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Tilting Theory 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 field 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 quiver 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 representation theory of deals with the investigation of “ ” mod Λ Λ called irreducible maps, and the dotted lines indicate a certain the module category of finitely generated left -modules. τ “ ” – functor 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.
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