Abelian categories. triangulated categories: Some examples.

Transcription

1 Abelian categories versus triangulated categories: Some examples Claus Michael Ringel Trondheim, 000 For Idun Reiten on the occasion of her 70th birthday

2 Comparison Abelian categories Rigidity Factorizations are usually uniquley determined Triangulated categories Flexibilty Factorization are usually not unique Few symmetries A lot of symmetries There are too many abelian categories There are few triangulated categories

3 What is an abelian category? an additive category satisfying some conditions a Frobenius category? an additive category with an exact structure satisfying some conditions a triangulated category? an additive category with additional data satisfying some conditions

4 Mutual constructions From abelian categories to triangulated categories: Example: If A is abelian, form the derived categories D(A) or D b (A) From triangulated categories to abelian categories: Example: If T is triangulated, consider functor categories, say all functors from T to modk, or functors preserving some structure In both cases, starting with small categories, we obtain quite large ones Aim: to look for abelian/triangulated categories which are close to each other, say one obtained from the other by factoring out some finitely generated ideal The typical examples to have in mind: Let Γ be a self-injective artin algebra: modγ is an abelian category mod Γ/ Γ is a triangulated category In the converse direction C a cluster category, T a cluster tilting object in C: C is a trinagulated category Then C/ T is abelian

5 From: Some Remarks concerning Tilting Modules and Tilted Algebras (in: Handbook of Tilting Theory) Footnote : We have mentioned that the cluster theory brought many surprises Here is another one! One knows since long time many examples of abelian categories A with an object M such that the category A/ M (obtained by setting zero all maps which factor through add M) becomes a triangulated category: just take A = modr, where R is a self-injective artin algebra R and M = R R The category modr/ R R = modr is the stable module category of R But we are not aware that non-trivial examples where known of a triangulated category D with an object N such that D/ N becomes abelian Cluster tilting theory is just about this

6 Outline of the Lecture: Part I From abelian categories to triangulated categories A abelian, I fg ideal in A and A/I triangulated Part II From triangulated categories to abelian categories T triangulated, I fg ideal in T and T /I abelian

7 The source of the examples: Let Γ be an artin algebra mod Γ the category of all fg Γ-modules, this is an abelian category G(Γ) the subcategory of all Gorenstein-projective Γ-modules P(Γ) the subcategory of all projective Γ-modules G(Γ)/P(Γ) is a triangulated category Recall: A Gorenstein-projective module M is of the form M = Imd 0, where P = (P,d ) is an exact complex with all modules P i projective, such that also Hom(P,Γ) is exact We are mainly interested in d-gorenstein algebras Γ (ie the injective dimension of Γ as a left and as a right module is at most d) Lemma Let Γ be d-gorenstein The following assertions are equivalent: (i) M is Gorenstein-projective, (ii) M = Ω d (N) for some Γ-module N (iii) M is torsionless and Ext i (M,Γ) = 0 for i < d (iv) Ext i (M,Γ) = 0 for i The Gorenstein-projectives (as defined by Enochs-Jenda) are the maximal Cohen-Macaulay modules (Buchweitz, Auslander-Buchweitz)

8 An example Γ α β γ αγβ = 0 = βαγ P(Γ) modγ G(Γ)

9 Remark The study of Gorenstein-projective modules is often subsumed under the heading of relative homological algebra This may lead to a misdirection! Given an artin algebra Γ, the subcategory G(Γ) is one of the basic objects attached to Γ, it is a kind of nucleus of the module category modγ (We really deal with an absolute concept, which does not rely on any choice)

10 G(Γ) is an additive category When is G(Γ) an abelian category? Given any additive category A, we may ask whether it is abelian or not Warning If G(Γ) is abelian, the embedding G(Γ) modγ may not be exact and usually is not exact Theorem (Auslander-Solberg, Kong) Let Γ be an artin algebra Then G(Γ) is an abelian category iff the dominent dimension of Γ is at least and Γ is -Gorenstein, thus iff there is an exact sequence 0 Γ Γ I 0 I I 0 with I 0,I projective and injective, and I injective Auslander-Solberg: Γ -Gorenstein, dom dim Γ, then G(Γ) is abelian The converse is part of the thesis of Fan Kong (SJTU)

11 When is G(Γ) abelian and the embedding G(Γ) modγ exact? Answer: Only in case Γ is self-injective (and thus G(Γ) = modγ) Namely: If G(Γ) is abelian, then it is equivalent to a module category modλ and the embedding is given by a functor Hom Λ (M, ): modλ modγ where M is a generator-cogenerator with End(M) op = Γ and with further properties (see below) But if Hom Λ (M, ) is exact, then M has to be projective Thus Λ has a projective cogenerator, therefore Λ is self-injective, and Hom Λ (M, ) is a Morita-equivalence

12 Our example Λ α β αβ = 0 Γ α β γ αγβ = 0 = βαγ modλ G(Γ) M = Hom(M, )

13 Kong s theorem: main steps of the proof Theorem Let Γ be an artin algebra such that G = G(Γ) is abelian Then there is an exact sequence 0 Γ Γ I 0 I I 0 with I 0,I projective and injective, and I injective (a) G is closed under kernels, since P G (b) Ω X G for any module X (c) If P is projective and U P is a submodule in G, then P/U is torsionless In particular, all modules of projective dimension at most are torsionless (d) The torsionless modules are closed under extensions (e) Hom(X,Γ) = 0 implies Ext (X,Γ) = 0 (f) P projective = I(P) projective (g) M = I(Γ)/Γ has projective dimension at most, thus is torsionless, thus I(M) is projective (thus domdimγ )

14 Endomorphism rings of generator-cogenerators These are the artin algebras Γ with domdimγ Also, these are the QF- rings which are maximal quotient rings This topic is very central in representation theory But priority questions are highly contested! Or: Therefore priority questions are highly contested! Some references: Thrall (definition of QF- rings rings, 948) Findlay-Lambek (maximal ring of quotients, 958) Morita (958) Tachikawa (definition of the dominant dimension, 964) Morita-Tachikawa Bruno Müller Maurice Auslander (Wedderburn correspondance) etc

15 Endomorphism rings of generator-cogenerators Theorem There is a bijection between Pairs (Λ,M) where Λ is a basic artin algebra and M is a basic generatorcogenerator in mod Λ Pairs (Γ,I), where Γ is a basic artin algebra with domdimγ and I is a minimal faithful projective-injective Γ-module The bijection is given by (Λ,M) End(M) op, (Γ,I) End(I) op, and the functor Hom(M, ): modλ modγ yields yields an equivalence between mod Λ and the full subcategory C(Γ) of all Γ-modules N with an exact sequence 0 N I 0 I where I 0,I are projective-injective Γ-modules

16 Auslander-Solberg Let Λ be an artin algebra with Auslander-Reiten translation τ An indecomposable Λ-module is called transjective provided its τ ± -orbit contains a projective or an injective module periodic provided τ t M = M for some t Theorem(Auslander-Solberg) Let Λ be an artin algebra and M a generatorcogenerator Then: End(M) is -Gorenstein if and only if τm addm if and only if the indecomposable direct summands in add M are the transjective modules and the modules from a finite number of periodic τ-orbits In particular: If End(M) is -Gorenstein, then Λ is transjective-finite (or, in the terminology of Auslander-Solberg, D Tr-injective) Observe: If Γ is -Gorenstein and domdimγ, then C(Γ) = G(Γ) Thus Hom(M, ) furnishes an equivalence between mod Λ and G(Γ)

17 -Gorenstein algebras with dominant dimension Let P(a) be indecomposable projective with minimal injective coresolution ( ) 0 P(a) I 0 I I 0 Then either I = 0, thus also I = 0 and P(a) is injective, orelsei isindecomposable andthesequence isaminimalprojectiveresolution If Γ is -Gorenstein with dominant dimension, we thus get a bijection between the indecomposable projective modules P(a) which are not injective and the indecomposable injective modules I(b) which are not projective Write Γ = End( Λ M) op, with M generator-cogenerator If X is indecomposable injective, then P(a) = Hom(M, X) is projective-injective, If X M is indecomposable and not injective, a minimal injective copresentation 0 X J 0 J yields the left part of the sequence ( ), namely 0 Hom(M,X) Hom(M,J 0 ) Hom(M,J ) and the cokernel is just DHom(τ X,M) = I(b) Thus, the bijection P(a) I is given by X τ X

18 Abelian categories as Frobenius categories The Auslander-Solberg-Kong theorem provides examples of the form A abelian, I = M A/I triangulated where M is in A and M is the ideal of all maps which factor through addm Recipe for obtaining triangulated categories mod Λ/ M Start with any transjective-finite artin algebra Λ Let M 0 be the direct sum of all transjective indecomposable Λ-modules For i t, let O i be a periodic τ-orbit say of lengths s i : O i = {M i,τm i,,τs i M i = M i }, let M i = s i j=0 τ j M i Let M = M 0 M M t and Γ = End(M) op Then Hom(M, ) yields an equivalence between modλ and G(Γ), under this equivalence, add M is mapped onto P(Γ) Thus G/P modλ/ M

19 Our example Λ α β αβ = 0 Γ α β γ αγβ = 0 = βαγ M = modλ Hom(M, ) G(Γ) Instead of M, we also could take M = and deal with Γ = End(M ) op (an artin algebra with 5 vertices) For Γ, we have G(Γ ) = P(Γ )

20 Another example Let Λ be a local self-injective algebra of length 4 and Loewy length, for example k[α,β]/ α,β,αβ βα Let M = Λ R where R is any regular Λ-module (not necessarily indecomposable) Recall that all regular Λ-modules R satisfy τr = R Let Γ = End(M) op Then Γ is -Gorenstein with dominant dimension, G(Γ) is equivalent to modλ under Hom(M, ) Under this equivalence, P(Γ) corresponds to add M, thus the Auslander-Reiten quiver of G/P is obtained from that of modλ by deleting the indecomposable projective module Λ as well as the indecomposable direct summands of R

21 G(Γ) modλ R Λ G/P

22 Transjective-finite artin algebras All representation-finite artin algebras are transjective-finite All self-injective artin algebras are transjective-finite (Auslander-Solberg): The Auslander algebra of a local uniserial algebra is transjective-finite Further examples: with P() projective-injective τ I() = P(), τ I() = P() Summery: For any transjective finite artin algebra Λ, and M the direct sum of all transjective indecomposable modules, mod Λ/ M is equivalent to some G(Γ)/P(Γ), thus a triangulated category And factoring out periodic τ-orbits, we obtain again categories of the form G(Γ)/P(Γ) thus again triangulated categories

23 A general result Zhu Bin has pointed out to me that there is the following general result due to Beligiannis and Jørgensen: Proposition Let k be a field Let T a triangulated k-category with finite-dimensional Hom-spaces, split idempotents and Auslander-Reiten translation functor τ Let X be a functorially finite full subcategory with τx = X Then T / X is triangulisable Observe that this means: Given such a triangulated category T, we may factor out any finite τ-orbit and obtain again a triangulated category Thus we may obtain a wealth of triangulated categories as factor categories Compare this with the situation of dealing with an abelian category A It is rather seldom the case that given a subcategory X of A, the factor category A/ X is abelian again

24 Comparison Abelian categories There are few abelian categories Triangulated categories There are too many triangulated categories

25 What is an abelian category? an additive category satisfying some conditions a Frobenius category? an additive category with an exact structure satisfying some conditions a triangulated category? an additive category with additional data satisfying some conditions

26 What is an abelian category? an additive category satisfying some conditions a Frobenius category? an additive category with a full subcategory (the projective-injectives) satisfying some conditions a triangulisable category? an additive category satisfying some conditions and we are interested just in additive categories and subquotients of them

27 Part II From triangulated categories to abelian categories We are looking for: (joint work with Pu Zhang) T triangulated, I fg ideal in T T /I abelian Cluster categories and cluster-tilting objects provide such examples with I idempotent Here we want to present examples where I is nilpotent: Thus T and T /I have the same indecomposable objects, but there is a slight difference between the homomorphisms

28 The setting k a field, Q a finite connected directed quiver A = k[ǫ] = k[t]/t the algebra of dual numbers over k Γ = kq k k[ǫ] = kq[ǫ] = kq[t]/t Γ-modules = representations of Q over A Γ-modules are just pairs (M, ǫ), where M is a kq-module and ǫ: M M an endomorphism with ǫ = 0, thus a differential kq-module Recall: A differential kq-module (M,ǫ) is called perfect provided M is a projective kq-module There is the homology functor H: modγ modkq, if M = (M i,m α ) i Q0,α Q is a representation of kq over k[ǫ], then H(M) = (H(M i ),H(M α )) i,α

29 The Gorenstein-projective Γ-modules Γ is a -Gorenstein algebra (the injective dimension of Γ as a left or as a right module is equal to ) Note that for a -Gorenstein algebra Γ, the module Γ Γ is a cotilting module (of injective dimension at most ) It follows: Lemma Let M be a Γ-module The following conditions are equivalent: (i) M is Gorenstein projective (ii) M is torsionless (ii) Ext (M,Γ) = 0 (iv) M is a perfect differential kq-module P the category of the projective Γ-modules G the category of Gorenstein-projective Γ-modules Again, we are interested in G/P (a triangulated category)

30 Main theorem Main theorem The homology functor H is a full and dense functor H: G modkq It kills the modules in P, and the kernel I of the induced functor H: G/P modkq is a finitely generated ideal I which satisfies I = 0 Thus, here we have: G/P a triangulated category, mod kq an abelian category, and a full and dense functor H with nilpotent kernel

31 The theorem asserts: H: G/P modkq is an equivalence Actually: H: G \P modkq has nice properties: it maps indecomposable objects to indecomposable objects, it maps non-isomorphic objects to non-isomorphic objects, it yields a bijection between the indecomposables in G\P and in modkq Recall that a lot is known about the structure of modkq, thus about G/P Thus, we are looking for the inverse construction: G/P η mod kq

32 The construction η Let N be a kq-module 0 Ω 0 N P 0 (N) N 0 0 Ω 0 N k A ηn N 0 0 Ω 0 N ΩN 0 The map ηn N is the Gorenstein approximation of N P 0 N Ω 0 N P 0 N Ω 0 N Ω 0 N k A

33 The embedding of the AR-quiver of kq into the AR-quiver of G For any vertex x of Q, let P 0 (x) and I 0 (x) be the indecomposable projective or injective kq-module, respectively The AR-quiver of G/P is obtained from the AR-quiver of modkq by adding for every arrow α: i j in Q an arrow I 0 (i) P 0 (j) (a ghost map ) To obtain the AR-quiver of G, we have to add the indecomposable projective Γ-modules P(x) Easy: radp(x) belongs to G and is indecomposable, actually H(rad P(x)) = S(x) (the simple module)

34 j Q i modq S(i) I 0 (j) P 0 (i) S(j) G/P ηs(i) ηi 0 (j) P 0 (i) S(j)

35 G/P ηs(i) ηi 0 (j) P 0 (i) S(j) G P ǫ (i) ηs(i) P 0 (i) ηi 0 (j) S(j) P ǫ (j)

36 A second example: The linearly oriented quiver Q of type A The AR-quiver of modkq as a subquiver of ZA : The universal covering of G(kQ[ǫ]):

37 Theorem The kernel of H: G/P modkq is generated by the maps ηi(x) P(y), where x,y are vertices of Q Actually: any arrow α: x y in Q yields a specific map g α : ηi(x) P(y) (a ghost map ), and the kernel of H is generated by these ghost maps

38 R ring P R category of finitely generated projective R-modules C b (P R ) the category of perfect complexes, K b (P R ) the corresponding homotopy category If R has finite global dimension, then K b (P R ) is just D b (modr) For R = kq, we obtain the following commutative diagram C b (P kq ) π K b (P kq ) γ G kq[ǫ] π γ G kq[ǫ] The vertical functors π are just the stabilization functors The horizontal functors γ are forgetful functors (forgetting the grading, pushdown functor in covering theory)

39 C b (P kq ) π K b (P kq ) γ G kq[ǫ] π γ G kq[ǫ] The category C b (P kq ) is locally bounded, thus γ is dense H 0 : C b (P kq ) modkq (the homology at the position 0) provides a bijection between the shift orbits of the indecomposable objects in K b (P kq ) and the indecomposable kq-modules

40 Remarks on the homology functor H We have seen: H has nice properties when restricted G \P It maps indecomposable modules to indecomposables, non-isomorphic modules to non-isomorphic ones In general, one cannot expect such a pleasant behavior Example Q quiver of type A with two sources and a sink Then the AQ module M = (k A k) is indecomposable, but H(M) = (k 0 k) is decomposable Example Q the Kronecker quiver (two arrows from to ) There is a P -family of indecomposable AQ modules M with M = k and M = A For all of them H(M) is the simple module S()

41 Remark Buchweitz (987): Given a Gorenstein algebra Γ, then D b (modγ)/dperf b (modγ) G(Γ) Orlov (00) proposed the name triangulated category of singularities for this Verdier quotient For Γ = kq[ǫ], we see: the triangulated category of singularities is just D b (modkq)/[]

42 Recall our aim: To look for triangulated/abelian categories which are close to each other, say one obtained from the other by factoring out some finitely generated ideal Here we have such a pair of categories which differ only by a square-zero ideal: always circular G/P for Γ = kq[ǫ] modkq globally directed for Dynkin quivers: directed

43 References M Auslander, R Buchweitz The homology thetheory of Cohen-Macaulay approximations Soc Math France Mem 8 (989), 57 M Auslander, I Reiten Cohen-Macaulay and Gorenstein Artin algebras In Representation theory of finite groups and finite-dimensional algebras, Birkhäuser (99), 45 M Auslander, Ø Solberg: Gorenstein algebras and algebras with dominant dimension at least Comm Alg (99), R Buchweitz: Maximal Cohen-Macaulay modules and Tate-cohomology Ms (unpublished) D Happel: Triangulated categories in the representation theory of finite dimensional algebras, LMS Lecture Notes Vo9, Cambridge University Press (988) D Happel: On Gorenstein algebras In Representation theory of finite groups and finite-dimensional algebras, Birkhäuser (99) P Jørgensen: Quotients of cluster categories Proc Roy Soc Edinburgh Sect A 40 (00), 65-8 F Kong: Comparison between D Tr-selfinjective algebras and representationfinite algebras Preprint D Orlov: D Orlov Triangulated categories of singularities and D-branes in Landau-Ginzburg models Proc Steklov Inst Math 46 () (004), 748 (also: arxiv:math/0004) C M Ringel: Artin algebras of dominant dimension at least ringel/opus/domdimpdf C M Ringel, P Zhang: Representations of quivers over the algebra of dual numbers arxiv:94