Introduction to τ-tilting theory

Osamu Iyamaa and Idun Reitenb,1

aGraduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan; and bDepartment of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway

Edited by Bernard Leclerc, University of Caen, Caen, France, and accepted by the Editorial Board January 22, 2014 (received for review September 17, 2013) From the viewpoint of mutation, we will give a brief survey of when dealing with categorification of cluster algebras. For cluster tilting theory and cluster-tilting theory together with a motivation categories (10), and more generally, for 2-Calabi–Yau trian- from cluster algebras. Then we will give an introduction to τ-tilting gulated categories (11), a class of objects called cluster-tilting theory which was recently developed. objects have the desired property (Theorem 9). For path algebras kQ, this result for cluster categories was translated to the fact 1. Introduction that the same holds for support tilting pairs (which generalize Let Λ be a finite dimensional algebra over an algebraically closed tilting modules) (12, 13) (Theorem 7). More generally, for k, for example k is the field of complex numbers. We always 2-Calabi–Yau triangulated categories we can translate to the assume that Λ is associative and has an identity. An important property that for the associated 2-Calabi–Yau tilted algebras class of algebras is the path algebras kQ for a finite Q, that the support τ-tilting pairs have the desired property for is, a finite-directed graph. The k-basis for kQ is the paths in Q. complements (section 5). This raised the question of a possible a b extension to support τ-tilting pairs for other classes of alge- For example, when Q is 1!2!3, then the paths are bras. In fact it turned out to be the case for any finite di- fe ; e ; e ; a; b; bag.Heree ; e ; e are the trivial paths, starting 1 2 3 1 2 3 mensional algebra (9) (Theorem 16). and ending at 1,2,3 respectively. Then multiplication in kQ is in- For any finite dimensional k-algebra Λ, in particular for the duced by composition of paths, so that for example b · a = ba, path algebras of Dynkin quivers, we have an associated AR-quiver, a · b = 0, a · e = a, a · e = 0. When Q is acyclic (i.e., contains no 1 2 where the vertices correspond to the isomorphism classes of the oriented cycles), then kQ is a finite dimensional k-algebra. Λ indecomposable modules. The arrows correspond to what is The of deals with the investigation of “ ” mod Λ Λ called irreducible maps, and the dotted lines indicate a certain the category of finitely generated left -modules. τ “ ” – called AR-translation (see section 4.1 for details). For We refer to refs. 1 3 for background on representation theory. a b One of the basic properties of mod Λ is the Krull–Schmidt example, for the path algebra kQ of the quiver 1!2!3, we property: Any module X is isomorphic to a finite direct sum have the following AR-quiver: X1⊕⋯⊕Xm of indecomposable modules X1; ...; Xm,whichare uniquely determined up to isomorphism and permutation. We say that X is basic if X1; ...; Xm are pairwise nonisomorphic, and we write jXj = m in this case. We assume that Λ is basic as a Λ-module. Then Λ is isomorphic to kQ=I for a finite quiver Q and an ideal I in kQ. In this case, let f1; ...; ng be the set of vertices in Q. Then jΛj = n holds, and we have a decomposition In this case there are 5 tilting modules Λ = P1⊕⋯⊕Pn,wherePi = Λei is an indecomposable projective Λ-module. Moreover the simple Λ-modules are S1; ...; Sn, where Si = Pi=rad Pi for the radical rad Pi of Pi.Afamous theorem of Gabriel tells us that a path algebra kQ of a con- nected finite acyclic quiver Q has only a finite number of nonisomorphic indecomposable modules if and only if the un- derlying graph of Q is one of the simply laced Dynkin diagrams where the edges indicate mutation. This picture is incomplete in A , D , E , E , E . n n 6 7 8 the sense that it is impossible to replace for example P1 in the An important class of modules has been the tilting modules, tilting module P ⊕P ⊕P to get another tilting module. Con- – 1 2 3 introduced more than 30 y ago (4 6) as a simultaneous general- sider now a slightly bigger category called the cluster category ization of progenerators in classical Morita theory and Bernstein- (section 3) with the following AR-quiver: Gelfand-Ponomarev (BGP) reflections for quiver representations = ⊕ ⊕ (7). They are a certain class of basic modules T T1 ⋯ Tn with Significance n = jΛj, where each Ti is indecomposable. BGP-reflections can be realized by a special class of tilting modules called “APR-tilting Mutation is an ingredient in the construction of cluster alge- modules” (8). One of the important properties of tilting modules bras. It deals with replacing an element in the free generating is that when we remove some direct summand Ti from T to get set (x1, x2, ..., xn) of the rational function field Q(x1, x2, ..., xn) T=Ti = ⊕j≠iTj (which is called an almost complete tilting module), p by another element. In module categories for finite dimen- then there is at most one indecomposable module Ti such that “ τ ” pl = ⊕ p sional algebras, what is called support -tilting modules have Ti Ti and ðT TiÞ Ti is again a tilting module (which is called n p a similar behavior. They are built up by parts, and there is an mutation of T). Then Ti (and Ti if it exists) are said to be the operation of mutation. This plays an important role in cate- = complement(s) of T Ti. Thus, any almost complete tilting module gorifying cluster algebras. has either one or two complements (Theorem 2),andmutationis possible only when there are two complements. Author contributions: O.I. and I.R. designed research, performed research, and wrote To make mutation always possible, it is desirable to enlarge the paper. our class of tilting modules to get the more regular property that The authors declare no conflict of interest. almost complete ones always have two complements. This is This article is a PNAS Direct Submission. B.L. is a guest editor invited by the Editorial accomplished by introducing τ-tilting modules, or more precisely, Board. support τ-tilting pairs (9). Such a property is of central interest 1To whom correspondence should be addressed. Email: [email protected].

9704–9711 | PNAS | July 8, 2014 | vol. 111 | no. 27 www.pnas.org/cgi/doi/10.1073/pnas.1313075111 Downloaded by guest on September 28, 2021 Recall that g : U′ → Y is a right ðaddUÞ-approximation if U′ ∈ addU and any map h : U → Y factors through g. Moreover SPECIAL FEATURE it is called (right) minimal if any map a : U′ → U′ satisfying ga = g is an isomorphism. Let again Q be the quiver 1 → 2 → 3 and kQ the corresponding path algebra. The vertex 3 of the quiver Q is a sink, that is, no arrow starts at 3. Then a reflection (or mutation)ofQ (7) at In this category there is an important class of objects called vertex 3 is defined by reversing all arrows ending at 3, to get the cluster-tilting objects, which generalize tilting modules. In this ′ = μ : → ← quiver Q 3ðQÞ 1 2 3 in this example. We have the fol- case there are 14 cluster-tilting objects lowing close relationship between mod kQ and mod kQ′. Theorem 4 (7). Let Q be a finite acyclic quiver with a sink i ′dμ and Q iðQÞ. Then there is an equivalence of categories : → F ESi ESi′ , where ESi (respectively, ESi′ ) is the full subcategory of modkQ (respectively, mod kQ′) consisting of modules without di- rect summands Si (respectively, S′i). −1 It was shown in ref. 8 that the kQ-module T = ðkQ=SiÞ⊕τ Si has the endomorphism algebra kQ′, and the above functor F is isomorphic to HomkQðT;− Þ. This T is a special case of what was later defined to be tilting modules. An important notion in tilting theory is a torsion pair, that is, a pair ðT ; FÞ of full subcategories of mod Λ such that (i) HomΛðT ; FÞ= 0 and (ii) for any X in mod Λ, there exists an exact sequence 0 → Y → X → Z → 0 with Y ∈ T and Z ∈ F. In this case T is a torsion class, that is, a subcategory closed under ⊕ extensions, factor modules, and isomorphisms; and F is a tor- where the symbol is omitted and the edges indicate mutation. sionfree class, that is, a subcategory closed under extensions, This picture is complete in the sense that we can replace any submodules, and isomorphisms. For any M in mod Λ, we have element in any cluster-tilting object to get another cluster-tilting ⊥ MATHEMATICS a torsion class MdfX ∈ mod ΛjHomΛðX; MÞ = 0g and a tor- object. This corresponds to the fact that we have exactly two ⊥ sionfree class M dfX ∈ mod ΛjHomΛðM; XÞ = 0g. Let Fac M complements for almost complete cluster-tilting objects in clus- (respectively, Sub M) be the subcategory of mod Λ consisting of ter categories (Theorem 9). factor modules (respectively, submodules) of direct sums of copies 2. Tilting Theory of M. The equivalence in Theorem 4 has the following interpre- tation in tilting theory. In this section we include some main results on tilting modules Theorem 5 (4, 5). Let T be a tilting module over a finite di- and related objects, focusing on when we have a well-defined op mensional k-algebra Λ and ΓdEndΛðTÞ . operation of mutation on them. We refer to refs. 2, 3, 14, and 15 ⊥ for more background on tilting theory. (a) T dFacT and FdT give a torsion pair ðT ; FÞ in mod Λ, and Xd⊥DT and YdSubDT (where D is the k-dual) give 2.1. Properties of Tilting Modules. Let as before Λ be a finite a torsion pair ðX; YÞ in mod Γ. dimensional k-algebra with n = jΛj.AΛ-module T is a partial (b) We have equivalences of categories HomΛðT; −Þ : T → Y and 1 tilting module if T has projective dimension at most one and ExtΛðT; −Þ : F → X. 1 ExtΛðT; TÞ = 0. It is a tilting module if moreover jTj = n.This condition can be replaced by the condition: There is an exact Note that in the above special case S3 is the only indecomposable kQ-module not in T = Fac T,andF = add S3. sequence 0 → Λ → T0 → T1 → 0withT0 and T1 in addT.HereaddT denotes the subcategory of mod Λ consisting of direct sum- There is a close connection between tilting modules and tor- mands of finite direct sums of copies of T.Thefollowing sion classes. We need an important notion of functorially finite subcategories (19). For torsion classes T , being functorially finite result due to Bongartz is basic in tilting theory. modΛ = Fac Theorem 1 (6). Any partial tilting module is a direct summand is equivalent to the existence of M in such that T M. We denote by tiltΛ the set of isomorphism classes of basic tilting of a tilting module. Λ Λ It is natural to ask how many tilting modules exist for a given -modules, and by ff-tors the set of faithful functorially finite torsion classes in mod Λ. partial tilting module. There is an explicit answer for an almost Theorem 6 – Λ complete tilting module, that is, a partial tilting module U satisfying (19 22). Let be a finite dimensional k-algebra. = − Then there is a bijection tilt Λ → ff-tors Λ given by T↦Fac T. jUj n 1. An indecomposable module X is a complement for an ; Λ almost complete tilting module U if U⊕ X is a tilting module. The Let ðT PÞ be a pair of -modules, where P is projective, Λ HomΛðP; TÞ = 0 and T is a partial tilting Λ-module. Then ðT; PÞ following result is basic in this paper, where a -module X is called Λ faithful if the annihilator ann Xdfa ∈ ΛjaX = 0g of X is zero. is called a support partial tilting pair for . It is called a support Theorem 2 (16–18). Let U be an almost complete tilting module. tilting pair (respectively, almost complete support tilting pair)if jTj + jPj = n (respectively, jTj + jPj = n − 1), where n = jΛj. For an (a) U has either one or two complements. indecomposable module X, we say that ðX; 0Þ [respectively, (b) U has two complements if and only if U is faithful. ð0; XÞ]isacomplement for an almost complete support tilting pair ðT; PÞ if ðT⊕X; PÞ [respectively, ðT; P⊕XÞ] is a support When there are exactly two complements, then there is a nice tilting pair. We then have the following result for path algebras. relationship between them. Theorem 7 (12, 13). Let Λ = kQ be a path algebra of an acyclic Theorem 3 (16, 17). Let X and Y be two complements for an quiver Q. Then any almost complete support tilting pair has pre- almost complete tilting module U. After we interchange X and Y if cisely two complements. f g necessary, there is an exact sequence 0!X !U′!Y !0 Note that Theorem 7 is not true for finite dimensional k-algebras a b where f is a minimal left ðaddUÞ-approximation and g is a minimal in general. For example, let Q be the quiver 1!2!3and right ðaddUÞ-approximation. ΛdkQ=hbai. In this case there are eight support tilting pairs for Λ

Iyama and Reiten PNAS | July 8, 2014 | vol. 111 | no. 27 | 9705 Downloaded by guest on September 28, 2021 3. Cluster Algebras and 2-Calabi–Yau Categories In this section we discuss cluster categories and 2-Calabi–Yau triangulated categories, which were motivated by trying to cate- gorify the essential ingredients of the cluster algebras of Fomin– Zelevinsky (32, 33). We restrict to considering what is called “acyclic cluster algebras,” with no coefficients. They are associated with a finite acyclic quiver with n vertices. Essential concepts in the theory of cluster algebras are clusters, cluster variables, seeds, and quiver mutation. where the symbol ⊕ is omitted and edges indicate mutation. For Let Q be a finite connected quiver having no loops or two- μ example, the almost complete support tilting pair ðP1⊕P3; 0Þ cycles. For a vertex i of Q,wedefineanewquiver iðQÞ having does not have two complements. no loops or two-cycles, called mutation of Q,asfollows:(i) There are two important quivers whose vertices are tilting For any pair a : j → i and b : i → k of arrows in Q,wecreate modules. One is the Hasse quiver of tilt Λ, where we regard tilt Λ a new arrow ½ba : j → k. (ii) Reverse all arrows starting or as a partially ordered set by defining T ≥ U when Fac T ⊃ Fac U. ending at i. (iii) Remove a maximal disjoint set of two-cycles. → → μ → ← Thus, we draw an arrow T → U if T > U and there is no V ∈ tilt Λ For example, if Q is 1 2 3, then 3ðQÞ is 1 2 3and μ satisfying T > V > U. The other is the exchange quiver of tiltΛ.We 2ðQÞ is . draw an arrow X⊕U → Y⊕U when X and Y are complements of = Q ; ...; U and there is an exact sequence 0 → X → U′ → Y → 0 as given in Fix a function field F ðx1 xnÞ in n variables over the Q ; ...; ; Theorem 3. Here we have the following. field of rational numbers . Then ðfx1 xng QÞ is an initial ; ...; Theorem 8 (23). The Hasse quiver and the exchange quiver seed, consisting of the pair of the free generating set fx1 xng coincide. of F over Q and the acyclic quiver Q. Then for each i = 1; ...; n we define a new seed 2.2. Generalizations of Tilting Modules. Tilting modules discussed in μ ; ...; ; d ; ...; p ; ...; ; μ ; the previous subsection are often called “classical” tilting mod- iðfx1 xng QÞ x1 xi xn iðQÞ ules, and there are many generalizations. Here we are mainly p + = m1 m2 interested in cases where there is some kind of mutation, which is where xi , with m1 and m2 being certain monomials in ; ...; ; xi ; ...; given via approximation sequences. x1 xi−1 xi+1 xn determined by the quiver in the seed. We μ ; ...; μ We have a more general class of tilting modules of finite perform this process for all 1 n, and then on the new μ2 = = ; ...; projective dimension, which by definition are Λ-modules T with seeds, etc. It is easy to see that i id holds for i 1 n. This the following properties: (i) T has finite projective dimension, (ii) gives rise to a graph where the vertices are the seeds, and the j edges between them are induced by the μ . The graph may be ExtΛðT; TÞ = 0 for j > 0, and (iii) there is an exact sequence i 0 → Λ → T → ⋯ → T → 0, where all T are in add T (14, 24). finite or infinite. 0 m i The n element subsets of F obtained in this way are called The notion of almost complete tilting modules and their com- clusters, and the elements of clusters are called cluster variables. plements are defined in the same way as before. In this case The associated is the subalgebra of F generated by there are in general more than two complements, and all of the all of the cluster variables. There is only a finite number of clusters complements are usually connected by approximation sequences (or equivalently cluster variables) if and only if the quiver Q (25). There are other aspects of such tilting modules which have is Dynkin. been investigated more, like connections with functorially finite One approach to the study of cluster algebras is via so-called – subcategories, with applications to algebraic groups and Cohen categorification. This means that we want to find some nice cat- – Macaulay representations (26 28). egories, like module categories or triangulated categories, where One of the important properties of tilting modules of finite we have some objects with similar properties as clusters and projective dimension is that they induce a derived equivalence cluster variables. In particular we are looking for indecompos- Λ op between and EndΛðTÞ . From this viewpoint of derived able objects corresponding to cluster variables. Morita theory, the notion of tilting complexes introduced in As we have seen in section 1, the tilting modules almost have ref. 29 is most important. Also here there are usually more than μ the desired properties, but the operation i on tilting modules is two complements for almost complete ones. From the viewpoint not always defined. This motivated the introduction of cluster of mutation theory, the notion of silting complexes (30) is more categories CQ associated with a finite acyclic quiver (10). These natural than that of tilting complexes. We denote by Kbðproj ΛÞ categories have more indecomposable objects than mod kQ, and the homotopy category of bounded complexes of finitely generated also more morphisms between the old objects. They are defined projective modules. A tilting complex (respectively, silting complex) as orbit categories DbðkQÞ=τ½−1, where DbðkQÞ is the bounded Kb proj Λ ; = of mod kQ and τ : DbðkQÞ → DbðkQÞ is a de- is an object P in ð Þ satisfying HomKbðproj ΛÞ ðP P½iÞ 0 rived AR-translation. For example, the AR-quiver of the cluster for any i ≠ 0 (respectively, i > 0) and thick P = Kbðproj ΛÞ.Here category of type A was given in section 1. The category C is thick Kb proj Λ 3 Q P is the smallest full subcategory of ð Þ which Hom-finite, and also triangulated (34), and we have a functorial ± contains P and is closed under cones, ½ 1, direct summands, isomorphism DExt1 ðX; YÞ’Ext1 ðY; XÞ (10). In general a CQ CQ and isomorphisms. There is a mutation theory analogous to the k-linear Hom-finite C is called 2-Calabi–Yau case of tilting modules (31). One has operations of left and 1 ; 1 ; if there is a functorial isomorphism DExtCðX YÞ’ExtCðY XÞ. right mutations obtained using approximation sequences, which Hence CQ is 2-Calabi–Yau. shows that there are always infinitely many complements for an We can regard the tilting kQ-modules as special objects in CQ, almost complete silting complex. Moreover there is a natural partial but this class of objects is not large enough since mutation is not order whose Hasse quiver coincides with the exchange quiver. If always possible. Instead, if we consider all tilting modules coming we restrict to silting complexes which are “two-term,” then al- from algebras derived equivalent to kQ, we actually get a class of most complete silting complexes have exactly two complements objects such that mutation is always possible. These objects can (9) (Corollary 24). be described directly as follows.

9706 | www.pnas.org/cgi/doi/10.1073/pnas.1313075111 Iyama and Reiten Downloaded by guest on September 28, 2021 We say that an object T in a 2-Calabi–Yau triangulated 4. τ-Tilting Theory 1 ; = SPECIAL FEATURE category C is rigid if ExtCðT TÞ 0. It is maximal rigid if it is In this section we discuss the central results of τ-tilting theory, rigid, and moreover if T⊕X is rigid for some X in C,thenX is including those analogous to results from tilting theory. Let as add 1 ; = in T.ArigidobjectT in C is cluster-tilting if ExtCðT XÞ 0 before Λ be a finite dimensional k-algebra. for some X in C implies that X is in add T. Clearly any cluster- tilting object is maximal rigid, and the converse is true for the 4.1. Definition and Basic Properties. We have dualities D = Homk – op p cluster category (10), and also for more general 2-Calabi Yau ð−; kÞ : mod Λ↔mod Λ and ð−Þ dHomΛð−; ΛÞ : proj Λ ↔ p triangulated categories [Corollary 32(b)]. A basic object U in proj Λop which induce an equivalence νdDð−Þ : proj Λ → inj Λ C is an almost complete cluster-tilting object if there exists an called Nakayama functor. For X in mod Λ with a minimal pro- indecomposable object X in C such that U⊕ X is a basic cluster d1 d0 tilting object. In this case X is as before called a comple- jective presentation P1 !P0 !X !0, we define Tr X in mod Λop τ mod Λ ment for U. and X in by the exact sequences We have the following result for cluster-tilting objects in d p ν 2-Calabi–Yau triangulated categories, which improves Theorem p 1 p τ ν d1 ν : P0 !P1 !Tr X !0 and 0 ! X ! P1 ! P0 2 for tilting modules. Theorem 9 (10, 11). Let C be a 2-Calabi–Yau triangulated cat- Then τ (respectively, Tr) gives bijections between the isomor- egory. Any almost complete cluster-tilting object in C has precisely Λ two complements. phism classes of indecomposable nonprojective -modules and the isomorphism classes of indecomposable noninjective Λ-mod- It was mentioned in section 1 that when we deal with support op μ ules (respectively, nonprojective Λ -modules). More strongly, tilting modules over path algebras kQ, then the mutation i is = ; ...; τ and Tr give an equivalence and a duality, respectively, between defined for all i 1 n. To prove this, one uses the corre- mod Λ sponding result for cluster-tilting objects in cluster categories and the stable categories. We denote by the stable category modulo projectives and by mod Λ the costable category modulo interprets the condition Ext1 ðT; TÞ = 0 and jTj = n for T in CQ injectives. Then Tr gives a duality Tr : mod Λ↔ mod Λop called a cluster category as a condition for mod kQ. the transpose,andτ gives an equivalence τ : mod Λ → mod Λ called Further basic results about cluster categories were developed. the AR-translation. Moreover we have a functorial isomorphism They were of interest in themselves, and also helped to establish a closer connection with acyclic cluster algebras, in a series of 1 ; ; τ MATHEMATICS papers including refs. 35–39. This led to the result that there is Hom ΛðX YÞ’DExtΛðY XÞ a map C → A for the acyclic cluster algebra A associated with Q X Y mod Λ AR-duality the quiver Q, called cluster character, inducing a bijection be- for any and in called . There is an important class of modules, which were introduced tween indecomposable rigid objects in C and the cluster varia- Q by Auslander–Smalo (19) (see also refs. 1 and 2) almost at the bles in A. We now give some classes of 2-Calabi–Yau triangulated cat- same time when tilting modules were introduced in ref. 4, then later almost forgotten: We call T in mod Λ τ-rigid if HomΛðT; τTÞ = 0. In egories with cluster-tilting objects, in addition to the cluster 1 ; = categories. this case we have ExtΛðT TÞ 0 (i.e., T is rigid) by AR-duality. Now let us call T in mod Λ τ-tilting if T is τ-rigid and jTj = jΛj. (1) Let WQ be the Coxeter group associated with a finite acyclic We call T in mod Λ support τ-tilting if there exists an idempotent ∈ quiver Q (40, 41). For any element w WQ, there is associ- e of Λ such that T is a τ-tilting ðΛ=heiÞ-module. – ated a 2-Calabi Yau triangulated category Cw as follows. Let For example, any local algebra Λ has precisely two basic support Π be the preprojective algebra associated with Q (section τ-tilting modules Λ and 0. It follows from AR-duality that any ; ...; 4.4). Let e1 en denote the idempotent elements associ- partial tilting module is τ-rigid, and any tilting module is τ-tilting. ; ...; ated with the vertices 1 n. Then we associate with i the However, the converse is far from being true. In fact, a self- = Π − Π Π = ⋯ ideal Ii ð1 eiÞ in , and the ideal Iw Ii1 Iit with an injective algebra Λ usually has a lot of τ-tilting modules (e.g., ∈ = ⋯ element w WQ with a reduced expression w si1 sit .Let section 4.4) even though it has a unique basic tilting module Λ. Π = Π= Sub Π Π w Iw and w be the category of w-modules which a b For another example, let Q be the quiver 1 2 3 and are submodules of a finite direct sum of copies of Πw.Then ! ! ΛdkQ= ba . Then T = P ⊕S ⊕P is a τ-tilting Λ-module which the stable category Cwd Sub Πw is a 2-Calabi–Yau triangu- h i 1 1 3 lated category. In particular, the stable category mod Π, is not a tilting Λ-module. where Π is the preprojective algebra of a Dynkin quiver, The next observation gives a more precise relationship be- belongs to this class. It was studied by Geiss et al. (42), who tween tilting modules and τ-tilting modules, where a Λ-module X Λ also independently studied a subclass of Cw (43). We will see is called sincere if any simple -module appears in X as a com- in section 4.4 that, in the Dynkin case, the ideals Iw play a role position factor. Clearly any faithful module is sincere. in τ-tilting theory. Proposition 10 (9). (2) There is a generalization of the cluster categories CQ to so- called generalized cluster categories CΛ associated with alge- (a) Tilting modules are precisely faithful (support) τ-tilting modules. Λ : Λ ≤ bras with gl dim 2andCQ;W associated with quivers (b) Any τ-tilting (respectively, τ-rigid) Λ-module T is a tilting (re- with potential ðQ; WÞ (44). Under a certain finiteness condition spectively, partial tilting) ðΛ=annTÞ-module. on Λ and ðQ; WÞ, the generalized cluster category is a 2-Calabi– (c) τ-tilting modules are precisely sincere support τ-tilting modules. Yau triangulated category with cluster-tilting object. (3) When R is a 3-dimensional isolated Gorenstein singular- When T is a ðΛ=heiÞ-module for some idempotent e, it is often ity with k ⊂ R, then the stable category CM R of maximal useful to compare properties of T as ðΛ=heiÞ-module with prop- Cohen–Macaulay R-modules is a 2-Calabi–Yau triangu- erties of T as a Λ-module. lated category. More concrete examples are invariant rings Proposition 11 (9). Let e be an idempotent of Λ and T ∈ G mod Λ= τ Λ τ R = k½½X; Y; Z , where G is a finite subgroup of SL3ðkÞ ð heiÞ. Then T is a -rigid -module if and only if T is a -rigid acting freely on k3nf0g. For other examples, see ref. 45. These ðΛ=heiÞ-module. 2-Calabi–Yau triangulated categories have quite different There is a nice relationship between torsion classes and sup- origin from those in (2) above, but it was shown in ref. 46 port τ-tilting modules, which improves Theorem 6. We denote by that they are often equivalent. sτ-tiltΛ the set of isomorphism classes of basic support τ-tilting

Iyama and Reiten PNAS | July 8, 2014 | vol. 111 | no. 27 | 9707 Downloaded by guest on September 28, 2021 Λ-modules, and by f-torsΛ the set of functorially finite torsion Note that it is easy to understand when we have a complement classes in mod Λ. of ðU; PÞ of the form ð0; XÞ and how to find it. This occurs precisely Theorem 12 (9). There is a bijection sτ-tiltΛ → f-torsΛ given by when U is not a sincere ðΛ=heiÞ-module for an idempotent e of Λ T↦Fac T. satisfying add P = addðΛeÞ. In this case there exists a primitive This gives a natural way of extending a τ-rigid module to a idempotent e′ of Λ such that U is a sincere ðΛ=he + e′iÞ-module, τ-tilting module, similar to Bongartz’s result (Theorem 1): For and ð0; Λe′Þ is a complement of ðU; PÞ. any τ-rigid module U, we have an associated functorially finite The next result gives a relationship between complements of ⊥ torsion class ðτUÞ.ByTheorem 12, there exists a support the form ðX; 0Þ, which is an analog of Theorem 3 for almost τ-tilting module T such that Fac T= ⊥ðτUÞ. It is easy to show that complete tilting modules. T is a τ-tilting module satisfying U ∈ add T. Consequently we Theorem 17 (9). Let ðX; 0Þ and ðY; 0Þ be two complements of have the following result. an almost complete support τ-tilting pair ðU; PÞ. After we in- Theorem 13 (9). Any τ-rigid module is a direct summand of a terchange X and Y if necessary, there is an exact sequence τ-tilting module. f g X !U′!Y !0, where f is a minimal left ðaddUÞ-approxi- The next result gives several characterizations of τ-tilting mation and g is a minimal right ðaddUÞ-approximation. modules. This is important because it has a result from ref. 47 A main difference from Theorem 3 is that f above is not nec- listed in Corollary 32 as a special case. essarily injective. Theorem 14 (9). The following conditions are equivalent for a Now we introduce an important partial order on the set τ-rigid Λ-module T. sτ-tiltΛ.Wehaveabijectionsτ-tiltΛ → f-torsΛ given by (a) Tisτ-tilting. T↦Fac T in Theorem 12. Then inclusion in f-torsΛ gives rise to (b) Tismaximalτ-rigid, i.e., if T⊕Xisτ-rigid for some Λ-module X, a partial order on sτ-tiltΛ, i.e., we define then X ∈ add T. (c) ⊥ðτTÞ = Fac T. T ≥ U (d) If HomΛðT; τXÞ = 0 and HomΛðX; τTÞ = 0, then X ∈ add T. if and only if Fac T ⊃ Fac U. Clearly ðΛ; 0Þ is the maximum ele- τ τ In addition to -rigid and support -tilting modules, it is also ment and ð0; ΛÞ is the minimum element of sτ-tiltΛ. We give convenient to consider associated pairs of modules. basic properties. ; ∈ mod Λ ∈ proj Λ Let ðT PÞ be a pair with T and P . We call Proposition 18 (9). ðT; PÞ a τ-rigid pair if T is τ-rigid and HomΛðP; TÞ = 0. We call − † : τ Λ → τ‐ Λop ðT; PÞ a support τ-tilting pair if ðT; PÞ is τ-rigid and jTj + jPj = jΛj. (a) The bijection ð Þ s -tilt s tilt reverses the partial The notion of support τ-tilting module and that of support order. ; ′; ′ τ Λ τ-tilting pair are essentially the same since, for an idempotent e (b) Let ðT QÞ and ðT Q Þ be support -tilting pairs for which Λ ; Λ τ are mutations of each other. Then we have either ðT; QÞ > of , the pair ðT eÞ is support -tilting if and only if T is a ′; ′ ; < ′; ′ τ-tilting ðΛ=heiÞ-module (compare with Proposition 11). More- ðT Q Þ or ðT QÞ ðT Q Þ. over e is uniquely determined by T in this case. Now we have a recipe to calculate mutation of support – τ Next we discuss the left right symmetry of support -tilting τ-tilting pairs, which is slightly more complicated than that for modules. This is somewhat surprising since it does not have an tilting modules because the map f in Theorem 17 is not nec- mod Λ = analog for tilting modules. We decompose T in as T essarily injective. Let ðU; PÞ be an almost complete support ⊕ Tpr Tnp where Tpr is a maximal projective direct summand of τ-tilting pair with a completion ðT; QÞ.Wegiveamethodto τ ; Λ T.Fora -rigid pair ðT PÞ for ,let calculate another completion ðT′; Q′Þ of ðU; PÞ from ðT; QÞ, whichisdividedintotwocases(A)and(B)below,depending ; †d ⊕ p ; p : ; ′; ′ ðT PÞ TrTnp P Tpr on which of ðT QÞ and ðT Q Þ is bigger. This can be checked without knowing ðT′; Q′Þ since ðT; QÞ > ðT′; Q′Þ holds if and We denote by τ-rigidΛ the set of isomorphism classes of basic only if T = U⊕X for an indecomposable module X satisfying τ-rigid pairs of Λ. X ∉ FacU. † Theorem 15 (9). ð−Þ gives bijections τ-rigidΛ↔τ‐rigid Λop and ; > ′; ′ op †† (A) Assume ðT QÞ ðT Q Þ holds. Then we take a minimal left sτ-tiltΛ↔sτ‐tilt Λ such that ð−Þ = id. f ðaddUÞ-approximation X !U′ of X. If f is not surjective, 4.2. Mutation and Partial Order. Now we give a basic theorem in the then the cokernel of f is an indecomposable module Y mutation theory for support τ-tilting modules, which improves (Theorem 17), and we have ðT′; Q′Þ = ðU⊕Y; PÞ.Iff is sur- Theorem 2 for tilting modules. We say that a τ-rigid pair ðU; PÞ is jective, then there exists an indecomposable projective mod- ′; ′ = ; ⊕ almost complete support τ-tilting if jUj + jPj = jΛj − 1. In this case, ule Y such that ðT Q Þ ðU P YÞ. ; < ′; ′ ; † > ′; ′ † for an indecomposable module X, we say that ðX; 0Þ [respec- (B) Assume ðT QÞ ðT Q Þ holds. Since ðT QÞ ðT Q Þ holds ′; ′ † ; † tively, ð0; XÞ]isacomplement of ðU; PÞ if ðU⊕X; PÞ [respectively, by Proposition 18(a), we can calculate ðT Q Þ from ðU PÞ U; P⊕X ] is a support τ-tilting pair, which we call a completion by using the left approximation, as we observed in (A). Ap- ð Þ − † ′; ′ † ′; ′ of ðU; PÞ. plying ð Þ to ðT Q Þ , we obtain ðT Q Þ. Theorem 16 Λ a b (9). Let be a finite dimensional k-algebra. Then For example, let Q be the quiver 1!2!3andΛd any almost complete support τ-tilting pair has precisely two kQ=hbai. First we caclulate μ ; ðΛ; 0Þ.SinceP2 ∉ FacðΛ=P2Þ, complements. ðP2 0Þ we use (A) above. The minimal left ðadd Λ=P2Þ-approximation Two completions ðT; QÞ and ðT′; Q′Þ of an almost complete of P is P → P , which has cokernel S .Thus,μ ; ðΛ; 0Þ = τ ; 2 2 1 1 ðP2 0Þ support -tilting pair ðU PÞ are called mutations of each other. P ⊕S ⊕P ; ′; ′ = μ ; ′; ′ = μ ð 1 1 3 0Þ. We write ðT Q Þ ðX;0ÞðT QÞ [respectively, ðT Q Þ ð0;XÞ μ ⊕ ; Next we calculate ð0;P3ÞðP1 S1 P3Þ. We must use (B) in this ðT; QÞ]ifðX; 0Þ [respectively, ð0; XÞ] is a complement of ðU; PÞ † p p p p case. We have ðP ⊕S ; P Þ = ððTr S Þ⊕P ; P Þ = ðS′⊕P ; P Þ, giving rise to ðT; QÞ. 1 1 3 1 3 1 2 3 1 a b ′ Λop where Si is the simple -module associated with the ver- For example, let Q be the quiver 1!2!3andΛdkQ=hbai. p p p p μ p ′⊕ ; = ′; ⊕ μ Λ; = ⊕ ; μ Λ; = ⊕ ⊕ ; tex i. Now we have ðP ;0ÞðS2 P3 P1Þ ðS2 P1 P3Þ. Thus, Then ðP ;0Þð 0Þ ðP2 P3 P1Þ, ðP ;0Þð 0Þ ðP1 S1 P3 0Þ, 3 μ 1 Λ; = ⊕ ⊕ ; 2 μ ⊕ ; = ′; p⊕ p † = ⊕ ⊕ ; and ðP3;0Þð 0Þ ðP1 P2 S2 0Þ. ð0;P3Þ ðP1 S1 P3Þ ðS2 P1 P3Þ ðP1 S1 P3 0Þ.

9708 | www.pnas.org/cgi/doi/10.1073/pnas.1313075111 Iyama and Reiten Downloaded by guest on September 28, 2021 The following result shows a strong connection between mu- The following result shows that g-vectors determine τ-rigid tation and partial order. pairs, which is an analog of (ref. 51; Theorem 2.3). SPECIAL FEATURE Theorem 19 (9). For T; U ∈ sτ-tiltΛ, the following conditions Theorem 27 (9). We have an injective map from the set of iso- are equivalent. morphism classes of τ-rigid pairs for Λ to K0ðprojΛÞ given by T; P ↦gT − gP (a) T and U are mutations of each other, and T > U. ð Þ . (b) T > U and there is no V ∈ sτ-tiltΛ such that T > V > U. 4.4. Examples. In this section, we use a convention to describe 1 We define the exchange quiver of sτ-tiltΛ as the quiver whose modules via their composition series. For example, 2 is an in- Λ set of vertices is sτ-tiltΛ and we draw an arrow from T to U if U is decomposable -module X with a simple submodule S2 such that = = a mutation of T such that T > U. The following analog of The- X S2 S1. For example, each simple module Si is written as i. a b orem 8 is an easy consequence of Theorem 19. Let Q be the quiver 1!2!3 and Λ = kQ=hbai. Then Corollary 20 τ Λ (9). The exchange quiver of s -tilt coincides with P = 1 , P = 2 , and P = 3 in this case. The Hasse quiver of τ Λ 1 2 2 3 3 the Hasse quiver of the partially ordered set s -tilt . sτ-tiltΛ is the following. Another application of Theorem 19 is the following. Corollary 21 (9). If the support τ-tilting quiver has a finite connected component, then it is connected. We end this subsection with the following generalization of Theorem 16, where Γ is given naturally by using Bongartz com- pletion in Theorem 13. Theorem 22 (48). Let ðU; PÞ be a τ-rigid pair for Λ, and τ Λ τ Λ τ s -tiltðU;PÞ the subset of s -tilt consisting of support -tilting pairs which have ðU; PÞ as a direct summand. Then there is a bijection τ Γ → τ Λ Γ s -tilt s -tiltðU;PÞ for some finite dimensional k-algebra with jΓj = jΛj − jUj − jPj. When ðU; PÞ is an almost complete support τ-tilting pair, the algebra Γ is local and hence sτ-tiltΓ = fΓ; 0g holds. Thus, Theorem where the symbol ⊕ is omitted. For example, 12 is 1 ⊕ 2 16 is a special case of Theorem 22. 233 2 3 MATHEMATICS ⊕ 3 = P1⊕P2⊕P3. a a a 4.3. Connection with Silting Complexes. In this subsection, we ob- Let Q be the quiver 1!2!3!1 and Λ = kQ=ha2i. serve that there are bijections between support τ-tilting modules Then the Hasse quiver of sτ-tiltΛ is the following. and a certain class of silting complexes defined in section 2.2. Let Λ be a finite dimensional k-algebra. We call a complex P = ðPi; diÞ in Kbðproj ΛÞ two-term if Pi = 0 for all i ≠ 0; − 1. We denote by 2-siltΛ the set of isomorphism classes of basic two-term silting complexes in Kbðproj ΛÞ. Two-term tilting complexes were studied by Hoshino et al. (49). We have the following connection between support τ-tilting modules and two-term silting complexes. Theorem 23 (9, 50). Let Λ be a finite dimensional k-algebra. Then there exists a bijection 2-siltΛ → sτ-tiltΛ given by P↦H0ðPÞ. The natural bijection 2-siltΛ → 2‐silt Λop given by P ↦ HomΛðP; ΛÞ corresponds to the bijection in Theorem 15. Immediately we have the following consequence. Corollary 24 (9, 50). Let Λ be a finite dimensional k-algebra. Then any two-term almost complete silting complex has precisely two complements which are two-term. This coincides with the exchange graph of cluster-tilting objects τ The following property of -rigid pairs is an analog of a prop- in the cluster category of type A3 given in section 1, which is a erty of silting complexes (ref. 31; Lemma 2.25). consequence of Theorem 31 below. Proposition 25 (9). Let ðT; QÞ be a τ-rigid pair for Λ and Now we deal with an important class of algebras. Let Q be → → → Q1 Q0 T 0 a minimal projective presentation. Then Q0 and a Dynkin quiver. Define a new quiver Q by adding a new arrow ⊕ * Q1 Q have no nonzero direct summands in common. a : j → i to Q for each arrow a : i → j in Q. We call proj Λ Let K0ð Þ be the of the additive * + category proj Λ of finitely generated projective Λ-modules. It is X p p a free abelian group with a basis consisting of the isomorphism ΠdKQ= ðaa − a aÞ : classes P1; ...; Pn of indecomposable projective Λ-modules. Let a arrow in Q Q1 → Q0 → T → 0 be a minimal projective presentation of T in mod Λ. The element the preprojective algebra of Q. Any preprojective algebra Π of Dynkin type is a finite di- Π gT dQ − Q mensional algebra which is selfinjective. In particular is a 0 1 unique tilting Π-module. On the other hand we will see that there is a large family of support τ-tilting Π-modules. in K0ðproj ΛÞ is called the index or the g-vector of T. The follow- ing result is an immediate consequence of Theorem 23 and For a Dynkin quiver Q, we denote by WQ the associated Weyl ; ...; a general result for silting complexes (ref. 31; Theorem 2.27). group, i.e., WQ is presented by generators s1 sn with the following relations: Theorem 26 (9). Let ðT1⊕⋯⊕Tℓ; Qℓ+1⊕⋯⊕QnÞ be a support τ-tilting pair for Λ with T and Q indecomposable. Then gT1 ; ⋯; • 2 = i i si 1, Tℓ Qℓ + Qn g ; g 1 ; ⋯; g form a basis of K0ðproj ΛÞ. • sisj = sjsi if there is no arrow between i and j in Q,

Iyama and Reiten PNAS | July 8, 2014 | vol. 111 | no. 27 | 9709 Downloaded by guest on September 28, 2021 • sisjsi = sjsisj if there is precisely one arrow between i and j The Hasse quiver of sτ-tiltΠ for m = n = 3 is the following. in Q. = ∈ We say that an expression w si1 ⋯siℓ of w WQ with i1; ...; iℓ ∈ f1; ...; ng is reduced if ℓ is the smallest possible num- ber for w.Suchℓ is called the length of w and denoted by ℓðwÞ.We have a partial order on WQ, called the right order: We define − w ≥ w′ if and only if ℓðwÞ = ℓðw′Þ + ℓðw′ 1wÞ. When Q is a Dynkin quiver, there is the following nice con- nection between elements of WQ and support τ-tilting Π-mod- ules. A similar result holds for tilting modules over preprojective algebras of non-Dynkin type (40, 41). Theorem 28 (52). Let Π be a preprojective algebra of Dynkin type and Ii is an ideal Πð1 − eiÞΠ in Π.

(a) There exists a bijection WQ → sτ-tiltΠ given by = ↦ d = w si1 ⋯siℓ Iw Ii1 ⋯Iiℓ , where w si1 ⋯siℓ is any reduced ex- pression of w. (b) The right order on WQ coincides with the partial order on sτ-tiltΠ. 5. Connection with Cluster-Tilting Theory In section 2 we have seen how the result that almost complete The Hasse quiver of sτ-tiltΠ for type A is the following. 2 cluster-tilting objects in cluster categories have exactly two com- plements implied a similar result for support tilting modules over path algebras. However, we have seen that this does not work for any finite dimensional algebra. The motivation for considering support τ-tilting modules comes from the investigation of another class of algebras associated with the cluster category CQ,ormore generally, with any 2-Calabi–Yau triangulated category C with The Hasse quiver of sτ-tiltΠ for type A is the following. cluster-tilting object. 3 Let C bea2-Calabi–Yau triangulated category and T a cluster- op tilting object in C. Then Λ = EndCðTÞ is by definition a 2-Calabi– Yau tilted algebra (a cluster-tilted algebra if C is a cluster category CQ). We then have the following. Theorem 30 (35, 54). The functor ð−ÞdHomCðT; − Þ induces an equivalence of categories C=½T½1 → mod Λ, where C=½T½1 is the factor category of C by the ideal ½T½1 consisting of morphisms factoring through add T½1. In this setting we can express the condition for an object X in C to be τ-rigid in terms of the Λ-module HomCðT; XÞ. This leads us to the next result. For an object X in C, we decompose X = X′⊕X″ where X″ is a maximal direct summand of X which belongs to add T½1. We denote by c-tiltC (respectively, rigid C, m-rigidC) the set of isomorphism classes of basic cluster-tilting (respectively, rigid, maximal rigid) objects in C. Theorem 31 (9). The correspondence X↦ðX′; X″½−1Þ gives bi- jections rigid C → τ-rigidΛ and c-tiltC → sτ-tiltΛ. Moreover we have c-tiltC = m-rigidC = fU ∈ rigid Cj jUj = jTjg. As a consequence we get as a special case the following important results. Corollary 32. Let C be a 2-Calabi–Yau triangulated category with a cluster-tilting object T. (a) Any basic almost complete cluster-tilting object has precisely two These quivers coincide with the Hasse quivers of the symmetric complements (11). (b) An object U in C is cluster-tilting if and only if it is maximal rigid groups S and S respectively. 3 4 a a a a if and only if it is rigid and jUj = jTj (47). Finally let Q be the cyclic quiver 1!2!⋯!n!1 and Λ = kQ=hami,wheren and m are arbitrary positive integers We also include an application to a result from ref. 51. For ≥ satisfying m n. Then we have the following result, where a basic cluster-tilting object T = T1⊕⋯⊕Tn in C, we denote by τ Λ τ -tilt is the set of isomorphism classes of basic -tilting K0ðadd TÞ the Grothendieck group of the additive category Λ -modules. add T, which is a free abelian group with a basis T1; ...; Tn. For Theorem 29 (53). There are bijections between the following sets. an object X in C, there exists a triangle T″ → T′ → X → T″½1 in C (a) τ-tiltΛ. with T′; T″ in add T. In this case we define the index (or g-vector) X d ′ − ″ add Λ (b) sτ-tiltΛ − τ-tiltΛ. of X as gT T T in K0ð TÞ. Specializing Theorem 26 to (c) The set of triangulations of an n-gon with a puncture. being 2-Calabi–Yau tilted, we can translate to considering clus- – (d) The setP of sequences ða1; ...; anÞ of nonnegative integers sat- ter-tilting objects in 2-Calabi Yau triangulated categories to get n = isfying i=1ai n. the following.

9710 | www.pnas.org/cgi/doi/10.1073/pnas.1313075111 Iyama and Reiten Downloaded by guest on September 28, 2021 Corollary 33 (51). Let C be a 2-Calabi–Yau triangulated cate- ACKNOWLEDGMENTS. O.I. was supported by Japan Society for the Promo- = ⊕ ⊕ tion of Science Grants-in-Aid for Scientific Research 24340004, 23540045, SPECIAL FEATURE gory, and T and U U1 ⋯ Un be basic cluster-tilting objects with and 22224001. I.R. was supported by FRINAT Grant 19660 from the Research U1 ; ...; Un add Ui indecomposable. Then gT gT form a basis of K0ð TÞ. Council of Norway.

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