama s lemma, ΩC/ΩCJ Λ cannot be initely generated over k, and thus there exists some graded simple S Λ such that S[n] (ΩC/ΩCJ Λ or ininitely many n > 0. The induced epimorphisms π n : ΩC S[n] do not actor through the inclusion ΩC P C, since the image o ΩC is contained in the radical o P C. Thus the pushout o 0 ΩC P C C 0 along π n is a nonsplit extension o C by S[n]. Moreover, this extension is stable i and only i π n is not a split epimorphism. Thus it only remains to consider the case where π n splits or almost all n. Here, S[n] ΩC or ininitely many n > 0. We irst show that there exists a nonzero nonsplit morphism u : S A or some.g. nonprojective module A Λ. Such a map can be constructed by choosing two values o n (say, n 0 and n 1 or which S[n] ΩC, and then taking the appropriate shit o the composite S[n 0 ] ΩC P C P C /S[n 1 ] = A[n 0 ]. Clearly, A is nonprojective since S[n 1 ] ΩC P C J Λ. Since A and P C are.g., Hom Λ (P C, A[n] = 0 or n >> 0. Thus the maps u[n]π n : ΩC A[n] do not actor through P C or n >> 0. It ollows that the pushouts o 0 ΩC P C C 0 along the maps u[n]π n are nonsplit stable extensions o C by A[n] or all suiciently large values o n. 6

Theorem 3.4. Let α : gr Λ be an equivalence that commutes with the grading shit. Then, or any.g. nonprojective C Λ, ΩC is.g. i and only i ΩαC is.g. In particular, Γ is right noetherian i and only i Λ is. Proo. We assume that ΩC is not.g., so that by Proposition 3.3 there exists a.g. nonprojective A Λ such that Ext 1 Λ(C, A[n] contains nonzero stable extensions or ininitely many n > 0. Let us also assume that ΩαC is.g. In order to obtain a contradiction, it suices to show that the existence o a nonzero stable extension in Ext 1 Λ(C, A or A, C.g. nonprojective implies the existence o a nonsplit extension in Ext 1 Γ(αC, αa. Let ξ : 0 A B g C 0 be a nonsplit extension with 0. We may assume that A and C are indecomposable. We thus have the start o a minimal projective resolution or F mod(gr Λ (, A (, (, B (,g (, C F 0. Applying the equivalence ᾱ o unctor categories induced by α, we get the irst three terms o a minimal projective resolution or ᾱf in mod(gr Γ (, αa (,α (, αb (,αg (, αc ᾱf 0. Inside mod(gr Γ, ᾱf has a minimal projective presentation (, αb P 0 (,v (, αc ᾱf 0 with v v = αg. Regarding 0 K αb P 0 αc 0 as a pushout o the projective cover o αc yields the short exact sequence 0 ΩαC P αc K αb P 0 0, rom which we see that K must be.g. We thereore obtain the start o a projective resolution or ᾱf in mod(gr Γ (, K (,u (, αb (,v (, αc ᾱf 0. Comparing these two projective resolutions or ᾱf, we see that there must be a split epimorphism π : K αa with splitting i such that ui = α(. We now orm the pushout 0 K π u v αb P 0 αc 0 0 αa D αc 0. I the bottom sequence splits, π must actor through u, say π = hu. Thus 1 αa = πi = hui = hα(. It ollows that α( splits, but this contradicts the act that does not split. Hence we have produced a nonzero element in Ext 1 Γ(αC, αa. The inal statement is now immediate. Remarks. (1 Note that the above proo only requires the weaker assumption that α commutes with the grading shit on isomorphism classes o modules. That is, α(m[1] = (αm[1] or all nonprojective M. It would be interesting to know i this condition is even necessary. (2 More generally, suppose A and B are exact Krull-Schmidt categories with enough projectives that are stably equivalent. Along the lines o the above theorem, one can also ask whether A being abelian implies that B is also abelian? Corollary 3.5. Let α : gr Λ be an equivalence that commutes with the grading shit. I C Λ is.p. and nonprojective, then so is αc. Thus α induces an equivalence between the stable categories o.p. graded modules over Λ and Γ. 4. Stable extensions In the proo o Theorem 3.4, we showed that a graded stable equivalence α associates a nonsplit extension in gr Γ to any stable extension in gr Λ. In this section we begin a more careul study o the eect o a graded stable equivalence on short exact sequences. Our results will generalize classical results or inite dimensional algebras rom [4, 18], but our arguments are necessarily more involved due to the absence o a duality between gr Λ and gr Λ op. Our irst main result in this direction is the ollowing. 7

Theorem 4.1. Suppose α : gr Λ is an equivalence, and 0 A B P g C 0 is an exact sequence in gr Λ, where A, B, C have no projective summands, is right minimal, g 0, P is projective, and C is.p. Then there exists an exact sequence 0 αa u αb Q v αc 0 where Q is projective and u = α( and v = α(g. First, we point out that the term right minimal used here has exactly the same meaning as in [6]. In act, we will now show how this notion can be extended to any skeletally small Krull-Schmidt category A. To do so, we will pass to the unctor category Mod(A o contravariant additive unctors rom A to the category o abelian groups. As in [6], we deine a morphism : A B in A to be right minimal i s = or some s End A (A implies that s is an automorphism. Clearly, as we have a ull and aithul unctor A Mod(A given by A (, A, we see that : A B is right minimal i and only i (, : (, A (, B is. Proposition 4.2. Let A be a Krull-Schmidt category. For any morphism : A B in A the ollowing are equivalent. (1 is right minimal. (2 (, is the projective cover o its image in Mod(A. (3 For any (nonzero split monomorphism g : A A, we have g 0. Moreover, any morphism : A B in A has a decomposition (unique up to isomorphism = (, 0 : A A B with : A B right minimal. We say that : A B is the right minimal version o. The proo relies on a modiication o a result o Auslander [1] that states that any initely presented unctor in Mod(A has a minimal projective presentation. Proposition 4.3. For a Krull-Schmidt category A, any initely generated unctor F in Mod(A has a projective cover. Proo. Suppose : (, A F is an epimorphism and let Γ = End A (A, which is a semiperect ring. Then (A : (A, A = Γ F (A is an epimorphism o right Γ-modules. Since Γ is semiperect and add(a is equivalent to proj-γ, there exists a decomposition A = A A such that (A (A,A is a projective cover o F (A, while (A (A,A = 0. Writing = (, : (, A (, A F, we clearly have (A (A,A = (A and (A (A,A = (A. We thus see that (A = 0, and consequently (A = 0, which by Yoneda s lemma implies that = 0. Now since is surjective, must also be surjective. Finally, we must show that : (, A F is an essential epimorphism. The proo o this act is identical to that given in the proo o I.4.1 o [3], so we omit it here. Proo o Proposition 4.2. (3 (2: I (, is not the projective cover o its image, then there is some nonzero direct summand A o A such that (, vanishes on (, A (, A. Then clearly vanishes on A. (2 (1: Since (, is a projective cover o its image, it is right minimal, and thus so is. (1 (3: Suppose that g : A A is a split monomorphism such that g = 0. Then, writing A = g(a A, we can deine an endomorphism h o A by projection onto A with kernel g(a. Since g = 0, (1 A h = 0, but h is not an automorphism o A and thus is not right minimal. For the last remark, note that or an arbitrary : A B we can obtain the right minimal version o by taking the right minimal version (, o (, : (, A (, B, which is just the projective cover o the image o (,. Remark. (3 Analogously, using the category Mod(A op o covariant additive unctors rom A to abelian groups, we can obtain dual results or let minimal morphisms. Briely, : A B is let minimal i and only i (, is a projective cover o its image i and only i g 0 or all split epimorphisms g : B B. In order to prove Theorem 4.1 we will need two lemmas. Lemma 4.4. Suppose that A, B and C are nonprojective Λ-modules, C is.p., and we have a sequence A B g C with right minimal and g 0, such that (, A (, (, B (,g (, C is an exact 8

sequence o unctors. Then there exists an exact sequence 0 A Y where ρ = 0, p : P C is a projective cover, and Y ΩC. ρ r 1 r 2 B P (g,p C 0, Proo. Since (g, p : B P C is surjective and C is.p., its kernel K is.g. and we have an exact sequence 0 K u B P (g,p C 0. Since g = 0, we can actor it through the projective cover p : P C to obtain g = pq or a map q : A P. Thus (g, p = 0 and actors through u : K B P via ( q a map i : A K. Meanwhile, we have a projective resolution o the unctor F = coker(, g in mod(gr Λ, which we can compare to the given exact sequence o unctors as ollows: ( q (, K (,u (, B (,g (, C F 0. (,i (, A (, (, B (,g (, C F 0 Since is right minimal, (, is the projective cover o its image. It ollows that (, i, and hence i too, is a ( split monomorphism. Thus K = i(a Y and writing u = u1 ρ with respect to this decomposition, we clearly have ρ = 0. Since ( q = ui, we get a short exact sequence 0 A Y u 2 r ρ q r B P (g,p C 0. To see that Y ΩC, we take the pushout o the above sequence with respect to the projection A Y Y : 0 A Y ρ q r B P (g,p C 0 0 Y j (b,p W C 0. As j = bρ + p r, we have j = 0. Since Y has no projective summands, Lemma 3.2 implies that it is a direct summand o ΩC. q Lemma 4.5. Suppose 0 A B P (g,p C 0 is exact with P projective and C.p. Suppose A = A A such that = A induces the right minimal version o. Then, in the pushout o the given short exact sequence along the projection π : A A, the induced map j rom A to the pushout B has j right minimal. Proo. Let h : ΩC A be the connecting morphism so that πh is the connecting morphism or the pushout sequence. We thus have exact sequences o projectives (, ΩC (,h (, A (, (, B and (, ΩC (,πh (, A (,j (, B in mod(gr Λ. Thus, to show that j is right minimal it suices to show that (, j is the projective cover o its image, or equivalently that (, πh is a radical morphism. However, i this were not the case, then there would be a split epimorphism s : A A 0 such that sπh splits. I t is a splitting or s, we would then have t = 0, contradicting the right minimality o. Proo o Theorem 4.1. Choosing any maps u : αa αb and v : αb αc such that u = α and v = αg, the sequence αa u αb v αc satisies the hypotheses o Lemma 4.4. Thus we obtain a short exact sequence 0 αa Y u ρ q r αb P (v,p αc 0, 9

where ρ = 0 and Y ΩαC. We consider the ollowing commutative exact pushout diagram. 0 Y 0 αa Y u ρ q r 0 Y αb P (v,p αc 0 (1,0 0 αa i V (t,p 0 σ αc 0 0 0 Clearly, it suices to show that t induces an isomorphism between αb and V. For, i t 1 : αb( V denotes the induced isomorphism, and V = V Q, we can replace V by αb and the maps i and σ by t 1 1 0 0 1 Q i and ( t1 0 0 1 Q σ respectively. The commutativity o the above diagram then shows that these new maps rom αa to αb and rom αb to αc dier rom u and v by maps that actor through projectives. As σt = v, we have σ 0 and t 0. Furthermore, i is right minimal by Lemma 4.5. Applying α 1, we obtain maps j : A α 1 V, t : B α 1 V, and τ : α 1 V C, liting α 1 (i, α 1 (t and α 1 (σ respectively. For ease o reerence, we illustrate these maps in the ollowing commutative diagram in the stable category. A B g C A t j α 1 V τ C Since we have a short exact sequence 0 αa 4.4 applies to the sequence A j α 1 V 0 A Y i V σ αc 0 with i right minimal and σ 0, Lemma τ C, yielding a short exact sequence j ρ r 1 r 2 α 1 (τ,π V P C C 0 with π : P C C a projective cover and ρ = 0. As τt = g, we have g = τt + πw or a map w : B P C, and clearly p = πl or some l : P P C. This leads to the ollowing commutative diagram with exact rows. q 0 A s1 s 2 B P t (g,p 0 w l C 0 0 A Y α 1 V P C j ρ (τ,π C 0 r 1 r 2 We now have j = t = js 1, and since j is right minimal and A has no projective summands, s 1 is an ( ( automorphism o A. It now ollows that s = s1 is a split monomorphism and thus that t 0 is a s 2 10 w l

also monomorphism with cokernel isomorphic to Y. We have a commutative square A Y j ρ r 1 r 2 ( s 2s 1 1,1 α 1 V P C Y = Y which shows that the automorphism o Y obtained ( by going down and to the right actors through a projective. Hence Y must be projective. Thus t 0 splits, and t : B α 1 V must be an isomorphism. Since t = αt, t must also induce an isomorphism between αb and V. w l 5. Syzygies and stable Grothendieck groups Our present goal is to show that a stable equivalence between graded algebras commutes with the syzygy regarded as an operator on isomorphism classes o modules. As in the inite dimensional case, the possibility o nodes occurring as summands o the syzygy prevents this rom holding in complete generality. However, we will see that we can always say something about the nonprojective summands o the syzygies that are not nodes. We thus introduce the notation C to denote the maximal direct summand o C (unique up to isomorphism containing no projective modules or nodes as direct summands. In order to prove our main result, we will need the ollowing lemma. Lemma 5.1. Suppose i : X i Y i (1 i n are nonzero radical morphisms between indecomposable objects in a Krull-Schmidt category A. Then the map = n i=1 i : n i=1 X i n i=1 Y i is a let and right minimal radical morphism. Proo. I u i : Y i Y = n j=1 Y j and π i : X = n j=1 X j X i are the canonical inclusions and projections, we have = n i=1 u i i π i, which belongs to the radical o A since each i does. Now suppose that W = 0 where X = W Z. We may assume that W is indecomposable, and thus that it has a local endomorphism ring. Hence, W satisies the exchange property, and thus there exist direct summands X j o X j or each 1 j n such that X = W X 1 X n. By the uniqueness o the direct sum decomposition o X, some X i must be 0. However, letting v i : Y Y i be the projection, this implies that v i = 0 and hence that i = v i Xi = 0, a contradiction. The proo o let minimality is analogous. Theorem 5.2. Let α : gr Λ be an equivalence that commutes with the grading shit. I C Λ is.p. and indecomposable, then ΩαC = αωc. In particular, i Λ and Γ have no nodes, we have ΩαC = αωc. Proo. Assume that ΩC 0. For each indecomposable direct summand D i o ΩC, there exists a nonisomorphism i : D i A i with A i indecomposable and i 0. Let : ΩC A := A i be deined by extending i by zero on the remaining summands o ΩC, and orm the pushout We claim that 0 ΩC 0 A P C C 0 h B g C 0. (, ΩC (, (, A (,h (, B (,g (, C F 0 is the start o a minimal projective resolution o F = coker(, g in mod(gr Λ. Clearly, it suices to check that each o the maps is right minimal. By the above lemma applied to in gr Λ, we see that (, is right minimal. We also see that (, is a radical map between projectives, and thus it ollows that its cokernel is a projective cover. Hence (, h must be right minimal. Finally, we claim that (, h is also a radical morphism, rom which it will ollow that (, g is right minimal. Indeed, otherwise there would be a map t : B B such that th is a split epimorphism. But since is let minimal, we have th 0, contradicting h = 0. Applying ᾱ we get the start o a minimal projective resolution (, αωc (, αa (, αb (, αc ᾱf 0. 11

But, as in the proo o Theorem 3.4, ᾱf also has a projective resolution (, ΩαC (, K (, αb (, αc ᾱf 0, or some.g. module K. Comparing these resolutions yields αωc ΩαC. In act, αωc can have no nodes as summands since ΩC has no nodes or simple injectives as summands. Thus, we have αωc ΩαC. Completely analogously we have α 1 ΩαC ΩC, and it ollows that αωc = ΩαC. I ΩC = 0, and ΩαC 0, the above argument shows that the latter is a direct summand o αωc = 0, which is a contradiction. Remark. (4 The only place in the above proo where we have used the assumption that α commutes with the grading shit is in the application o Theorem 3.4 to conclude that ΩαC is.g. I we assume rom the beginning that Λ and Γ are right noetherian, then we have ΩαC = αωc even i α does not commute with the grading shit. Likewise, this proo carries over to the case where A and B are any stably equivalent abelian Krull-Schmidt categories with enough projectives. Corollary 5.3. Let α : gr Λ be a stable equivalence (that does not necessarily commute with the grading shit between two right noetherian algebras without nodes. Then pd Γ αc = pd Λ C or all.g. nonprojective Λ-modules C. For a second corollary, we turn our attention to initely presented graded modules. We say that a graded algebra Λ is right graded coherent i every.g. graded right ideal o Λ is.p. As in the nongraded setting, this is easily seen to be equivalent to either (1 every.g. graded submodule o a.p. graded right Λ-module is.p.; or (2 the syzygy o any.p. graded right module is.p.; or (3 the category.p.gr Λ is abelian. Corollary 5.4. Assume either that Λ and Γ are.g. as algebras so that the simples are.p. or that they have no nodes, and let α : gr Λ be an equivalence that commutes with the grading shit. Then Λ is right graded coherent i and only i Γ is. Proo. I Λ is right graded coherent, ΩC is.p. whenever C is.p. Thus, or all C.p., ΩC and Ω 2 C are.g. By Theorem 3.4, ΩαC and ΩαΩC are.g. over Γ. We have ΩC = ΩC S, where S Λ is a direct sum o nodes. Thus αωc = αωc T = ΩαC T, where T Γ = αs is a direct sum o nodes and simple injectives. Clearly, T is.g. since αωc is. Since ΩαC is also.g., there exists a inite direct sum T Γ o nodes such that αωc T = ΩαC T. Taking the syzygy o each side, and noting that ΩX = ΩX, we have ΩαΩC ΩT = Ω 2 αc ΩT. Since T and T are.g. and semisimple, they are.p., and hence their syzygies are.g. As ΩαΩC is.g., so is Ω 2 αc. Thus ΩαC is.p., and we conclude that Γ is right graded coherent. Remarks. (5 The ollowing example shows that the restrictions on the algebras in the above corollary are indeed necessary. Let Λ be the graded quiver algebra o the quiver Q consisting o one vertex and ininitely many loops {x i } i N, with x i in degree i, with the relations x i x j = 0 or all i and j. The single vertex o Q corresponds to a node o Λ, and separating it yields a stably equivalent algebra Γ that is the path algebra o the quiver consisting o two vertices and ininitely many parallel arrows {x i } i N, with x i in degree i. x 2. Λ 1.. Γ 1 x 1 Notice that Γ is hereditary, so any.g. right ideal is projective and thus also.p. On the other hand, the right ideal x 1 Λ is a simple Λ-module which is clearly not.p. We thus see that Γ is right graded coherent, while Λ is not. (6 We do not know whether the above corollary would still hold under the weaker hypothesis o a stable equivalence α :.p.gr Λ.p.gr Γ between categories o.p. modules. In essence, this is an interesting special case o the question raised in Remark (2 o Section 3. x 1 x 2. 2 12

As another application o Theorem 5.2, we can strengthen the results rom the previous section on short exact sequences. In act, this will then allow us to show that graded stably equivalent algebras have isomorphic stable Grothendieck groups, generalizing a theorem o the second-named author or stable equivalences between artin algebras [18]. Recall that the stable Grothendieck group o a skeletally small abelian Krull-Schmidt category A with enough projectives is deined as the quotient K0(A s = L(A/R(A, where L(A is the ree abelian group on the isomorphism classes [X] o objects o A and R(A is the subgroup generated by all elements o the orm [A] [B] + [C] or which there exists an exact sequence 0 A P B Q g C 0 with P and Q projective, and where A, B may be 0. Clearly, to show that a graded stable equivalence α : gr Λ between right noetherian algebras induces an isomorphism K0(gr s Λ = K0(gr s Γ it suices to show that it preserves short exact sequences o the above orm. The ollowing theorem takes care o the majority o such sequences. 1 2 q 1 q 2 Theorem 5.5. Suppose 0 A 1 A 2 B P (g,p C 0 is a short exact sequence in gr Λ, where P is projective, A 1 and B have no projective summands, and B and C are.p. Further, assume that 1 is right 1 2 q 1 q 2 minimal, 2 = 0 and g 0. Then there exists a short exact sequence 0 αa 1 Y αb P (g,p αc 0 in gr Γ, where P is projective, Y = αa 2, 1 = α( 1, g = α(g, and 2 = 0. Proo. We start by taking the pushout o the given sequence along the projection A 1 A 2 A 1 to get the ollowing commutative exact diagram. 0 0 A 2 A 2 0 A 1 A 2 1 2 q 1 q 2 B P (g,p C 0 (1,0 0 A 1 i r t B 1 Q (σ,π C 0 0 0 Here, Q is projective and B 1 has no projective summands. Clearly t and σ are nonzero. By Lemma 4.5, i is right minimal, and thus we may apply Theorem 4.1 to the bottom row to obtain an exact sequence i r 0 αa 1 αb 1 Q (σ,π αc 0 in gr Γ, where i = α(i and σ = α(σ. At the same time, Lemma 4.4 applied to the sequence (, αa 1 (,α1 (, αb (,αg (, αc yields an exact sequence 0 αa 1 Y 1 2 q 1 q 2 αb P (g,p αc 0 in gr Γ with p a projective cover, 1 = α( 1, 2 = 0 and g = α(g. Let t 0 : B B 1 be the appropriate component o t. Since σt 0 = g, or any lit t 0 o αt 0 we have σ t 0 = α(σt 0 = α(g = g. Thus σ t 0 g = p s 13

or some s : αb P, and we obtain a commutative diagram with exact rows. 0 αa 1 Y (s 1,s 2 1 2 q 1 q 2 t 0 0 s 1 αb P (g,p 1 p1 0 p 2 αb 1 P 0 αa 1 i r αb 1 Q (σ,p (σ,π αc 0 αc 0 αc 0 Here, ( p1 p 2 : P αb 1 Q exists since P is projective, and (s 1, s 2 : αa 1 Y αa 1 is the map induced on the kernels by the commutativity o the right hand rectangle. We easily see that i s 1 = t 0 1, and since t 0 1 = i we obtain i s 1 = α(t 0 1 = α(i = i. Since we know that i, and hence i, is right minimal, s 1 must be an isomorphism. O course, as αa has no projective summands, this implies that s 1 is an isomorphism. Hence the map (s 1, s 2 is a split epimorphism with kernel isomorphic to Y. γ From the middle column o the above diagram, we now obtain an exact sequence 0 Y αb t 0 P αb 1 Q 0 where the s all represent maps that actor through projectives. Similarly, in gr Λ we have a short exact sequence 0 A 2 yields a projective resolution 2 B P t0 0 (, B (,t0 (, C coker(, t 0 0. B 1 Q 0 with 2 = 0. The latter I we apply ᾱ we get a projective resolution o the same orm or the cokernel o (, αt 0 = (, t 0. An isomorphic projective resolution must be induced by the above short exact sequence starting in Y, and thus it ollows that γ = 0. We now wish to apply Lemma 3.2 to this sequence. I Y has projective summands we may irst orm the pushout along the projection Y Y, as the resulting extension is easily seen to be unstable as well. Thus it ollows that Y is isomorphic to a direct summand o ΩαB 1. As we are assuming that B is.p., we see that αb and αb 1 are as well, and thus the complement to Y must be isomorphic to ΩαB by the snake lemma applied to the appropriate analogue o the diagram in the proo o 3.2. Hence we have Y ΩαB = ΩαB 1 and similarly the other exact sequence yields A 2 ΩB = ΩB 1. Thereore, by Theorem 5.2 αa 2 αωb = αωb 1 = ΩαB1 = Y ΩαB, and it ollows that αa2 = Y. Theorem 5.6. Suppose that α : gr Λ is an equivalence where Λ and Γ are right noetherian and without nodes. Then K s 0(gr Λ = K s 0(gr Γ. Proo. Let 0 A Q B P g C 0 be a short exact sequence in gr Λ with P, Q projective, and such that A and B have no projective summands. We consider several cases. First, i g = 0, then we have an exact sequence o projectives 0 (, ΩC (, A (, B 0 in mod(gr Λ. This sequence must split, and we obtain A = B ΩC. Hence αa = αb ΩαC by Theorem 5.2, and noting that we always have [ΩαC] = [αc] in the stable Grothendieck group, we obtain the relation [αa] + [αc] = [αb] or K0(gr s Γ. Similarly, i = 0, we can use the exact sequence 0 (, ΩB (, ΩC (, A 0 to conclude that ΩC = ΩB A. Again by Theorem 5.2, applying α shows that the analogous relation holds in K0(gr s Γ. Thus we may now assume that and g are nonzero, and apply the previous theorem. Remark. (7 Similarly, i Λ and Γ are graded right coherent, then we obtain an isomorphism K s 0(.p.gr Λ = K s 0(.p.gr Γ. 14

6. Projective extensions We have already seen in Theorem 4.1 that or C Λ.p., a graded stable equivalence α : gr Λ associates a nonsplit extension o αc by αa to any nonsplit stable extension o C by A. We now undertake a more in-depth comparison o the extension groups Ext 1 Λ(C, A and Ext 1 Γ(αC, αa, parallel to the analysis o the last two sections. Assuming that Λ and Γ have no nodes and that C Λ is.p., α always induces an isomorphism (ΩC, A = (ΩαC, αa, and these homomorphism groups naturally arise as quotients o the aorementioned Ext-groups. We thus concentrate on the kernels o these quotient maps, that is, on the extensions o A by C corresponding to maps ΩC A that actor through a projective. Much o what ollows makes sense in any abelian category C with enough projectives, and we thus start o in this general setting and later specialize to the category Gr Λ, and to gr Λ when C is.p. While projective covers may not exist in C, we will still write ΩC or the kernel o some ixed epimorphism π C : P C C with P C projective. Deinition 6.1. Let ξ : 0 A B g C 0 be an extension in Ext 1 C(C, A with connecting morphism h : ΩC A, which is unique up to the addition o a map that actors through the inclusion ΩC P C. We say that ξ is projective i h = 0. Clearly, this deinition depends only on the equivalence class o the extension ξ, and is independent o the choices o π C and h. We let K(C, A Ext 1 C(C, A denote the subset o equivalence classes o projective extensions. The ollowing lemma implies that K(, is an additive sub-biunctor o Ext 1 C(, [7]. The proo is routine and not essential to our discussion, so we omit it. Lemma 6.2. The class o projective extensions is closed under taking direct sums and orming pushouts or pullbacks. We can now study the relative homology o C with respect to the sub-biunctor K(, o Ext 1 C(, as in [7]. We say that an object C o C is K-projective i K(C, A = 0 or all objects A in C. Clearly, this is equivalent to all projective extensions 0 A B C 0 splitting. Lemma 6.3. An object C is K-projective i and only i Ext 1 C(C, P = 0 or all projectives P. Proo. The orward direction is clear since any extension 0 P B C 0 is projective. In act, K(C, P = Ext 1 C(C, P or all C and all projectives P. Conversely, i ξ : 0 A B C 0 is projective with connecting morphism h : ΩC A, then h actors through an epimorphim π A : P A A with P A projective. It ollows that ξ arises as the pushout via π A o an extension in Ext 1 C(C, P A = 0. Hence ξ splits. I C is a.g. right Λ-module, we shall say that C is K-projective in gr Λ i K(C, A = 0 or all.g. A. The above lemma now reduces to say that C is K-projective i and only i Ext 1 Λ(C, Λ[i] = 0 or all i Z. Proposition 6.4. Let α : gr Λ be an equivalence that commutes with the grading shit, and suppose that Λ and Γ have no nodes. Then a.p. nonprojective indecomposable C Λ is K-projective i and only i αc is. Proo. I C is not K-projective, then according to the preceding lemma, there exists a nonsplit extension ξ : 0 P B g C 0 or some indecomposable projective P. I B is projective, then ΩC is isomorphic to a summand o P and hence projective. It ollows rom Theorem 5.2 that ΩαC is also projective, rom which we deduce that αc is not K-projective. We may thus assume that B is not projective. Furthermore, rom the short exact sequence 0 ΩC P C P B 0, we observe that ΩB = ΩC. Now, we have a minimal projective resolution 0 (, B (,g (, C F 0 or the unctor F = coker(, g in mod(gr Λ. Applying ᾱ, we obtain a minimal projective resolution 0 (, αb (,αg (, αc ᾱf 0 in mod(gr Γ, and it ollows that there is a nonsplit short exact sequence ζ : 0 A u v αb P 0 αc 0 with v = αg, which corresponds to the minimal projective resolution o ᾱf in mod(gr Γ. We know that A = ker v is.g. since αc is.p. and αb P 0 is.g. This short exact sequence, in turn, gives rise to a long exact sequence o unctors on gr Γ (, ΩA 0 0 (, ΩαB (, ΩαC (, A (, αb (, αc ᾱf 0. 15