RECOLLEMENTS AND SINGULARITY CATEGORIES. Contents

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1 RECOLLEMENTS AND SINGULARITY CATEGORIES DONG YANG Abstract. This is a report on my ongoing joint work with Martin Kalck. The recollement generated by a projective module is described. Application to singularity categories is discussed. Contents 1. Recollements 1 2. Recollements generated by projectives 2 3. Consequences of Theorem Classical singularity categories 6 5. DG algebras and their derived categories 9 6. Ginzburg dg algebras 11 References 13 This is the note of my talk given at the Morningside Center on 27 May I would like to thank Bin Zhu, Jie Xiao and Bangming Deng for the invitation. 1. Recollements A recollement of triangulated categories is a diagram of triangulated categories and triangle functors i j! T i =i! T j! =j T i! j such that 1) (i, i = i!, i! ) and (j!, j! = j, j ) are adjoint triples; 2) j!, i = i!, j are fully faithful; 3) j i = 0; 4) for every object X of T there are two triangles i! i! X X j j X Σi i! X 1

2 2 DONG YANG and j! j! X X i i X Σj! j X, where the four morphisms are the units and counits. This notion was introduced by Beilinson Bernstein Deligne Gabber [2]. 2. Recollements generated by projectives Let A be a finite-dimensional algebra, and e A an idempotent. Consider the standard diagram i D(A/AeA) i =i! D(A) j! =j j! D(eAe), i! j where i =? L A A/AeA, i = RHom A/AeA (A/AeA,?), i! =? L A/AeA A/AeA, j! =? L eae ea, j! = RHom A (ea,?), j =? L A Ae, i! = RHom A (A/AeA,?), j = RHom eae (Ae,?). It is not difficult to check that 1) 3) and j-part of 2) in the definition of recollement hold. According to [6], the ideal AeA is called a stratifying ideal if the canonical morphism Ae L eae ea AeA is an isomorphism or, equivalently, the quotient map A A/AeA induces a fully faithful triangle functor D(A/AeA) D(A) (i.e. the i-part of 2) holds). Theorem 2.1 (Cline Parshall Scott [6]). Let A be an algebra. The standard diagram associated to an idempotent e A is a recollement if and only if AeA is a stratifying ideal. Theorem 2.2 (Cline Parshall Scott [5]). Let A be a finite-dimensional quasi-hereditary algebra, and I be the partially ordered set indexing the isoclasses of simple modules. Let J I be an ideal, i.e. x J, y > x implies that y J, and e = e J the corresponding idempotent. Then AeA is a stratifying ideal. Proof. Under the assumptions, the right A-module AeA lies in add(ea). Thus from the triangle in D(A) AeA A A/AeA ΣAeA we see that A/AeA, considered as an object of D(A), has no self-extensions. Consequently, the restriction functor D(A/AeA) D(A) is fully faithful.

3 3 Example 2.3. Let A be given by the quiver with relation 1 α 2, αβ = 0. β Then A is quasi-hereditary with respect to the order 2 > 1. Thus the standard diagram associated to e 2 is a recollement. However, the ideal Ae 1 A is not stratifying, because Ext 2 A/Ae 1 A (A/Ae 1A, A/Ae 1 A) = 0, while Ext 2 A (A/Ae 1A, A/Ae 1 A) = k. Example 2.4 (Dlab Ringel). Let A be a finite-dimensional hereditary algebra, and I be an index set of the isoclasses of simple A-modules. Then A is quasi-hereditary for any partial order on I. Consequently, for any idempotent e A, the associated standard diagram is a recollement. Let A be an algebra and e A an idempotent. Theorem 2.1 says that when the ideal AeA is not stratifying the associated standard diagram is not a recollement. Paying some price, we can extend it to a recollement. We need dg algebras and their derived categories (see Section 5). Let Q be a graded quiver such that the vertex set Q 0 is finite. Let kq be the complete path algebra of Q over a fixed field k, let e i denote the trivial path at the vertex i, and let m be the ideal of kq generated by the arrows. Let d : kq kq be a continuous homogeneous linear map of degree 1 satisfying d(ab) = d(a)b + ( 1) p ad(b) for a, b kq, where a is homogeneous of degree p, d(e i ) = 0, for any i Q 0, d(ρ) m 2, for any ρ Q 1. Let à be the dg algebra ( kq, d). Let us call it a free dg algebra. If there is a quasiisomorphism à A, we say that A admits a cofibrant minimal model Ã. Let e A be an idempotent. We lift it to the sum of the corresponding trivial paths of some vertices of Q. Theorem 2.5. Let A be an algebra admitting a cofibrant minimal model à and e A an idempotent. There is a recollement i D(Ã/ÃeÃ) i =i! D(A) j! =j j! D(eAe). i! j This is an immediate consequence of

4 4 DONG YANG Lemma 2.6. Let à be a free dg algebra, and e the sum of trivial paths corresponding to some vertices. Then the standard diagram associated to e i D(Ã/ÃeÃ) i =i! D(Ã) j! =j j! D(eÃe) i! j is a recollement. Corollary 2.7. There is a triangle equivalence D(Ã)/ Tria(eÃ) = D(Ã/ÃeÃ), inducing a triangle equivalence between the idempotent completion of per(ã)/ thick(eã) and per(ã/ãeã). Here for a triangulated category T and an object X T, thick(x) denotes the smallest triangulated subcategory of T containing X and closed under taking direct summands. Remark 2.8. This generalizes a result in [13, Section 7], see also [7]. Let A be an algebra. Assume that A is finite-dimensional, or 3. Consequences of Theorem 2.5 A is semilocal complete noetherian of finite global dimension. Then A admits a cofibrant minimal model Ã. This follows from a bar-cobar formalism, see [7]. Precisely, let S be the direct sum of a complete set of representatives of simple A-modules, and let A = Ext (S, S) be the A -Koszul dual of A, i.e. algebra equipped with A -structure. Let Q be the graded Ext-quiver, i.e. A is the Yoneda its vertices are the isoclasses of simple modules, and the number of arrows of degree p 0 from S i to S j equals the dimension of Ext p+1 (S i, S j ). Then à = ( kq, d), where Q is the graded dual quiver of Q, and d is given by the A -structure on Ext >0 (S, S). In the following two cases à can be explicitly described A is Koszul, A is some algebra arising from geometry, e.g. A is the complete preprojective algebra of a non-dynkin quiver, or A is a 3-Calabi Yau Jacobian algebra (see Section 6), since in these cases the A -Koszul dual of A can be given explicitly. Another class of algebras which we can describe their cofibrant minimal models is those of global dimension 2. Proposition 3.1 (Keller [13]). Let A be an algebra of global dimension 2. Assume that A = kq / I, where I = {r 1,..., r s } is a set of minimal relations and I is the closure of the ideal generated by I. Let Q be the graded quiver such that

5 Q 0 = Q 0, Q 1 consists of two types of arrows, which respectively lie in degree 0 and degree 1, degree 0: arrows from Q 1, degree 1: ρ t : i j for r t : i j, t = 1,..., s. Let d : kq kq the continuous homogeneous linear map of degree 1 satisfying d(ab) = d(a)b + ( 1) p ad(b) for a, b kq, where a is homogeneous of degree p, d(e i ) = 0, for any i Q 0, d(ρ) = 0, for any ρ Q 1, d(ρ t ) = r t, for any t = 1,..., s. Then à = ( kq, d) is the cofibrant minimal model of A. Example 3.2. Let A be given by the quiver with relation 5 1 α 2, αβ = 0. β The global dimension of A is 2. By Proposition 3.1, the cofibrant minimal model à = ( kq, d) of A is given by Q = 1 α β 2 γ, d(γ) = αβ, d(α) = 0 = d(β). Here α and β are of degree 0 and γ is of degree 1. From now on we assume further that A is of finite global dimension and admits a cofibrant minimal model Ã. Let e A be an idempotent and we lift it to an idempotent of Ã. Let B = Ã/ÃeÃ. Recall from Theorem 2.5 that there is a recollement i D(B) i =i! D(A) j! =j j! D(eAe). i! j Ming Fang pointed out that one should be able to recover the dominance dimension of A from B if taking e such that ea is maximal injective. Let D sg (A, e) := (per(a)/ thick(ea)), the idempotent completion of the triangle quotient per(a)/ thick(ea). Theorem 3.3. a) i induces a triangle equivalence D sg (A, e) per(b). b) j induces a fully faithful triangle functor D fd (A)/i D fd (B) D fd (eae).

6 6 DONG YANG c) B = (1 e)a (1 e). d) B is homologically smooth, i.e. B per(b op B). Consequently, per(b) D fd (B). e) H p (B) = 0 for any p > 0 and H 0 (B) = A/AeA. f) If A/AeA is finite-dimensional, then H p (B) is finite-dimensional for any p Z and per(b) is Hom-finite, i.e. all the Hom-spaces in per(b) are finite-dimensional. Conjecture 3.4. There is a triangle equivalence per(b)/d fd (B) thick(ae)/ per(eae). Theorem 3.5. The Grothendieck group of per(a)/ thick(ea) is free. Its rank equals the rank of the number of isoclasses of simple A/AeA-modules, and it is generated by the indecomposable direct summands of (1 e)a. This can be proved on the Koszul dual side using a result of [14, 16] on positive A - algebras. 4. Classical singularity categories Let R be a local complete noetherian commutative ring. Let mod R denote the category of finitely generated R-modules. According to Buchweitz [3], the singularity category D sg (R) of R is the idempotent completion of the triangle quotient D b (mod R)/ per(r): D sg (R) := (D b (mod R)/ per(r)). Assume that R is Gorenstein. Then the category MCM(R) of maximal Cohen Macauley modules is Frobenius with projective-injective objects being the free modules of finite rank and there is a triangle equivalence D sg (R) = MCM(R) (Buchweitz [3]). If further R is an isolated singularity, then MCM(R) is Hom-finite (Auslander?). If R = k[[x 1,..., x d ]]/f is a hypersurface singuularity, then D sg (R) = MF (f), the stable category of matrix factorization of f (Eisenbud). Take T = R M MCM(R) basic, and let A = End R (T ). If gldim(a) <, then A is a non-commutative crepant resolution of R in the sense of Van den Bergh [18] (see [7]). The following three cases are of particular interest MCM(R) has finitely many isoclasses of indecomposable objects, and T is the direct sum of a complete set of representatives. T = R m, where m is the maximal ideal of R. T is a higher cluster tilting object in MCM(R). From now on assume that A has finite global dimension. Let à be the cofibrant minimal model of A. Let e = 1 R be the identity of R, considered as an element of A and Ã. Then

7 7 eae = R and A/AeA = End MCM(R) (T ). By Theorem 2.5, there is a recollement i D(Ã/ÃeÃ) i =i! D(A) j! =j j! D(R). i! j The restricted functor j! =? L R ea : per(r) per(a) is fully faithful and takes R to ea. Thus the triangle quotient per(a)/ thick(ea) measures the difference between the singularity R and its resolution A. This is an invariant of R studied by Kalck Burban [4]. Theorem 4.1 (Burban Kalck [4]). The triangle quotient per(a)/ thick(ea) is idempotent complete (hence the singularity category D sg (A, e) defined in the preceding section is per(a)/ thick(ea)). Moreover, its Grothendieck group is free of rank the number of indecomposable direct summands of M. On the other hand, the functor j =? L A Ae takes A to Ae = T and ea to eae = R. Therefore it induces a triangle functor per(a)/ thick(ea) thick(t )/ per(r) D b (mod R)/ per(r), i.e. D sg (A, e) D sg (R) = MCM(R). Recall from Theorem 3.3 that per(b) = D sg (A, e) and per(b) D fd (B). Theorem 4.2 (Thanhoffer de Völcsey Van den Bergh [7]). Assume that R is an isolated singularity. Then a) c) d) e) of Theorem 3.3 hold. Further, f) per(b) is Hom-finite. g) The above triangle functor per(b) MCM(R) induces a triangle equivalence per(b)/d fd (B) MCM(R). Example 4.3 (Burban Kalck [4]). Let R = k[[x, y]]/xy be the 1-dimensional singularity of type A 1. Then MCM(R) has three indecomposable objects up to isomorphism. The Auslander Reiten quiver of MCM(R) is where M + = R/y and M = R/x. M + R M

8 8 DONG YANG Let T = R M + M. Then A = End R (T ) is given by the quiver with relations + α β γ δ, δα = 0, βγ = 0. The cofibrant minimal model à = ( kq, d) of A is given by (see Proposition 3.1) ζ + α β η γ δ, d(ζ) = δα, d(η) = βγ. Here α, β, γ, δ are of degree 0 and ζ, η are in degree 1. The dg algebra B = Ã/Ãe à is the path algebra of the graded quiver + ζ, η where ζ and η are of degree 1. Let T = R M + (or R M ). Then T is a cluster-tilting object in MCM(R), and A = End R (T ) is given by the quiver with relations 1 α β 2 γ, γα = 0, βγ = 0. The simple module S 1 is a 3-spherical object, so (1 e 2 )A (1 e 2 ) = Ext (S 1, S 1 ) = k[s]/s 2 is the graded algebra with s in degree 3. The dual bar construction yields B = Ã/Ãe 2à = k[t] with t in degree 2. Thus by Theorem 4.2 g) we obtain that MCM(R) is triangle equivalent to per(b)/d fd (B), which is the cluster category of type A 1. Example 4.4. Let R be a Kleinian singularity. Namely, R = S G is the algebra of G- invariant functions, where the finite group G SL 2 (C) acts naturally on S = C[[x, y]]. It is known that R is a hypersurface singularity R = C[[x, y, z]]/f and that R admits an ADE classification, as shown in the following table. G type f cyclic A n (n 1) x 2 + y n+1 + z 2 binary dihedral D n (n 4) x 2 y + y n 1 + z 2 binary tetrahedral E 6 x 3 + y 4 + z 2 binary octahedral E 7 x 3 + xy 3 + z 2 binary icosahedral E 8 x 3 + y 5 + z 2 There are only finitely many indecomposable objects in MCM(R) up to isomorphism. Let T be the direct sum of them. Then A = End R (T ) is isomorphic to the smash product S G, and is Morita equivalent to the complete preprojective algebra of an/any Euclidean quiver Q of type X, where X is the type of R. The cofibrant minimal model à of A is the

9 derived preprojective algebra of Q (and the 2-Calabi Yau completion of kq) in the sense of Keller [13]. Let Q be the quiver obtained from Q by deleting an/the extending vertex (i.e. the vertex corresponding to the summand R of T ). Then the associated dg algebra B is the derived preprojective algebra of Q (and the 2-Calabi Yau completion of the kq ). The stability conditions on D fd (B) was considered by Bridgeland. Example 4.5 (Amiot Iyama Reiten [1], Thanhoffer de Völcsey Van den Bergh [7]). Let G SL 3 (C) be a finite subgroup. It acts naturally on S = C[[x, y, z]]. Let R = S G be the algebra of invariants. Then T = R S is a cluster tilting object in MCM(R). The endomorphism algebra A = End R (T ) is isomorphic to the smash product S G. It is Morita equivalent to a 3-Calabi Yau Jacobian algebra J(Q, W ), where Q is the McKay quiver, and W is a generic potential. For example, for G = diag(ζ 3, ζ 3, ζ 3 ), the McKay quiver Q is 9 and for G = diag(ζ 5, ζ5 2, ζ2 5 ), the McKay quiver Q is Here ζ n is an n-th root of unity. Since A = J(Q, W ) is 3-Calabi Yau, it follows that the Ginzburg dg algebra à = Γ(Q, W ) is the cofibrant minimal model of A, see Section 6. Let (Q, W ) be the quiver with potential obtained from (Q, W ) by deleting the vertex corresponding to the summand R of T. Then the associated dg algebra B is the Ginzburg dg algebra Γ(Q, W ). Thus by Theorem 4.2 g), the category MCM(R) is triangle equivalent to Amiot s generalized cluster category MCM(R) = C (Q,W ) := per( Γ(Q, W ))/D fd ( Γ(Q, W )). 5. DG algebras and their derived categories We follow [9, 10]. Let k be a field. A dg (k-)algebra A is a graded algebra endowed with the structure of a complex with differential d such that the graded Leibniz rule holds: d(ab) = d(a)b + ( 1) p ad(b),

10 10 DONG YANG where a, b A and a is homogeneous of degree p. A (right) dg module M over A is a graded (right) A-module endowed with the structure of a complex with differential d M such that d(ma) = d M (m)a + ( 1) p md(a), where m M is homogeneous of degree p and a A. Let A be a dg algebra. Let M and N be two dg A-modules. A homomorphism of dg modules from M to N is a chain map of complexes from M to N which commutes with the A-actions. It is null-homotopic if as a chain map it is homotopic to 0, and it is a quasi-isomorphism if as a chain map it is a quasi-isomorphism. Let H(A) be the homotopy category of dg A-modules: its objects are dg A-modules, and its morphism space Hom H(A) (M, N) is defined as the quotient of the space of homomorphisms of dg A-modules from M to N by the subspace of null-homotopic homomorphisms. H(A) is naturally a triangulated category with suspension functor being the shift of complexes. Let acyc(a) H(A) be the subcategory of acyclic dg A-modules. The derived category D(A) is defined as the triangle quotient D(A) := H(A)/ acyc(a). In fact, this definition can be extended to a recollement ([9, Theorem 3.1]) acyc(a) H(A) π i D(A) Here π is the canonical projection, p is the functor of taking projective resolution and i is the functor of taking injective resolution. Two nice and important triangulated subcategories of D(A) are per(a) = thick(a A ), D fd (A) = {M D(A) H p (M) is finite-dimensional}. p Z p For an (ordinary) algebra A, considered as a dg algebra concentrated in degree 0, we have D(A) = D(Mod A), per(a) = H b (proj A), D fd (A) = D b (mod A), where mod A is the category of finite-dimensional A-modules. If the dg algebra A has finite-dimensional total cohomology, then clearly per(a) D fd (A). Dually,

11 11 Lemma 5.1 (Keller [12]). If A is homologically smooth, then per(a) D fd (A). Let A and B be two dg algebras. Given a dg B-A-bimodule M, we can define a pair of adjoint triangle functors between homotopy categories H(B)? B M Hom A (M,?) H(A) We derive them to adjoint triangle functors between derived categories? L B M D(B) p? B M π B H(B) π A H(A) D(A) Hom A (M,?) i RHom A (M,?) Lemma 5.2. A quasi-isomorphism B A induces a triangle equivalence? L B A : D(B) D(A). 6. Ginzburg dg algebras Let k be a field. Let Q be a finite quiver (possibly with loops and 2-cycles). We denote its set of vertices by Q 0 and its set of arrows by Q 1. The trivial path corresponding to a vertex i will be denoted by e i. The complete path algebra kq is the completion of the path algebra kq with respect to the ideal generated by the arrows of Q. A potential on Q is an element of the closure of the space generated by all non trivial cycles of Q. For an arrow ρ and a cycle c of Q, we define ρ (c) = c=upv vu taken over all decompositions of the cycle c (where u and v are possibly trivial paths). Writing W = c:cycle λ cc, we define ρ (W ) = c:cycle λ c ρ (c). Let W be a potential on Q. The (complete) Ginzburg dg algebra Γ(Q, W ) of the quiver with potential (Q, W ) is constructed as follows [8]: Let Q be the graded quiver with the same vertices as Q and whose arrows are the arrows of Q (they all have degree 0), an arrow ρ : j i of degree 1 for each arrow ρ : i j of Q, a loop t i : i i of degree 2 for each vertex i of Q.

12 12 DONG YANG The underlying graded algebra of Γ(Q, W ) is the completion of the graded path algebra k Q in the category of graded vector spaces with respect to the ideal generated by the arrows of Q. Thus, the n-th component of Γ(Q, W ) consists of elements of the form p λ pp, where p runs over all paths of degree n. The differential of Γ(Q, W ) is the unique continuous linear endomorphism homogeneous of degree 1 which satisfies the graded Leibniz rule d(uv) = (du)v + ( 1) p udv, for all homogeneous u of degree p and all v, and takes the following values on the arrows of Q: d(ρ) = 0 for each arrow ρ of Q, d(ρ ) = ρ W for each arrow ρ of Q, d(t i ) = e i ( ρ Q 1 [ρ, ρ ])e i for each vertex i of Q. The A -Koszul dual of Γ(Q, W ) is the Kontsevich Soibelman A -algebra [17] associated to (Q, W ). It is easy to see Lemma 6.1. Let i Q 0 be a vertex of Q, and e = e i be the corresponding trivial path. Let (Q, W ) be the quiver with potential obtained from (Q, W ) by deleting the vertex i. Then Γ(Q, W )/ Γ(Q, W )e Γ(Q, W ) = Γ(Q, W ). The (complete) Jacobian algebra J(Q, W ) of the quiver with potential (Q, W ) is by definition the 0-th cohomology of the Ginzburg dg algebra Γ(Q, W ). Concretely we have J(Q, W ) = kq/ ρ W, ρ Q 1. Theorem 6.2. Let (Q, W ) be a quiver with potential. a) (Keller [11]) The Ginzburg dg algebra Γ(Q, W ) is (topologically) homologically smooth. In particular, per( Γ(Q, W )) D fd ( Γ(Q, W )). The Amiot cluster category is defined as C (Q,W ) := per( Γ(Q, W ))/D fd ( Γ(Q, W )). b) (Keller [13]) The triangulated category D fd ( Γ(Q, W )) is 3-Calabi Yau. c) (Ginzburg [8]) If D fd (J(Q, W )) is 3-Calabi Yau, then the canonical projection Γ(Q, W ) J(Q, W ) is a quasi-isomorphism. The relation among mutation of quivers with potential, derived equivalence of Ginzburg dg algebras, change of t-structures in 3-Calabi Yau triangulated categories, mutation of cluster-tilting objects and nearly Morita equivalence of nearby Jacobian algebras can be found in [15].

13 13 References [1] Claire Amiot, Osamu Iyama, and Idun Reiten, Stable categories of Cohen-Macauley modules and cluster categories, arxiv: [2] Alexander A. Beilinson, Joseph Bernstein, and Pierre Deligne, Analyse et topologie sur les espaces singuliers, Astérisque, vol. 100, Soc. Math. France, 1982 (French). [3] Ragnar-Olaf Buchweitz, Maximal Cohen-Macaulay modules and Tate-Cohomology over Gorenstein rings, preprint [4] Igor Burban and Martin Kalck, Singularity category of a non-commutative resolution of singularities, arxiv: [5] Edward Cline, Brian Parshall, and Leonard L. Scott, Finite-dimensional algebras and highest weight categories, J. reine ang. Math. 391 (1988), [6], Stratifying endomorphism algebras, Mem. Amer. Math. Soc. 124 (1996), no [7] Louis de Thanhoffer de Völcsey and Michel Van den Bergh, Explicit models for some stable categories of maximal Cohen-Macaulay modules, arxiv: [8] Victor Ginzburg, Calabi-Yau algebras, arxiv:math/ v3 [math.ag]. [9] Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, [10], On differential graded categories, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp [11], Pseudocompact dg algebras and derived categores, appendix to derived equivalence from mutations of quivers with potential, arxiv: v2. [12], Calabi-Yau triangulated categories, Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, pp [13], Deformed Calabi Yau-completions, doi: /crelle , in press. Also arxiv: [14] Bernhard Keller and Pedro Nicolás, Simple dg modules over positive dg algebras, arxiv: v2. [15] Bernhard Keller and Dong Yang, Derived equivalences from mutations of quivers with potential, Adv. Math. 226 (2011), no. 3, [16] Steffen Koenig and Dong Yang, On tilting complexes providing derived equivalences that send simpleminded objects to simple objects, arxiv: [17] Maxim Kontsevich and Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arxiv: [18] Michel Van den Bergh, Non-commutative crepant resolutions, The legacy of Niels Henrik Abel (Berlin), Springr, Berlin, 2004, pp