Introduction to Τ-Tilting Theory

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In this category there is an important class of objects called Let again Q be the quiver 1 → 2 → 3 and kQ the corre- cluster-tilting objects, which generalize tilting modules. In this sponding path algebra. The vertex 3 of the quiver Q is a sink, case there are 14 cluster-tilting objects that is, no arrow starts at 3. Then a reflection (or mutation) of Q at vertex 3 is defined by reversing all arrows ending at ′ a1b1c1 a2b1c1 3, to get the quiver Q = µ3(Q) : 1 −→ 2 ← 3 in this example [BGP]. We have the following close relationship between a1a4b1 a2a4b1 modkQ and modkQ′. Theorem 4. [BGP] Let Q be a finite acyclic quiver with a sink a1a3c1 a1a3b3 a1a4b3 ′ i and Q := µi(Q). Then there is an equivalence of categories F : ES →ES′ , where ES (respectively, ES′ ) is the full subcat- a3b3c2 a4b3c2 a2a4c2 i i i i egory of modkQ (respectively, modkQ′) consisting of modules ′ a3b2c1 a3b2c2 without direct summands Si (respectively, Si). It was shown in [APR] that the kQ-module T =(kQ/Si)⊕ −1 ′ a2b2c1 a2b2c2 τ Si has the endomorphism algebra kQ , and the above functor F is isomorphic to HomkQ(T, −). This T is a special case where the symbol ⊕ is omitted and the edges indicate mu- of what was later defined to be tilting modules. tation. This picture is complete in the sense that we can An important notion in tilting theory is a torsion pair, replace any element in any cluster-tilting object to get an- that is, a pair (T , F) of full subcategories of modΛ such other cluster-tilting object. This corresponds to the fact that that (i) HomΛ(T , F) = 0 and (ii) for any X in modΛ, there we have exactly two complements for almost complete cluster- exists an exact sequence 0 → Y → X → Z → 0 with tilting objects in cluster categories (see Theorem 9). 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 torsionfree class, that is, 2. Tilting theory a subcategory closed under extensions, submodules and iso- In this section we include some main results on tilting morphisms. For any M in modΛ, we have a torsion class ⊥ modules and related objects, focusing on when we have a M := {X ∈ modΛ | HomΛ(X, M) = 0} and a torsionfree ⊥ well defined operation of mutation on them. We refer to class M := {X ∈ modΛ | HomΛ(M, X) = 0}. Let FacM [ASS, Rin1, Ha, AHK] for more background on tilting the- (respectively, SubM) be the subcategory of modΛ consisting ory. of factor modules (respectively, submodules) of direct sums of copies of M. The equivalence in Theorem 4 has the following 2.1. Properties of tilting modules. Let as before Λ be a fi- interpretation in tilting theory: nite dimensional k-algebra with n = |Λ|. A Λ-module T is a Theorem 5. [BB, HR] Let T be a tilting module over a finite partial tilting module if T has projective dimension at most dimensional k-algebra Λ and Γ := End (T )op. 1 Λ one and ExtΛ(T,T ) = 0. It is a tilting module if moreover (a) T := FacT and F := T ⊥ give a torsion pair (T , F) in |T | = n. This condition can be replaced by the condition: modΛ, and X := ⊥DT and Y := SubDT (where D is the There is an exact sequence 0 → Λ → T → T → 0 with T 0 1 0 k-dual) give a torsion pair (X , Y) in modΓ. and T in addT . Here addT denotes the subcategory of modΛ 1 (b) We have equivalences of categories Hom (T, −) : T → Y consisting of direct summands of finite direct sums of copies Λ and Ext1 (T, −) : F→X . of T . The following result due to Bongartz is basic in tilting Λ theory. Note that in the above special case S3 is the only indecomposable kQ-module not in T = FacT , and F = addS . Theorem 1. [B] Any partial tilting module is a direct summand 3 There is a close connection between tilting modules and of a tilting module. torsion classes. We need an important notion of functorially It is natural to ask how many tilting modules exist for finite subcategories [AS]. For torsion classes T , being functo- a given partial tilting module. There is an explicit answer rially finite is equivalent to the existence of M in modΛ such for an almost complete tilting module, that is, a partial tilting that T = FacM. We denote by tiltΛ the set of isomorphism module U satisfying |U| = n − 1. An indecomposable module classes of basic tilting Λ-modules, and by ff-torsΛ the set of X is a complement for an almost complete tilting module U if faithful functorially finite torsion classes in modΛ. U ⊕ X is a tilting module. The following result is basic in this Theorem 6. [AS, Ho, S, As] Let Λ be a finite dimensional k- paper, where a Λ-module X is called faithful if the annihilator algebra. Then there is a bijection tiltΛ → ff-torsΛ given by ann X := {a ∈ Λ | aX = 0} of X is zero. T 7→ FacT . Theorem 2. [HU1, RS, U] Let U be an almost complete tilting Let (T, P ) be a pair of Λ-modules, where P is projective, module. Hom (P,T ) = 0 and T is a partial tilting Λ-module. Then (a) U has either one or two complements. Λ (T, P ) is called a support partial tilting pair for Λ. It is called (b) U has two complements if and only if U is faithful. a support tilting pair (respectively, almost complete support When there are exactly two complements, then there is a tilting pair) if |T | + |P | = n (respectively, |T | + |P | = n − 1), nice relationship between them. where n = |Λ|. For an indecomposable module X, we say Theorem 3. [HU1, RS] Let X and Y be two complements for that (X, 0) (respectively, (0, X)) is a complement for an al- an almost complete tilting module U. After we interchange most complete support tilting pair (T, P ) if (T ⊕ X, P ) (re-

2 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Footline Author spectively, (T, P ⊕ X)) is a support tilting pair. We then have phisms. There is mutation theory analogous to the case of the following result for path algebras. tilting modules [AI]. One has operations of left and right mu- Theorem 7. [IT, Rin3] Let Λ = kQ be a path algebra of an tations obtained using approximation sequences, which shows acyclic quiver Q. Then any almost complete support tilting that there are always inﬁnitely many complements for an al- pair has precisely two complements. most complete silting complex. Moreover there is a natural partial order whose Hasse quiver coincides with the exchange Note that Theorem 7 is not true for ﬁnite dimensional quiver. If we restrict to silting complexes which are ‘two- k-algebras in general. For example, let Q be the quiver a b term’, then almost complete silting complexes have exactly 1 −→ 2 −→ 3 and Λ := kQ/hbai. In this case there are 8 two complements [AIR] (Corollary 24). support tilting pairs for Λ

(P1P2S2, 0) (P1S2,P3) 3. Cluster algebras and 2-Calabi-Yau categories In this section we discuss cluster categories and 2-Calabi-Yau (P1P2P3, 0) (P2S2,P1) (P2,P1P3) (0,P1P2P3) triangulated categories, which were motivated by trying to categorify the essential ingredients of the cluster algebras of (P P ,P ) (S ,P P ) Fomin-Zelevinsky [FZ1, FZ2]. We restrict to considering what 2 3 1 3 1 2 is called acyclic cluster algebras, with no coefficients. They are associated with a finite acyclic quiver with n vertices. Es- where the symbol ⊕ is omitted and edges indicate muta- sential concepts in the theory of cluster algebras are clusters, tion. For example, the almost complete support tilting pair cluster variables, seeds, and quiver mutation. (P ⊕ P , 0) does not have two complements. 1 3 Let Q be a finite connected quiver having no loops or 2- There are two important quivers whose vertices are tilt- cycles. For a vertex i of Q, we define a new quiver µi(Q) ing modules. One is the Hasse quiver of tiltΛ, where we re- having no loops or 2-cycles, called mutation of Q, as follows: gard tiltΛ as a partially ordered set by defining T ≥ U when (i) For any pair a : j → i and b : i → k of arrows in Q, we cre- FacT ⊃ FacU. Thus we draw an arrow T → U if T > U and ate a new arrow [ba] : j → k. (ii) Reverse all arrows starting there is no V ∈ tiltΛ satisfying T>V >U. The other is the or ending at i. (iii) Remove a maximal disjoint set of 2-cycles. exchange quiver of tiltΛ. We draw an arrow X ⊕ U → Y ⊕ U For example, if Q is 1 −→ 2 −→ 3, then µ (Q) is 1 −→ 2 ←− 3 and when X and Y are complements of U and there is an exact 3 sequence 0 → X → U ′ → Y → 0 as given in Theorem 3. Here µ2(Q) is 1 2 3 . we have the following. Fix a function field F = Q(x1,...,xn) in n variables over Theorem 8. [HU2] The Hasse quiver and the exchange quiver the field of rational numbers Q. Then ({x1,...,xn},Q) is an coincide. initial seed, consisting of the pair of the free generating set {x1,...,xn} of F over Q and the acyclic quiver Q. Then for 2.2. Generalizations of tilting modules. Tilting modules dis- each i = 1,...,n we define a new seed ∗ cussed in the previous subsection are often called ‘classical’ µi({x1,...,xn},Q):=({x1,...,xi ,...,xn}, µi(Q)), tilting modules, and there are many generalizations. Here we ∗ m1+m2 where xi = , with m1 and m2 being certain monomials are mainly interested in cases where there is some kind of xi mutation, which is given via approximation sequences. in x1,...,xi−1,xi+1,...,xn determined by the quiver in the seed. We perform this process for all µ1,...,µn, and then on We have a more general class of ‘tilting modules of finite 2 projective dimension’, which by definition are Λ-modules T the new seeds, etc. It is easy to see that µi = id holds for with the following properties: (i) T has finite projective di- i = 1,...,n. This gives rise to a graph where the vertices are j the seeds, and the edges between them are induced by the µi. mension, (ii) Ext (T,T ) = 0 for j > 0, (iii) there is an exact Λ The graph may be finite or infinite. sequence 0 → Λ → T → ··· → Tm → 0, where all Ti are 0 The n element subsets of F obtained in this way are called in addT [Miy, Ha]. The notion of almost complete tilting clusters, and the elements of clusters are called cluster vari- modules and their complements are defined in the same way ables. The associated cluster algebra is the subalgebra of F as before. In this case there are in general more than two generated by all the cluster variables. There is only a finite complements, and all the complements are usually connected number of clusters (or equivalently cluster variables) if and by approximation sequences [CHU]. There are other aspects only if the quiver Q is Dynkin. of such tilting modules which have been investigated more, One approach to the study of cluster algebras is via so- like connections with functorially finite subcategories, with called categorification. This means that we want to find some applications to algebraic groups and Cohen-Macaulay repre- nice categories, like module categories or triangulated cate- sentations [AB, AR, Rin2]. gories, where we have some objects with similar properties as One of the important properties of tilting modules of fi- clusters and cluster variables. In particular we are looking for nite projective dimension is that they induce a derived equiv- indecomposable objects corresponding to cluster variables. alence between Λ and End (T )op. From this viewpoint of Λ As we have seen in section 1, the tilting modules al- derived Morita theory, the notion of tilting complexes intro- most have the desired properties, but the operation µi on duced in [Ri] is most important. Also here there are usually tilting modules is not always defined. This motivated the more than two complements for almost complete ones. From introduction of cluster categories CQ associated with a fi- the viewpoint of mutation theory, the notion of silting com- nite acyclic quiver [BMRRT]. These categories have more plexes [KV] is more natural than that of tilting complexes. b indecomposable objects than modkQ, and also more mor- We denote by K (projΛ) the homotopy category of bounded phisms between the old objects. They are defined as orbit complexes of finitely generated projective modules. A tilt- categories Db(kQ)/τ[−1], where Db(kQ) is the bounded de- ing complex (respectively, silting complex) is an object P in b b b rived category of modkQ and τ : D (kQ) → D (kQ) is a K (projΛ) satisfying HomKb(projΛ)(P, P [i]) = 0 for any i 6= 0 derived AR-translation. For example, the AR-quiver of the b (respectively, i > 0) and thickP = K (projΛ). Here thickP is cluster category of type A3 was given in section 1. The cat- b the smallest full subcategory of K (projΛ) which contains P egory CQ is Hom-finite, and also triangulated [K], and we 1 1 and is closed under cones, [±1], direct summands and isomor- have a functorial isomorphism D ExtCQ (X,Y ) ≃ ExtCQ (Y, X)

Footline Author PNAS Issue Date Volume Issue Number 3 [BMRRT]. In general a k-linear Hom-finite triangulated cat- ers with potential (Q,W ) [Am]. Under a certain finiteness egory C is called 2-Calabi-Yau if there is a functorial isomor- condition on A and (Q,W ), the generalized cluster category 1 1 phism D ExtC(X,Y ) ≃ ExtC(Y, X). Hence CQ is 2-Calabi- is a 2-Calabi-Yau triangulated category with cluster-tilting Yau. object. We can regard the tilting kQ-modules as special objects (3) When R is a 3-dimensional isolated Gorenstein sin- in CQ, but this class of objects is not large enough since mu- gularity with k ⊂ R, then the stable category CMR of max- tation is not always possible. Instead, if we consider all tilting imal Cohen-Macaulay R-modules is a 2-Calabi-Yau triangu- modules coming from algebras derived equivalent to kQ, we lated category. More concrete examples are invariant rings G actually get a class of objects such that mutation is always R = k[[X,Y,Z]] , where G is a finite subgroup of SL3(k) act- possible. These objects can be described directly as follows. ing freely on k3\{0}. For other examples, see [BIKR]. These We say that an object T in a 2-Calabi-Yau triangulated 2-Calabi-Yau triangulated categories have quite different ori- 1 category C is rigid if ExtC(T,T ) = 0. It is maximal rigid if gin from those in (2), but it was shown in [AmIR] that they it is rigid, and moreover if T ⊕ X is rigid for some X in C, are often equivalent. then X is in addT . A rigid object T in C is cluster-tilting if 1 ExtC(T, X) = 0 for some X in C implies that X is in addT . τ Clearly any cluster-tilting object is maximal rigid, and the 4. -tilting theory converse is true for the cluster category [BMRRT], and also for In this section we discuss the central results of τ-tilting the- more general 2-Calabi-Yau triangulated categories (see Corol- ory, including those analogous to results from tilting theory. lary 32(b)). A basic object U in C is an almost complete Let as before Λ be a finite dimensional k-algebra. cluster-tilting object if there exists an indecomposable object X in C such that U ⊕ X is a basic cluster tilting object. In 4.1. Definition and basic properties. We have dualities D = op ∗ this case X is as before called a complement for U. Homk(−,k) : modΛ ↔ modΛ and (−) := HomΛ(−, Λ) : We have the following result for cluster-tilting objects in 2- projΛ ↔ projΛop which induce an equivalence ν := D(−)∗ : Calabi-Yau triangulated categories, which improves Theorem projΛ → injΛ called Nakayama functor. For X in modΛ with d1 d0 2 for tilting modules. a minimal projective presentation P1 −→ P0 −→ X −→ 0, we mod op mod Theorem 9. [BMRRT, IY] Let C be a 2-Calabi-Yau triangu- define Tr X in Λ and τX in Λ by the exact se- lated category. Any almost complete cluster-tilting object in C quences ∗ has precisely two complements. ∗ d1 ∗ νd1 P0 −→ P1 −→ Tr X −→ 0 and 0 −→ τX −→ νP1 −−→ νP0. It was mentioned in section 1 that when we deal with support tilting modules over path algebras kQ, then the mu- Then τ (respectively, Tr) gives bijections between the isomorphism classes of indecomposable non-projective Λ-modules tation µi is defined for all i = 1,...,n. To prove this, one uses and the isomorphism classes of indecomposable non-injective the corresponding result for cluster-tilting objects in cluster op categories and interprets the condition Ext1 (T,T ) = 0 and Λ-modules (respectively, non-projective Λ -modules). More CQ strongly, τ and Tr give an equivalence and a duality respec- |T | = n for T in a cluster category as a condition for modkQ. tively between the stable categories. We denote by modΛ Further basic results about cluster categories were devel- the stable category modulo projectives and by modΛ the oped. They were of interest in themselves, and also helped to costable category modulo injectives. Then Tr gives a dual- establish a closer connection with acyclic cluster algebras, in ity Tr : modΛ ↔ modΛop called the transpose, and τ gives a series of papers including [BMR, BMRT, CC, CK1, CK2]. an equivalence τ : modΛ → modΛ called the AR-translation. This led to the result that there is a map CQ → A for the acyclic cluster algebra A associated with the quiver Q, called Moreover we have a functorial isomorphism 1 cluster character, inducing a bijection between indecompos- HomΛ(X,Y ) ≃ D ExtΛ(Y,τX) able rigid objects in CQ and the cluster variables in A. We now give some classes of 2-Calabi-Yau triangulated for any X and Y in modΛ called AR-duality. categories with cluster-tilting objects, in addition to the clus- There is an important class of modules, which were intro- ter categories. duced by Auslander-Smalo [AS] (see also [ARS, ASS]) almost (1) [IR, BIRS] Let WQ be the Coxeter group associated at the same time when tilting modules were introduced in [BB], then later almost forgotten: We call T in modΛ τ-rigid with a finite acyclic quiver Q. For any element w ∈ WQ, 1 there is associated a 2-Calabi-Yau triangulated category Cw if HomΛ(T,τT ) = 0. In this case we have ExtΛ(T,T ) = 0 as follows. Let Π be the preprojective algebra associated with (i.e. T is rigid) by AR-duality. Now let us call T in modΛ τ-tilting if T is τ-rigid and |T | = |Λ|. We call T in modΛ Q (see section 4.4). Let e1,...,en denote the idempotent elements associated with the vertices 1,...,n. Then we asso- support τ-tilting if there exists an idempotent e of Λ such that ciate with i the ideal Ii = Π(1 − ei)Π in Π, and the ideal T is a τ-tilting (Λ/hei)-module. Iw = Ii · · · Ii with an element w ∈ WQ with a reduced Any local algebra Λ has precisely two basic support τ- 1 t tilting modules Λ and 0. It follows from AR-duality that any expression w = si1 · · · sit . Let Πw = Π/Iw and SubΠw be the category of Πw-modules which are submodules of a fi- partial tilting module is τ-rigid, and any tilting module is nite direct sum of copies of Πw. Then the stable category τ-tilting. But the converse is far from being true. In fact, Cw := SubΠw is a 2-Calabi-Yau triangulated category. In a selfinjective algebra Λ usually has a lot of τ-tilting mod- particular, the stable category modΠ, where Π is the prepro- ules (see e.g. section 4.4) even though it has a unique basic jective algebra of a Dynkin quiver, belongs to this class. It tilting module Λ. For another example, let Q be the quiver a b was studied by Geiss, Leclerc and Schröer [GLS2], who also 1 −→ 2 −→ 3 and Λ := kQ/hbai. Then T = P1 ⊕ S1 ⊕ P3 is a independently studied a subclass of Cw [GLS1]. We will see in τ-tilting Λ-module which is not a tilting Λ-module. section 4.4 that, in the Dynkin case, the ideals Iw play a role The next observation gives a more precise relationship in τ-tilting theory. between tilting modules and τ-tilting modules, where a Λ- (2) There is a generalization of the cluster categories CQ module X is called sincere if any simple Λ-module appears to so-called generalized cluster categories CΛ associated with in X as a composition factor. Clearly any faithful module is algebras Λ with gl. dim Λ ≤ 2 and CQ,W associated with quiv- sincere.

4 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Footline Author Proposition 10. [AIR] (a) Tilting modules are precisely faithful a τ-rigid pair (U, P ) is almost complete support τ-tilting if (support) τ-tilting modules. |U|+|P | = |Λ|−1. In this case, for an indecomposable module (b) Any τ-tilting (respectively, τ-rigid) Λ-module T is a tilting X, we say that (X, 0) (respectively, (0, X)) is a complement (respectively, partial tilting) (Λ/ ann T )-module. of (U, P ) if (U ⊕ X, P ) (respectively, (U, P ⊕ X)) is a support (c) τ-tilting modules are precisely sincere support τ-tilting τ-tilting pair, which we call a completion of (U, P ). modules. Theorem 16. [AIR] Let Λ be a finite dimensional k-algebra. When T is a (Λ/hei)-module for some idempotent e, it Then any almost complete support τ-tilting pair has precisely is often useful to compare properties of T as (Λ/hei)-module two complements. with properties of T as a Λ-module. Two completions (T,Q) and (T ′,Q′) of an almost com- Proposition 11. [AIR] Let e be an idempotent of Λ and T ∈ plete support τ-tilting pair (U, P ) are called mutations of ′ ′ mod(Λ/hei). Then T is a τ-rigid Λ-module if and only if T is each other. We write (T ,Q ) = µ(X,0)(T,Q) (respectively, ′ ′ a τ-rigid (Λ/hei)-module. (T ,Q ) = µ(0,X)(T,Q)) if (X, 0) (respectively, (0, X)) is a There is a nice relationship between torsion classes and complement of (U, P ) giving rise to (T,Q). a b support τ-tilting modules, which improves Theorem 6. We For example, let Q be the quiver 1 −→ 2 −→ 3 and Λ := denote by sτ-tiltΛ the set of isomorphism classes of basic sup- kQ/hbai. Then µ(P ,0)(Λ, 0) = (P2 ⊕ P3, P1), µ(P ,0)(Λ, 0) = port τ-tilting Λ-modules, and by f-torsΛ the set of functorially 1 2 (P1 ⊕ S1 ⊕ P3, 0), and µ(P3,0)(Λ, 0)=(P1 ⊕ P2 ⊕ S2, 0). finite torsion classes in modΛ. Note that it is easy to understand when we have a com- Theorem 12. [AIR] There is a bijection sτ-tiltΛ → f-torsΛ plement of (U, P ) of the form (0, X) and how to find it. This given by T 7→ FacT . occurs precisely when U is not a sincere (Λ/hei)-module for This gives a natural way of extending a τ-rigid module to an idempotent e of Λ satisfying addP = add(Λe). In this case ′ a τ-tilting module, similar to Bongartz’s result (Theorem 1): there exists a primitive idempotent e of Λ such that U is a ′ ′ For any τ-rigid module U, we have an associated functorially sincere (Λ/he + e i)-module, and (0, Λe ) is a complement of finite torsion class ⊥(τU). By Theorem 12, there exists a sup- (U, P ). port τ-tilting module T such that FacT = ⊥(τU). It is easy The next result gives a relationship between complements to show that T is a τ-tilting module satisfying U ∈ addT . of the form (X, 0), which is an analog of Theorem 3 for almost Consequently we have the following result. complete tilting modules. Theorem 13. [AIR] Any τ-rigid module is a direct summand of Theorem 17. [AIR] Let (X, 0) and (Y, 0) be two complements a τ-tilting module. of an almost complete support τ-tilting pair (U, P ). After we interchange X and Y if necessary, there is an exact sequence The next result gives several characterizations of τ-tilting f ′ g r modules. This is important because it has a result from [ZZ] X −→ U −→ Y → 0 for some positive integer r, where f is a listed in Corollary 32 as a special case. minimal left (addU)-approximation and g is a minimal right (addU)-approximation. Theorem 14. [AIR] The following conditions are equivalent for a τ-rigid Λ-module T . A main difference from Theorem 3 is that f above is not (a) T is τ-tilting. necessarily injective. We conjecture that the integer r appear- (b) T is maximal τ-rigid, i.e. if T ⊕ X is τ-rigid for some ing in the exact sequence above is always one. Λ-module X, then X ∈ addT . Now we introduce an important partial order on the set (c) ⊥(τT )= FacT . sτ-tiltΛ. We have a bijection sτ-tiltΛ → f-torsΛ given by (d) If HomΛ(T,τX) = 0 and HomΛ(X,τT ) = 0, then X ∈ T 7→ FacT in Theorem 12. Then inclusion in f-torsΛ gives addT . rise to a partial order on sτ-tiltΛ, i.e. we define In addition to τ-rigid and support τ-tilting modules, it is T ≥ U also convenient to consider associated pairs of modules. Let (T, P ) be a pair with T ∈ modΛ and P ∈ projΛ. We if and only if FacT ⊃ FacU. Clearly (Λ, 0) is the maximum call (T, P ) a τ-rigid pair if T is τ-rigid and HomΛ(P,T ) = 0. element and (0, Λ) is the minimum element of sτ-tiltΛ. We We call (T, P ) a support τ-tilting pair if (T, P ) is τ-rigid and give basic properties. |T | + |P | = |Λ|. The notion of support τ-tilting module and Proposition 18. [AIR] that of support τ-tilting pair are essentially the same since, † op for an idempotent e of Λ, the pair (T, Λe) is support τ-tilting (a) The bijection (−) : sτ-tiltΛ → sτ-tiltΛ reverses the partial order. if and only if T is a τ-tilting (Λ/hei)-module (cf. Proposition ′ ′ 11). Moreover e is uniquely determined by T in this case. (b) Let (T,Q) and (T ,Q ) be support τ-tilting pairs for Λ which are mutations of each other. Then we have either Next we discuss the left-right symmetry of support τ- ′ ′ ′ ′ tilting modules. This is somewhat surprising since it does (T,Q) > (T ,Q ) or (T,Q) < (T ,Q ). not have an analog for tilting modules. We decompose T in Now we have a recipe to calculate mutation of support τ- modΛ as T = Tpr ⊕ Tnp where Tpr is a maximal projective tilting pairs, which is slightly more complicated than that for direct summand of T . For a τ-rigid pair (T, P ) for Λ, let tilting modules because the map f in Theorem 17 is not necessarily injective. Let (U, P ) be an almost complete support † ∗ ∗ (T, P ) := ((Tr Tnp) ⊕ P ,Tpr). τ-tilting pair with a completion (T,Q). We give a method to calculate another completion (T ′,Q′) of (U, P ) from (T,Q), We denote by τ-rigidΛ the set of isomorphism classes of basic which is divided into two cases (A) and (B) below, depending τ-rigid pairs of Λ. on which of (T,Q) and (T ′,Q′) is bigger. This can be checked ′ ′ ′ ′ Theorem 15. [AIR] (−)† gives bijections τ-rigidΛ ↔ τ-rigidΛop without knowing (T ,Q ) since (T,Q) > (T ,Q ) holds if and and sτ-tiltΛ ↔ sτ-tiltΛop such that (−)†† = id. only if T = U ⊕X for an indecomposable module X satisfying X∈ / FacU. ′ ′ 4.2. Mutation and Partial order. Now we give a basic the- (A) Assume (T,Q) > (T ,Q ) holds. Then we take a min- f orem in the mutation theory for support τ-tilting modules, imal left (addU)-approximation X −→ U ′ of X. If f is not which improves Theorem 2 for tilting modules. We say that surjective, then the cokernel of f is a direct sum of copies

Footline Author PNAS Issue Date Volume Issue Number 5 of an indecomposable module Y (Theorem 17), and we have The following property of τ-rigid pairs is an analog of a (T ′,Q′)=(U ⊕ Y, P ). If f is surjective, then there exists property of silting complexes [AI, 2.25]. ′ ′ an indecomposable projective module Y such that (T ,Q )= Proposition 25. [AIR] Let (T,Q) be a τ-rigid pair for Λ and (U, P ⊕ Y ). Q1 → Q0 → T → 0 a minimal projective presentation. Then ′ ′ † ′ ′ † (B) Assume (T,Q) < (T ,Q ) holds. Since (T,Q) > (T ,Q ) Q0 and Q1 ⊕Q have no non-zero direct summands in common. ′ ′ † holds by Proposition 18(a), we can calculate (T ,Q ) from Let K0(projΛ) be the Grothendieck group of the addi- (U, P )† by using the left approximation, as we observed in tive category projΛ of finitely generated projective Λ-modules. (A). Applying (−)† to (T ′,Q′)†, we obtain (T ′,Q′). It is a free abelian group with a basis consisting of the iso- a b morphism classes P ,...,Pn of indecomposable projective Λ- For example, let Q be the quiver 1 −→ 2 −→ 3 and 1 modules. Let Q → Q → T → 0 be a minimal projective Λ := kQ/hbai. First we caclulate µ (Λ, 0). Since P ∈/ 1 0 (P2,0) 2 presentation of T in modΛ. The element Fac(Λ/P2), we use (A) above. The minimal left (addΛ/P2)- approximation of P2 is P2 → P1, which has cokernel S1. Thus T g := Q0 − Q1 µ(P2,0)(Λ, 0) =(P1 ⊕ S1 ⊕ P3, 0). Next we calculate µ(0,P3)(P1 ⊕ S1, P3). We must use (B) in K0(projΛ) is called the index or the g-vector of T . The † ∗ ∗ in this case. We have (P1 ⊕ S1, P3) = ((Tr S1) ⊕ P3 , P1 ) = following result is an immediate consequence of Theorem 23 ′ ∗ ∗ ′ op (S2 ⊕ P3 , P1 ), where Si is the simple Λ -module associ- and a general result for silting complexes [AI, 2.27]. ′ ∗ ∗ ∗ ated with the vertex i. Now we have µ(P3 ,0)(S2 ⊕ P3 , P1 ) = Theorem 26. [AIR] Let (T1 ⊕···⊕ Tℓ,Qℓ+1 ⊕···⊕ Qn) be a ′ ∗ ∗ ′ ∗ ∗ † support τ-tilting pair for Λ with Ti and Qi indecomposable. (S2, P1 ⊕ P3 ). Thus µ(0,P3)(P1 ⊕ S1, P3)=(S2, P1 ⊕ P3 ) = T1 Tℓ Qℓ+1 Qn proj (P1 ⊕ S1 ⊕ P3, 0). Then g , · · · ,g ,g , · · · ,g form a basis of K0( Λ). The following result shows a strong connection between The following result shows that g-vectors determine τ- mutation and partial order. rigid pairs, which is an analog of [DK, 2.3]. Theorem 19. [AIR] For T, U ∈ sτ-tiltΛ, the following condi- Theorem 27. [AIR] We have an injective map from the set of tions are equivalent. isomorphism classes of τ-rigid pairs for Λ to K0(projΛ) given (a) T and U are mutations of each other, and T > U. by (T, P ) 7→ gT − gP . (b) T > U and there is no V ∈ sτ-tiltΛ such that T>V >U. We define the exchange quiver of sτ-tiltΛ as the quiver 4.4. Examples. In this section, we use a convention to de- 1 whose set of vertices is sτ-tiltΛ and we draw an arrow from T scribe modules via their composition series. For example, 2 to U if U is a mutation of T such that T > U. The following is an indecomposable Λ-module X with a simple submodule analog of Theorem 8 is an easy consequence of Theorem 19. S2 such that X/S2 = S1. For example, each simple module Corollary 20. [AIR] The exchange quiver of sτ-tiltΛ coincides Si is written as i . a b with the Hasse quiver of the partially ordered set sτ-tiltΛ. Let Q be the quiver 1 −→ 2 −→ 3 and Λ= kQ/hbai. Then 1 2 Another application of Theorem 19 is the following. P1 = 2 , P2 = 3 and P3 = 3 in this case. The Hasse quiver Corollary 21. [AIR] If the support τ-tilting quiver has a finite of sτ-tiltΛ is the following. connected component, then it is connected. 1 2 1 1 We end this subsection with the following generalization 2 3 2 2 2 2 1 of Theorem 9, where Γ is given naturally by using Bongartz completion in Theorem 13. 1 Theorem 22. [J] Let (U, P ) be a τ-rigid pair for Λ, and 1 2 1 2 2 0 sτ-tilt(U,P )Λ the subset of sτ-tiltΛ consisting of support τ- 2 3 3 2 1 3 3 2 tilting pairs which have (U, P ) as a direct summand. Then there is a bijection sτ-tiltΓ → sτ-tilt(U,P )Λ for some finite 1 3 dimensional k-algebra Γ with |Γ| = |Λ| − |U| − |P |. 2 When (U, P ) is an almost complete support τ-tilting pair, 3 3 3 the algebra Γ is local and hence sτ-tiltΓ = {Γ, 0} holds. Thus 1 2 Theorem 9 is a special case of Theorem 22. where the symbol ⊕ is omitted. For example, 2 3 3 is 1 2 2 ⊕ 3 ⊕ 3 = P1 ⊕ P2 ⊕ P3. a a a 4.3. Connection with silting complexes. In this subsection, Let Q be the quiver 1 −→ 2 −→ 3 −→ 1 and Λ= kQ/ha2 = we observe that there are bijections between support τ-tilting 0i. Then the Hasse quiver of sτ-tiltΛ is the following. modules and a certain class of silting complexes defined in section 2.2. Let Λ be a finite dimensional k-algebra. We call 1 2 2 a complex P =(P i,di) in Kb(projΛ) two-term if P i = 0 for all 2 3 2 3 2 i 6= 0, −1. We denote by 2-siltΛ the set of isomorphism classes 1 of basic two-term silting complexes in Kb(projΛ). Two-term 2 2 2 tilting complexes were studied by Hoshino, Kato and Miyachi 1 2 3 1 3 1 [HKM]. We have the following connection between support 2 3 1 2 1 1 2 1 τ-tilting modules and two-term silting complexes. 3 Theorem 23. [AIR, DF] Let Λ be a finite dimensional k- 1 1 1 0 algebra. Then there exists a bijection 2-siltΛ → sτ-tiltΛ given 0 2 3 3 by P 7→ H (P ). 3 3 1 3 1 The natural bijection 2-siltΛ → 2-siltΛop given by P 7→ 2 HomΛ(P, Λ) corresponds to the bijection in Theorem 15. 3 3 3 Immediately we have the following consequence. Corollary 24. [AIR, DF] Let Λ be a finite dimensional k- This coincides with the exchange graph of cluster-tilting ob- algebra. Then any two-term almost complete silting complex jects in the cluster category of type A3 given in section 1, has precisely two complements which are two-term. which is a consequence of Theorem 31 below.

6 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Footline Author Now we deal with an important class of algebras. Let Q These quivers coincide with the Hasse quivers of the symmet- be a Dynkin quiver. Define a new quiver Q by adding a new ric groups S3 and S4 respectively. ∗ a a a a arrow a : j → i to Q for each arrow a : i → j in Q. We call Finally let Q be the cyclic quiver 1 −→ 2 −→· · · −→ n −→ 1 and Λ = kQ/hami, where n and m are arbtrary positive in- ∗ ∗ Π := KQ/h X (aa − a a)i teger satisfying m ≥ n. Then we have the following result, a:arrow in Q where τ-tiltΛ is the set of isomorphism classes of basic τ- tilting Λ-modules. the preprojective algebra of Q. Theorem 29. [Ad] There are bijections between the following Any preprojective algebra Π of Dynkin type is a finite di- sets. mensional algebra which is selfinjective. In particular Π is a (a) τ-tiltΛ. unique tilting Π-module. On the other hand we will see that (b) sτ-tiltΛ − τ-tiltΛ. there is a large family of support τ-tilting Π-modules. (c) The set of triangulations of an n-gon with a puncture. For a Dynkin quiver Q, we denote by WQ the associated (d) The set of sequences (a1,...,an) of non-negative integers n Weyl group, i.e. WQ is presented by generators s1,...,sn with satisfying Pi=1 ai = n. the following relations: 2 • si = 1, The Hasse quiver of sτ-tiltΛ for m = n = 3 is the follow- • sisj = sj si if there is no arrow between i and j in Q, ing. • sisj si = sj sisj if there is precisely one arrow between i and 1 2 2 j in Q. 2 3 2 3 2 3 1 2 3 1 2 3 2 We say that an expression w = si1 · · · siℓ of w ∈ WQ with i1,...,iℓ ∈ {1,...,n} is reduced if ℓ is the smallest possible 1 2 1 1 2 number. Such ℓ is called the length of w and denoted by ℓ(w). 3 2 2 2 2 We have a partial order on WQ, called the right order: We ′ ′ ′−1 define w ≥ w if and only if ℓ(w)= ℓ(w )+ ℓ(w w). 1 2 3 1 3 1 2 3 1 2 1 2 1 1 When Q is a Dynkin quiver, there is the following nice 3 1 2 3 1 2 3 1 2 1 2 connection between elements of WQ and support τ-tilting Π- modules. A similar result holds for tilting modules over pre- 3 3 3 1 1 1 1 0 projective algebras of non-Dynkin type [IR, BIRS]. 1 1 2

Theorem 28. [Miz] Let Π be a preprojective algebra of Dynkin 2 3 3 3 1 3 1 3 type and Ii is an ideal Π(1 − ei)Π in Π. 3 1 2 3 1 2 3 1 (a) There exists a bijection WQ → sτ-tiltΠ given by w = 2 si1 · · · siℓ 7→ Iw := Ii1 · · · Iiℓ , where w = si1 · · · siℓ is any re- 3 2 2 3 duced expression of w. 3 1 3 3 3 (b) The right order on WQ coincides with the partial order on sτ-tiltΠ. The Hasse quiver of sτ-tiltΠ for type A2 is the following. 5. Connection with cluster-tilting theory In section 2 we have seen how the result that almost com- 1 2 1 1 plete cluster-tilting objects in cluster categories have exactly two complements implied a similar result for support tilting 1 2 0 2 1 modules over path algebras. However we have seen that this 2 2 1 2 does not work for any ﬁnite dimensional algebra. The motiva- tion for considering support τ-tilting modules comes from the investigation of another class of algebras associated with the The Hasse quiver of sτ-tiltΠ for type A2 is the following. cluster category CQ, or more generally, with any 2-Calabi-Yau triangulated category C with cluster-tilting object. 1 2 2 1 Let C be a 2-Calabi-Yau triangulated category and T a op 1 cluster-tilting object in C. Then Λ = EndC(T ) is by deﬁni- 2 1 2 1 3 2 1 2 1 tion a 2-Calabi-Yau tilted algebra (a cluster-tilted algebra if C 1 is a cluster category CQ). We then have the following. 2 1 3 2 1 Theorem 30. [BMR, KR] The functor (−) := HomC(T, −) in- 1 2 1 duces an equivalence of categories C/[T [1]] → modΛ, where 2 31 2 2 31 2 1 3 2 1 3 2 1 2 1 C/[T [1]] is the factor category of C by the ideal [T [1]] consist- add 2 2 ing of morphisms factoring through T [1]. 3 2 1 In this setting we can express the condition for an object 1 2 3 1 3 2 2 31 2 2 31 2 2 13 2 3 1 2 0 X in C to be τ-rigid in terms of the Λ-module HomC(T, X). 3 2 1 3 2 1 3 2 1 This leads us to the next result. For an object X in C, we ′ ′′ ′′ 31 decompose X = X ⊕ X where X is a maximal direct sum- 3 2 1 mand of X which belongs to addT [1]. We denote by c-tiltC 2 3 3 2 (respectively, rigidC, m-rigidC) the set of isomorphism classes 2 13 2 31 2 3 2 3 3 2 1 3 2 1 of basic cluster-tilting (respectively, rigid, maximal rigid) ob- 2 3 3 2 jects in C. 3 Theorem 31. [AIR] The correspondence X 7→ (X′, X′′[−1]) 2 3 2 3 3 2 1 3 2 gives bijections rigidC → τ-rigidΛ and c-tiltC → sτ-tiltΛ. 3 Moreover we have c-tiltC = m-rigidC = {U ∈ rigidC | |U| = 3 2 3 2 1 |T |}.

Footline Author PNAS Issue Date Volume Issue Number 7 As a consequence we get as a special case the following category addT , which is a free abelian group with a basis important results. T1,...,Tn. For an object X in C, there exists a triangle T ′′ → T ′ → X → T ′′[1] in C with T ′,T ′′ in addT . In this case Corollary 32. Let C be a 2-Calabi-Yau triangulated category X ′ ′′ with a cluster-tilting object T . we deﬁne the index (or g-vector) of X as gT := T − T in (a) [IY] Any basic almost complete cluster-tilting object has K0(addT ). Specializing Theorem 26 to Λ being 2-Calabi-Yau precisely two complements. tilted, we can translate to considering cluster-tilting objects (b) [ZZ] An object U in C is cluster-tilting if and only if it is in 2-Calabi-Yau triangulated categories to get the following. maximal rigid if and only if it is rigid and |U| = |T |. Corollary 33. [DK] Let C be a 2-Calabi-Yau triangulated category, and T and U = U1 ⊕···⊕ Un be basic cluster-tilting We also include an application to a result from [DK]. For U1 Un objects with Ui indecomposable. Then gT ,...,gT form a ba- a basic cluster-tilting object T = T1 ⊕···⊕ Tn in C, we de- sis of K0(addT ). note by K0(addT ) the Grothendieck group of the additive

Ad. T. Adachi, τ-tilting modules over Nakayama algebras, arXiv:1309.2216. FZ1. S. Fomin, A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 AIR. T. Adachi, O. Iyama, I. Reiten, τ-tilting theory, to appear in Compos. Math., (2002), no. 2, 497–529. arXiv:1210.1036. FZ2. S. Fomin, A. Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. AI. T. Aihara, O. Iyama, Silting mutation in triangulated categories, J. Lond. Math. Soc. 154 (2003), no. 1, 63–121. 85 (2012), no. 3, 633–668. GLS1. C. Geiss, B. Leclerc, and J. Schröer, Cluster algebra structures and semicanonical Am. C. Amiot, Cluster categories for algebras of global dimension 2 and quivers with bases for unipotent groups, arXiv:math/0703039. potential, Annales de l’Institut Fourier, vol. 59, no.6 (2009) 2525–2590. GLS2. C. Geiss, B. Leclerc, and J. Schröer, Kac-Moody groups and cluster algebras, Adv. AmIR. C. Amiot, O. Iyama, I. Reiten, Stable categories of Cohen-Macaulay modules and Math. 228 (2011), no. 1, 329–433. cluster categories, arXiv:1104.3658. Ha. D. Happel, Triangulated categories in the representation theory of finite-dimensional AHK. L. Angeleri Hügel, D. Happel, H. Krause, Handbook of tilting theory, London Math- algebras, London Mathematical Society Lecture Note Series, 119. Cambridge Univer- ematical Society Lecture Note Series, 332. Cambridge University Press, Cambridge, sity Press, Cambridge, 1988. 2007. HR. D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), As. I. Assem, Torsion theories induced by tilting modules, Canad. J. Math. 36 (1984), no. 399–443. 5, 899–913. HU1. D. Happel and L. Unger, Almost complete tilting modules, Proc. Amer. Math. Soc. ASS. I. Assem, D. Simson, A. Skowroński, Elements of the representation theory of as- 107 (1989), no. 3, 603–610 sociative algebras. Vol. 1. Techniques of representation theory, London Mathematical HU2. D. Happel and L. Unger, On a partial order of tilting modules, Algebr. Represent. Society Student Texts, 65. Cambridge University Press, Cambridge, 2006. Theory 8 (2005), no. 2, 147–156. AB. M. Auslander, R-O. Buchweitz, The homological theory of maximal Cohen-Macaulay Ho. M. Hoshino, Tilting modules and torsion theories, Bull. London Ma th. Soc. 14 (1982), approximations, Colloque en l’honneur de Pierre Samuel (Orsay, 1987). Mem. Soc. no. 4, 334–336. Math. France (N.S.) No. 38 (1989), 5–37. HKM. M. Hoshino, Y. Kato, J. Miyachi, On t-structures and torsion theories induced by APR. M. Auslander, M.I. Platzeck and I. Reiten, Coxeter functors without diagrams, compact objects, J. Pure Appl. Algebra 167 (2002), no. 1, 15–35. Trans. Amer. Math. Soc. 250 (1979) 1–12. IT. C. Ingalls and H. Thomas, Noncrossing partitions and representations of quivers, Com- AR. M. Auslander, I. Reiten, Applications of contravariantly finite subcategories, Adv. pos. Math. 145 (2009), no. 6, 1533–1562. Math. 86 (1991), no. 1, 111–152. IR. O. Iyama, I. Reiten, Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau ARS. M. Auslander, I. Reiten, S. O. Smalo, Representation theory of Artin algebras, Cam- algebras, Amer. J. Math. 130 (2008), no. 4, 1087–1149. bridge Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge, IY. O. Iyama, Y. Yoshino, Mutations in triangulated categories and rigid Cohen-Macaulay 1995. modules, Inv. Math. 172 (2008), 117–168. AS. M. Auslander, S. O. Smalo, Almost split sequences in subcategories, J. Algebra 69 J. G. Jasso, Reduction of τ-tilting modules and torsion pairs, arXiv:1302.2709. (1981), no. 2, 426–454. K. B. Keller, On triangulated orbit categories, Doc. Math. 10 (2005), 551–581. BGP. I. N. Bernstein, I. M. Gelfand and V. A. Ponomarev, Coxeter functors and Gabriel’s KR. B. Keller, I. Reiten, Cluster-tilted algebras are Gorenstein and stably Calabi-Yau, Adv. theorem, Russ. Math. Surv. 28 (1973), 17–32. Math. 211 (2007), no. 1, 123–151. B. K. Bongartz, Tilted Algebras, Proc. ICRA III (Puebla 1980), Lecture Notes in Math. KV. B. Keller, D. Vossieck, Aisles in derived categories, Deuxiéme Contact Franco-Belge No. 903, Springer-Verlag 1981, 26–38. en Algébre (Faulx-les-Tombes, 1987). Bull. Soc. Math. Belg. Sér. A 40 (1988), no. 2, BB. S. Brenner and M. C. R. Butler, Generalization of the Bernstein-Gelfand-Ponomarev 239–253. reflection functors, Lecture Notes in Math. 839, Springer-Verlag (1980), 103–169. Miy. Y. Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 (1986), BMR. A.B. Buan, R. Marsh and I. Reiten, Cluster-tilted algebras, Trans. Amer. Math. no. 1, 113–146. Soc. 359(2007), no. 1, 323–332. Miz. Y. Mizuno, Classifying τ-tilting modules over preprojective algebras of Dynkin type, BMRT. A. B. Buan, R. Marsh, I. Reiten and G. Todorov, Clusters and seeds in acyclic to appear in Math. Z., arXiv:1304.0667. cluster algebras, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3049–3060, with an Ri. J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), appendix coauthored in addition by P. Caldero and B. Keller. no. 3, 436–456. BMRRT. A. B. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov, Tilting theory and Rin1. C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math- cluster combinatorics, Adv. Math. 204 (2006), no. 2, 572–618. ematics, 1099. Springer-Verlag, Berlin, 1984. BIKR. I. Burban, O. Iyama, B. Keller, I. Reiten, Cluster tilting for one-dimensional hyper- Rin2. C. M. Ringel, The category of modules with good filtrations over a quasi-hereditary surface singularities, Adv. Math. 217 (2008), no. 6, 2443–2484. algebra has almost split sequences, Math. Z. 208 (1991), no. 2, 209–223. BIRS. A. B. Buan, O. Iyama, I. Reiten, J. Scott, Cluster structures for 2-Calabi-Yau Rin3. C. M. Ringel, Appendix: Some remarks concerning tilting modules and tilted alge- categories and unipotent groups, Compos. Math. 145 (2009), no. 4, 1035–1079. bras. Origin. Relevance. Future. Handbook of tilting theory, 413–472, London Math. CC. P. Caldero, F. Chapoton, Cluster algebras as Hall algebras of quiver representations, Soc. Lecture Note Ser., 332, Cambridge Univ. Press, Cambridge, 2007. Comment. Math. Helv. 81 (2006), no. 3, 595–616. RS. C. Riedtmann and A. Schofield, On a simplicial complex associated with tilting mod- CK1. P. Caldero, B. Keller, From triangulated categories to cluster algebras, Invent. Math. ules, Comment. Math. Helv. 66 (1991), no. 1, 70–78. 172 (2008), no. 1, 169–211. CK2. P. Caldero and B. Keller, From triangulated categories to cluster algebras II, Ann. S. S. O. Smalo, Torsion theories and tilting modules, Bull. London Math. Soc. 16 (1984), no. 5, 518–522 Sci Ecole Norm. Sup. (4), 39 (2006), no. 6, 983–1009. CHU. F. Coelho, D. Happel, L. Unger, Complements to partial tilting modules, J. Algebra U. L. Unger, Schur modules over wild, finite-dimensional path algebras with three simple 170 (1994), no. 1, 184–205. modules, J. Pure Appl. Alg. 64. no.2 (1990) 205–222. DK. R. Dehy and B. Keller, On the combinatorics of rigid objects in 2-Calabi-Yau cate- ZZ. Y. Zhou, B. Zhu, Maximal rigid subcategories in 2-Calabi-Yau triangulated categories, gories, Int. Math. Res. Not. IMRN 2008, no. 11, Art. ID rnn029. J. Algebra 348 (2011), no. 1, 49–60. DF. H. Derksen, J. Fei, General Presentations of Algebras, arXiv:0911.4913.

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