STABLE EQUIVALENCES OF GRADED ALGEBRAS

Transcription

1 STABLE EQUIVALENCES OF GRADED ALGEBRAS ALEX S. DUGAS AND ROBERTO MARTÍNEZ-VILLA Abstract. We extend the notion o stable equivalence to the class o locally noetherian 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. For instance, a well-known result o Bernstein, Gel and, and Gel and states that the bounded derived category o coherent sheaves on projective n-space is equivalent to the stable category o initely generated Z-graded modules over the exterior algebra Λ(k n+1 ) o an (n + 1)- dimensional vector space [7]. Recently, this result has been generalized in various directions, which allow the exterior algebra to be replaced by more general inite-dimensional algebras. In [22], Saorín and the second author use Koszul duality theory to show that there is an equivalence o triangulated categories gr Λ D b (.p.gre(λ).l.gr E(Λ) or any inite-dimensional sel-injective Koszul algebra Λ whose Yoneda algebra E(Λ) is right graded coherent [22]. In parallel to the classical case, this derived category can be interpreted as the derived category o coherent sheaves over a noncommutative projective variety. Another generalization, due to Orlov [24], relates the stable category o graded Cohen-Macaulay modules over a graded noetherian Gorenstein algebra Λ to the bounded derived category o coherent sheaves on Proj Λ. Furthermore, stable categories o graded modules are also useul in describing derived categories o initedimensional algebras. A theorem o Happel s states that i Λ is a inite-dimensional k-algebra o inite global dimension, and ˆΛ is its repetitive algebra, then there are equivalences o triangulated categories ) D b (Λ) mod-ˆλ gr T Λ, where T Λ = Λ < DΛ is the trivial extension o Λ with the grading given by (T Λ) 0 = Λ and (T Λ) 1 = DΛ [12]. It thus seems worthwhile to extend the classical theory o stable equivalences o inite-dimensional algebras to categories o graded modules over graded algebras. Our methods in act apply to a wide class o graded algebras. 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 noetherian 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 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. MSC (2000): Primary 16W50, 16D90, 16G10; Secondary 16S38, 18A25. 1

2 arising, or example, in the integral representation theory o inite groups. Alternatively, i k happens to be artinian, we obtain the class o locally artinian graded algebras studied in [20]. Locally noetherian graded algebras thus provide a convenient generalization o these two rather distinct classes o algebras, and at the same time their categories o initely generated graded modules suiciently resemble module categories o artin algebras, thereby allowing much o the standard theory to generalize. As a result, we obtain a notion o stable equivalence that is applicable to any graded quotient o a path algebra o a quiver. In particular, our theory applies to preprojective algebras as well as quantum polynomial rings arising in noncommutative alegebraic geometry. Moreover, we point out that the graded stable categories o such rings are usually not triangulated, and thus cannot always be realized as derived categories. While there are several natural starting points in studying categories o graded modules, we opt or the category gr Λ o initely generated Z-graded right Λ-modules and degree 0 morphisms. We then deine the graded stable category o Λ to be 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. Although we shall deal exclusively with graded modules in this article, the general setting o locally noetherian graded algebras permits us to view the classical results on stable equivalence o artin algebras as a special case. Given an artin algebra Λ, we may o course regard it as a locally artinian graded algebra that is concentrated in degree 0. In this case, the category gr Λ can be identiied with mod(λ) (Z), where mod(λ) is the usual stable category. It is thus clear that i Λ and Γ are stably equivalent artin algebras, then they are also graded stably equivalent when considered as graded algebras concentrated in degree 0. Consequently, many o our results are in act generalizations o classical results due to Auslander, Reiten [4] and the second author [18]. While our proos still make use o unctor categories, they are necessarily a bit more complicated in order to navigate around the absence o initely generated injectives in the categories we consider. Furthermore, it ollows rom the above identiication gr Λ mod(λ) (Z) that i two artin algebras Λ and Γ, concentrated in degree 0, are graded stably equivalent, then they are stably equivalent in the usual sense. Whether or not an analogous result holds or artin algebras with more interesting gradings appears to be a diicult problem. According to [9], such questions can also be phrased in terms o covering theory, where they amount to asking whether a (nice) stable equivalence between Z-coverings o Λ and Γ induces a stable equivalence between Λ and Γ. 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 no 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, and we then proceed to show that these short exact sequences are actually preserved. 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 to the case where the monomorphism actors through a projective, 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. Finally, the last section is devoted to 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 noetherian k- algebras or a commutative semilocal noetherian ring k, complete with respect to its Jacobson radical m, unless noted otherwise. Recall that a positively graded k-algebra Λ = i 0 Λ i is locally noetherian i each 2

3 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. We will reer to such algebras as graded quiver algebras. 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. We call these algebras generalized graded quiver algebras ater [22]. 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 (however, we will occasionally need to rule out this possibility in order to ensure that the graded simple Λ-modules are initely presented, and this is the case i and only i Λ is initely generated as a k-algebra). 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 n i 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. 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, we note that most 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. For lack o a good reerence, we now take a moment to prove some elementary propositions on locally noetherian graded algebras. We will see that in most cases these acts ollow easily rom the analogous results or noetherian algebras, many o which can be ound in [14]. 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 3

4 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 noetherian 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. Proo. By Theorem (21.35) o [14], the Krull-Schmidt theorem holds or.g. Λ 0 -modules. Thus, it is easy to see that i M can be written as a direct sum o n nonzero modules, n cannot be larger than the number o indecomposable summands in a decomposition o the Λ 0 -module M/MΛ 1. For (b), notice that End Λ (M) is.g. as a k-module by the remarks above. I M is indecomposable, we know that End Λ (M) has no nontrivial idempotents, and is hence local according to Proposition (21.34) o [14]. 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 noetherian 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. Proo. We irst note that Λ 0 is semiperect by Theorem (23.8) o [14], since we have Λ 0 = EndΛ0 (Λ 0 ) = End Λ0 ( n i=1 P i) where the decomposition o Λ 0 as the direct sum o indecomposable projectives P i exists by the Krull-Schmidt theorem or.g. Λ 0 -modules. Let m 1,..., m n be homogeneous generators or M, deine M = m i Λ 0, and let π : P M be a projective cover over Λ 0. In act, P may be graded so that π is a morphism o graded modules when M is given the grading induced by that o M. Now consider the composite P Λ0 Λ π 1 M Λ0 Λ M, where is given by (m i λ) = m i λ. Clearly P Λ0 Λ is a graded projective Λ-module, the kernel o is contained in M Λ0 Λ 1, and the preimage o this submodule under π 1 is contained in P (rad Λ 0 ) Λ0 Λ 0 P Λ0 Λ 1 = (P Λ0 Λ)J Λ. Lemma 1.3 (Nakayama s lemma). Let Λ be a locally noetherian 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. Proo. Without loss o generality, we may suppose that M i = 0 or all i < 0 and M 0 0. Then M 0 = (MJ) 0 = M 0 (rad Λ 0 ), and since M 0 is.g. over Λ 0, the usual version o Nakayama s lemma implies that M 0 = 0, a contradiction. The second claim ollows rom the irst applied to M/N. 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 noetherian algebra are J-adically complete. Lemma 1.4. Let Λ be a locally noetherian 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 ). Proo. We have mλ 0 rad Λ 0 and (rad Λ 0 ) s mλ 0 or some integer s, since Λ 0 /mλ 0 is a inite-dimensional algebra over the residue ield k/m. Thus the powers o the two ideals J = rad Λ 0 Λ 1 and L = mλ 0 Λ 1 deine the same topology on Λ-modules, and it suices to show that the natural map M lim M/ML n is an isomorphism. Thus let N be a graded Λ-module with maps η n : N M/ML n or all n 1, such that η n 1 = π n η n, where π n is the projection M/ML n M/ML n 1. For a ixed degree d, we have induced maps η n : N d (M/ML n ) d and π n : (M/ML n ) d (M/ML n 1 ) d, which we regard as morphisms between k-modules. Let us assume that M i = 0 or all i < 0, so that n > d implies (ML n ) d M d m. In act, we have M d m n (ML n ) d M d m n d or all n > d. Since M d is a.g. k-module, it is complete with respect to 4

5 the iltration {M d m n } n 1, and thus the η n s induce a unique k-module morphism ϕ d : N d M d such that η n Nd = π n ϕ d or all n. It remains to show that the ϕ d deine a Λ-module morphism. Thus, let λ Λ e, and consider the pair o k-module morphisms x xλ ϕ d+e (xλ) and x ϕ d (x) ϕ d (x)λ rom N d to M d+e. Since each η n is a Λ-module morphism, these two composites induce the same maps when composed with the projections π n : M d+e (M/ML n ) d+e. By the argument o the preceding paragraph, it ollows that any map rom N d to M d+e with this property is unique, and the two composites above must be equal. 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. I we regard M as a Z-graded k-module, and E as a graded k-module concentrated in degree 0, then we can express this duality as DM = i Z Hom k (M[i], E) = i Z Hom k (M i, E). In addition, D restricts to dualities l..gr + Λ l.a.gr Λ op, 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. 5

6 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 [8] 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 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 noetherian 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. Proo. (i) (ii) : I S is projective, then Q = S is simple and the conclusion is obvious. Let π : P P/(soc P +(QJ)) be the projection, and notice that π actors through the projection rom Q to S, yielding 6

7 the ollowing commutative square. Q P h π S P/(soc P + (QJ)) g I g is a split monomorphism (or instance, i S is injective), then the projection rom Q to S would actor through, orcing to be an isomorphism, which is a contradiction. Consequently, we may assume that S is a node, and there is thus a map h such that g = πh. But the image o h will be in soc P, implying that g = πh = 0. Hence, π = 0 and (Q) ker(π) = soc P + (QJ). Now (QJ) (Q)J (QJ)J implies that (QJ) = (QJ)J. Since (QJ) is in l..gr + Λ, Nakayama s lemma yields (QJ) = 0, and the image o must be simple. (ii) (i) : Assume that S is not projective or injective, and that h : S M is a nonsplit monomorphism. Then h lits to a morphism : Q P M where π M : P M M is the projective cover o M. I were to split, it would induce a splitting o h. Thus, since cannot split, composing it with the projections o P M onto its indecomposable summands yields nonisomorphisms between indecomposable projectives. By (ii), it ollows that (Q) soc P M, and thus actors through π S : Q S, yielding = gπ S or g : S P M. Hence, hπ S = π M = π M gπ S, and thus h = π M g, as required. The equivalence o (ii), (iii) and (iv) is clear, and we shall omit the details. Now assume that S is initely copresented, so that we may talk about the almost split sequence starting in S. (i) (v) : I S is not injective, we have an almost split sequence 0 S B g C 0. Here, C must be.p., and thus B is.g. By (i), actors through the projective cover π : P B, via a map h : S P. Since does not split, neither can h, and thus h actors through. It ollows that π is an isomorphism, so B is projective. The converse is clear (even without any assumptions on S or Λ). 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/a)i 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 noetherian. 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. More precisely, this identiication yields an equivalence o categories (we reer the reader to [5] or more details). 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. Namely, or Hom Λ (X, Y ), F () = (, Xa ), where denotes the induced map X/Xa Y/Y a. We shall write gr S Λ or the ull subcategory o gr Λ consisting o those modules with no summand in add(s), and gr S Γ will denote the ull subcategory o gr Γ consisting o those modules with no summand isomorphic to (0, T, 0) or T add(s). Notice that α(x) gr S Γ or any Λ-module X. Furthermore, notice that α 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 gr S Γ is ull and dense, 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, 7

8 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. 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 g P C P A B (π C,0) (p,π A ) g C 0 C 0 i 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 (ΩC)J 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 ]. 8

9 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. 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 3.2 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, 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 u v αb P 0 αc 0 π 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. 9

10 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.g. injective modules. Our irst main result in this direction is the ollowing. 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 [5]. 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 [5], 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 10

11 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 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. 11

12 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, 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 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 Q i and ( ) t 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. Since we have a short exact sequence 0 αa 4.4 applies to the sequence A j α 1 V τ 0 A Y A A B t g C j α 1 V τ C i V σ αc 0 with i right minimal and σ 0, Lemma 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. 0 A s1 s 2 q 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 ) 12 w l

13 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 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. 13

14 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