aa r X i v : . [ m a t h . R T ] J un τ -TILTING THEORY TAKAHIDE ADACHI, OSAMU IYAMA AND IDUN REITEN
Dedicated to the memory of Dieter Happel
Abstract.
The aim of this paper is to introduce τ -tilting theory, which ‘completes’ (classical)tilting theory from the viewpoint of mutation. It is well-known in tilting theory that an almostcomplete tilting module for any finite dimensional algebra over a field k is a direct summand ofexactly 1 or 2 tilting modules. An important property in cluster tilting theory is that an almostcomplete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly2 cluster-tilting objects. Reformulated for path algebras kQ , this says that an almost completesupport tilting module has exactly two complements. We generalize (support) tilting modulesto what we call (support) τ -tilting modules, and show that an almost complete support τ -tiltingmodule has exactly two complements for any finite dimensional algebra.For a finite dimensional k -algebra Λ, we establish bijections between functorially finite tor-sion classes in mod Λ, support τ -tilting modules and two-term silting complexes in K b ( proj Λ).Moreover these objects correspond bijectively to cluster-tilting objects in C if Λ is a 2-CY tiltedalgebra associated with a 2-CY triangulated category C . As an application, we show that theproperty of having two complements holds also for two-term silting complexes in K b ( proj Λ).
Contents
Introduction 21. Background and preliminary results 41.1. Torsion pairs and tilting modules 41.2. τ -tilting modules 51.3. Silting complexes 61.4. Cluster-tilting objects 72. Support τ -tilting modules 82.1. Basic properties of τ -rigid modules 82.2. τ -rigid modules and torsion classes 102.3. Mutation of support τ -tilting modules 132.4. Partial order, exchange sequences and Hasse quiver 163. Connection with silting theory 204. Connection with cluster-tilting theory 234.1. Support τ -tilting modules and cluster-tilting objects 234.2. Two-term silting complexes and cluster-tilting objects 255. Numerical invariants 275.1. g -vectors and E -invariants for finite dimensional algebras 285.2. E -invariants for 2-CY tilted algebras 296. Examples 30Index 32References 34 The second author was supported by JSPS Grant-in-Aid for Scientific Research 21740010, 21340003, 20244001and 22224001.The second and third authors were supported by FRINAT grant 19660 from the Research Council of Norway.2010
Mathematics Subject Classification.
Key words and phrases. τ -tilting module, tilting module, torsion class, silting complex, cluster-tilting object, E -invariant. Introduction
Let Λ be a finite dimensional basic algebra over an algebraically closed field k , mod Λ thecategory of finitely generated left Λ-modules, proj
Λ the category of finitely generated projectiveleft Λ-modules and inj
Λ the category of finitely generated injective left Λ-modules. For M ∈ mod Λ,we denote by add M (respectively, Fac M , Sub M ) the category of all direct summands (respectively,factor modules, submodules) of finite direct sums of copies of M . Tilting theory for Λ, and itspredecessors, have been central in the representation theory of finite dimensional algebras since theearly seventies [BGP, APR, BB, HR, B]. When T is a (classical) tilting module (which always hasthe same number of non-isomorphic indecomposable direct summands as Λ), there is an associatedtorsion pair ( T , F ), where T = Fac T , and the interplay between tilting modules and torsion pairshas played a central role. Another important fact is that an almost complete tilting module U canbe completed in at most two different ways to a tilting module [RS, U]. Moreover there are exactlytwo ways if and only if U is a faithful Λ-module [HU1].Even for a finite dimensional path algebra kQ , where Q is a finite quiver with no orientedcycles, not all almost complete tilting modules U are faithful. However, for the associated clustercategory C Q , where we have cluster-tilting objects induced from tilting modules over path algebras kQ ′ derived equivalent to kQ , then the almost complete cluster-tilting objects have exactly twocomplements [BMRRT]. This fact, and its generalization to 2-Calabi-Yau triangulated categories[IY], plays an important role in the categorification of cluster algebras. In the case of clustercategories, this can be reformulated in terms of the path algebra Λ = kQ as follows [IT, Ri]: AΛ-module T is support tilting if T is a tilting (Λ / h e i )-module for some idempotent e of Λ. Usingthe more general class of support tilting modules, it holds for path algebras that almost completesupport tilting modules can be completed in exactly two ways to support tilting modules.The above result for path algebras does not necessarily hold for a finite dimensional algebra.The reason is that there may be sincere modules which are not faithful. We are looking for ageneralization of tilting modules where we have such a result, and where at the same time someof the essential properties of tilting modules still hold. It is then natural to try to find a class ofmodules satisfying the following properties:(i) There is a natural connection with torsion pairs in mod Λ.(ii) The modules have exactly | Λ | non-isomorphic indecomposable direct summands, where | X | denotes the number of nonisomorphic indecomposable direct summands of X .(iii) The analogs of basic almost complete tilting modules have exactly two complements.(iv) In the hereditary case the class of modules should coincide with the classical tilting modules.For the (classical) tilting modules we have in addition that when the almost complete ones have twocomplements, then they are connected in a special short exact sequence. Also there is a naturallyassociated quiver, where the isomorphism classes of tilting modules are the vertices.There is a generalization of classical tilting modules to tilting modules of finite projective di-mension [Ha, Miy]. But it is easy to see that they do not satisfy the required properties. Thecategory mod Λ is naturally embedded in the derived category of Λ. The tilting and silting com-plexes for Λ [Ri, AI, Ai] are also extensions of the tilting modules. An almost complete siltingcomplex has infinitely many complements. But as we shall see, things work well when we restrictto the two-term silting complexes.In the module case, it turns out that a natural class of modules to consider is given as follows.As usual, we denote by τ the AR translation (see section 1.2). Definition 0.1. (a) We call M in mod Λ τ -rigid if Hom Λ ( M, τ M ) = 0.(b) We call M in mod Λ τ -tilting (respectively, almost complete τ -tilting ) if M is τ -rigid and | M | = | Λ | (respectively, | M | = | Λ | − M in mod Λ support τ -tilting if there exists an idempotent e of Λ such that M isa τ -tilting (Λ / h e i )-module. -TILTING THEORY 3 Any τ -rigid module is rigid (i.e. Ext ( M, M ) = 0), and the converse holds if the projectivedimension is at most one. In particular, any partial tilting module is a τ -rigid module, and anytilting module is a τ -tilting module. Thus we can regard τ -tilting modules as a generalization oftilting modules.The first main result of this paper is the following analog of Bongartz completion for tiltingmodules. Theorem 0.2 (Theorem 2.10) . Any τ -rigid Λ -module is a direct summand of some τ -tilting Λ -module. As indicated above, in order to get our theory to work nicely, we need to consider support τ -tilting modules. It is often convenient to view them, and the τ -rigid modules, as certain pairs ofΛ-modules. Definition 0.3.
Let (
M, P ) be a pair with M ∈ mod Λ and P ∈ proj Λ.(a) We call (
M, P ) a τ -rigid pair if M is τ -rigid and Hom Λ ( P, M ) = 0.(b) We call (
M, P ) a support τ -tilting (respectively, almost complete support τ -tilting ) pair if( M, P ) is τ -rigid and | M | + | P | = | Λ | (respectively, | M | + | P | = | Λ | − M, P ) is basic if M and P are basic. Similarly we say that ( M, P ) is a directsummand of ( M ′ , P ′ ) if M is a direct summand of M ′ and P is a direct summand of P ′ .The second main result of this paper is the following. Theorem 0.4 (Theorem 2.18) . Let Λ be a finite dimensional k -algebra. Then any basic almostcomplete support τ -tilting pair for Λ is a direct summand of exactly two basic support τ -tiltingpairs. These two support τ -tilting pairs are said to be mutations of each other. We will define thesupport τ -tilting quiver Q(s τ -tiltΛ) by using mutation (Definition 2.29).When extending (classical) tilting modules to tilting complexes or silting complexes we havepointed out that we do not have exactly two complements in the almost complete case. Butconsidering instead only the two-term silting complexes, we prove that this is the case.The third main result is to obtain a close connection between support τ -tilting modules andother important objects in tilting theory. The corresponding definitions will be given in section 1. Theorem 0.5 (Theorems 2.7, 3.2, 4.1 and 4.7) . Let Λ be a finite dimensional k -algebra. We havebijections between (a) the set f-torsΛ of functorially finite torsion classes in mod Λ , (b) the set s τ -tiltΛ of isomorphism classes of basic support τ -tilting modules, (c) the set of isomorphism classes of basic two-term silting complexes for Λ , (d) the set c-tilt C of isomorphism classes of basic cluster-tilting objects in a 2-CY triangulatedcategory C if Λ is an associated 2-CY tilted algebra to C . Note that the correspondence between (b) and (d) improves results in [Smi, FL].By Theorem 0.5, we can regard s τ -tiltΛ as a partially ordered set by using the inclusion relationof f-torsΛ (i.e. we write T ≥ U if Fac T ⊇ Fac U ). Then we have the following fourth main result,which is an analog of [HU2, Theorem 2.1] and [AI, Theorem 2.35]. Theorem 0.6 (Corollary 2.34) . The support τ -tilting quiver Q(s τ -tiltΛ) is the Hasse quiver of thepartially ordered set s τ -tiltΛ . We have the following direct consequences of Theorem 0.5, where the second part is known by[IY], and the third one by [ZZ].
Corollary 0.7 (Corollaries 3.8, 4.5) . (a) Two-term almost complete silting complexes haveexactly two complements.
TAKAHIDE ADACHI, OSAMU IYAMA AND IDUN REITEN (b)
In a 2-Calabi-Yau triangulated category with cluster-tilting objects, any almost completecluster-tilting objects in a 2-CY category have exactly two complements. (c)
In a 2-Calabi-Yau triangulated category with cluster-tilting objects, any maximal rigid ob-ject is cluster-tilting.
Part (a) was first proved directly by Derksen-Fei [DF] without dealing with support τ -tiltingmodules. Here we obtain this result by combining a bijection in Theorem 0.5 with Theorem 0.4.Another important part of our work is to investigate to which extent the main properties oftilting modules mentioned above remain valid in the settings of support τ -tilting modules, two-termsilting complexes and cluster-tilting objects in 2-CY triangulated categories.A motivation for considering the problem of exactly two complements for almost completesupport τ -tilting modules was that the condition of τ -rigid module appears naturally when weexpress Ext C ( X, Y ) for X and Y objects in a 2-CY category C in terms of corresponding modules X and Y over an associated 2-CY tilted algebra (Proposition 4.4).There is some relationship to the E -invariants of [DWZ] in the case of finite dimensional Jacobianalgebras, where the expression Hom Λ ( M, τ N ) appears. Here we introduce E -invariants in section 5for any finite dimensional k -algebras, and express them in terms of dimension vectors and g -vectorsas defined in [DK], inspired by [DWZ].In the last section 6 we illustrate our results with examples.There is a curious relationship with interesting independent work by Cerulli-Irelli, Labardini-Fragoso and Schr¨oer [CLS], where the authors deal with E -invariants in the more general settingof basic algebras which are not necessarily finite dimensional. We refer to recent work by K¨onigand Yang [KY] for connection with t-structures and co-t-structures. Hoshino, Kato and Miyachi[HKM] and Abe [Ab] studied two term tilting complexes. Buan and Marsh have considered adirect map from cluster-tilting objects in cluster categories to functorially finite torsion classes forassociated cluster-tilted algebras. Acknowledgements
Part of this work was done when the authors attended conferences inOberwolfach (February 2011), Banff (September 2011), Shanghai (October 2011) and Trondheim(March 2012). Parts of the results in this paper were presented at conferences in Kagoshima(February 2012), Graz, Nagoya, Trondheim (March 2012), Matsumoto, Bristol (September 2012)and MSRI (October 2012). The authors would like to thank the organizers of these conferences.Part of this work was done while the second author visited Trondheim in March 2010, March 2011and March-April 2012. He would like to thank the people at NTNU for hospitality and stimulatingdiscussions. We thank Julian K¨ulshammer and Xiaojin Zhang for pointing out typos in the firstdraft. 1.
Background and preliminary results
In this section we give some background material on each of the 4 topics involved in our mainresults. This concerns the relationship between tilting modules and functorially finite subcategoriesand some results on τ -rigid and τ -tilting modules, including new basic results about them whichwill be useful in the next section. Further we recall known results on silting complexes, and oncluster-tilting objects in 2-CY triangulated categories.1.1. Torsion pairs and tilting modules.
Let Λ be a finite dimensional k -algebra. For a sub-category C of mod Λ, we let C ⊥ := { X ∈ mod Λ | Hom Λ ( C , X ) = 0 } , C ⊥ := { X ∈ mod Λ | Ext ( C , X ) = 0 } . Dually we define ⊥ C and ⊥ C . We call T in mod Λ a partial tilting module if pd Λ T ≤ ( T, T ) = 0. A partial tilting module is called a tilting module if there is an exact sequence0 → Λ → T → T → T and T in add T . Then any tilting module satisfies | T | = | Λ | .Moreover it is known that for any partial tilting module T , there is a tilting module U such that -TILTING THEORY 5 T ∈ add U and Fac U = T ⊥ , called the Bongartz completion of T . Hence a partial tilting module T is a tilting module if and only if | T | = | Λ | . Dually T in mod Λ is a ( partial ) cotilting module if DT is a (partial) tilting Λ op -module.On the other hand, we say that a full subcategory T of mod Λ is a torsion class (respectively, torsionfree class ) if it is closed under factor modules (respectively, submodules) and extensions. Apair ( T , F ) is called a torsion pair if T = ⊥ F and F = T ⊥ . In this case T is a torsion class and F is a torsionfree class. Conversely, any torsion class T (respectively, torsionfree class F ) gives riseto a torsion pair ( T , F ).We say that X ∈ T is Ext -projective (respectively, Ext -injective ) if Ext ( X, T ) = 0 (respec-tively, Ext ( T , X ) = 0). We denote by P ( T ) the direct sum of one copy of each of the indecom-posable Ext-projective objects in T up to isomorphism. Similarly we denote by I ( F ) the directsum of one copy of each of the indecomposable Ext-injective objects in F up to isomorphism.We first recall the following relevant result on torsion pairs and tilting modules. Proposition 1.1. [AS, Ho, Sma]
Let ( T , F ) be a torsion pair in mod Λ . Then the followingconditions are equivalent. (a) T is functorially finite. (b) F is functorially finite. (c) T = Fac X for some X in mod Λ . (d) F = Sub Y for some Y in mod Λ . (e) P ( T ) is a tilting (Λ / ann T ) -module. (f) I ( F ) is a cotilting (Λ / ann F ) -module. (g) T = Fac P ( T ) . (h) F = Sub I ( F ) .Proof. The conditions (a), (b), (c), (d), (e) and (f) are equivalent by [Sma, Theorem].(g) ⇒ (c) is clear.(e) ⇒ (g) There exists an exact sequence 0 → Λ / ann T a −→ T → T → T , T ∈ add P ( T ).For any X ∈ T , we take a surjection f : (Λ / ann T ) ℓ → X . It follows from Ext ( T ℓ , X ) = 0 that f factors through a ℓ : (Λ / ann T ) ℓ → T ℓ . Thus X ∈ Fac P ( T ).Dually (h) is also equivalent to the other conditions. (cid:3) There is also a tilting quiver associated with the (classical) tilting modules. The vertices arethe isomorphism classes of basic tilting modules. Let X ⊕ U and Y ⊕ U be basic tilting modules,where X and Y X are indecomposable. Then it is known that there is some exact sequence0 → X f −→ U ′ g −→ Y →
0, where f : X → U ′ is a minimal left ( add U )-approximation and g : U ′ → Y is a minimal right ( add U )-approximation. We say that Y ⊕ U is a left mutation of X ⊕ U . Thenwe draw an arrow X ⊕ U → Y ⊕ U , so that we get a quiver for the tilting modules. On the otherhand, the set of basic tilting modules has a natural partial order given by T ≥ U if and only if Fac T ⊇ Fac U , and we can consider the associated Hasse quiver. These two quivers coincide [HU2,Theorem 2.1].1.2. τ -tilting modules. Let Λ be a finite dimensional k -algebra. We have dualities D := Hom k ( − , k ) : mod Λ ↔ mod Λ op and ( − ) ∗ := Hom Λ ( − , Λ) : proj Λ ↔ proj Λ op which induce equivalences ν := D ( − ) ∗ : proj Λ → inj Λ and ν − := ( − ) ∗ D : inj Λ → proj Λcalled
Nakayama functors . For X in mod Λ with a minimal projective presentation P d P d X , we define Tr X in mod Λ op and τ X in mod Λ by exact sequences P ∗ d ∗ P ∗ Tr X τ X νP νd νP . TAKAHIDE ADACHI, OSAMU IYAMA AND IDUN REITEN
Then Tr and τ give bijections between the isomorphism classes of indecomposable non-projectiveΛ-modules, the isomorphism classes of indecomposable non-projective Λ op -modules and the iso-morphism classes of indecomposable non-injective Λ-modules. We denote by mod Λ the stablecategory modulo projectives and by mod
Λ the costable category modulo injectives. Then Tr givesthe
Auslander-Bridger transpose duality
Tr : mod Λ ↔ mod Λ op and τ gives the AR translations τ = D Tr : mod Λ → mod Λ and τ − = Tr D : mod Λ → mod Λ . We have a functorial isomorphismHom Λ ( X, Y ) ≃ D Ext ( Y, τ X )for any X and Y in mod Λ called
AR duality . In particular, if M is τ -rigid, then we haveExt ( M, M ) = 0 (i.e. M is rigid) by AR duality. More precisely, we have the following result,which we often use in this paper. Proposition 1.2.
For X and Y in mod Λ , we have the following. (a) [AS, Proposition 5.8] Hom Λ ( X, τ Y ) = 0 if and only if
Ext ( Y, Fac X ) = 0 . (b) [AS, Theorem 5.10] If X is τ -rigid, then Fac X is a functorially finite torsion class and X ∈ add P ( Fac X ) . (c) If T is a torsion class in mod Λ , then P ( T ) is a τ -rigid Λ -module.Proof. (c) Since T := P ( T ) is Ext-projective in T , we have Ext ( T, Fac T ) = 0. This implies thatHom Λ ( T, τ T ) = 0 by (a). (cid:3)
We have the following direct consequence (see also [Sk, ASS]).
Proposition 1.3.
Any τ -rigid Λ -module M satisfies | M | ≤ | Λ | .Proof. By Proposition 1.2(b) we have | M | ≤ | P ( Fac M ) | . By Proposition 1.1(e), we have | P ( Fac M ) | = | Λ / ann M | . Since | Λ / ann M | ≤ | Λ | , we have the assertion. (cid:3) As an immediate consequence, if τ -rigid Λ-modules M and N satisfy M ∈ add N and | M | ≥ | Λ | ,then add M = add N .Finally we note the following relationship between τ -tilting modules and classical notions. Proposition 1.4. [ASS, VIII.5.1](a)
Any faithful τ -rigid Λ -module is a partial tilting Λ -module. (b) Any faithful τ -tilting Λ -module is a tilting Λ -module. Silting complexes.
Let Λ be a finite dimensional k -algebra and K b ( proj Λ) be the categoryof bounded complexes of finitely generated projective Λ-modules. We recall the definition of siltingcomplexes and mutations.
Definition 1.5. [AI, Ai, BRT, KV] Let P ∈ K b ( proj Λ).(a) We call P presilting if Hom K b ( proj Λ) ( P, P [ i ]) = 0 for any i > P silting if it is presilting and satisfies thick P = K b ( proj Λ), where thick P is thesmallest full subcategory of K b ( proj Λ) which contains P and is closed under cones, [ ± Proposition 1.6. [AI, Theorem 2.27, Corollary 2.28](a)
For any P ∈ siltΛ , we have | P | = | Λ | . (b) Let P = L ni =1 P n be a basic silting complex for Λ with P i indecomposable. Then P , · · · , P n give a basis of the Grothendieck group K ( K b ( proj Λ)) . -TILTING THEORY 7 We call a presilting complex P for Λ almost complete silting if | P | = | Λ | −
1. There is a similartype of mutation as for tilting modules.
Definition-Proposition 1.7. [AI, Theorem 2.31] Let P = X ⊕ Q be a basic silting complex with X indecomposable. We consider a triangle X f Q ′ Y X [1]with a minimal left ( add Q )-approximation f of X . Then the left mutation of P with respect to X is µ − X ( P ) := Y ⊕ Q . Dually we define the right mutation µ + X ( P ) of P with respect to X . Thenthe left mutation and the right mutation of P are also basic silting complexes.There is the following partial order on the set siltΛ. Definition-Proposition 1.8. [AI, Theorem 2.11, Proposition 2.14] For
P, Q ∈ siltΛ, we write P ≥ Q if Hom K b ( proj Λ) ( P, Q [ i ]) = 0 for any i >
0, which is equivalent to P ⊥ > ⊇ Q ⊥ > where P ⊥ > is asubcategory of K b ( proj Λ) consisting of the X satisfying Hom K b ( proj Λ) ( P, X [ i ]) = 0 for any i > silting quiver Q(siltΛ) of Λ as follows: • The set of vertices is siltΛ. • We draw an arrow from P to Q if Q is a left mutation of P .Then the silting quiver gives the Hasse quiver of the partially ordered set siltΛ by [AI, Theo-rem 2.35], similar to the situation for tilting modules. We shall later restrict to two-term siltingcomplexes to get exactly two complements for almost complete silting complexes.1.4. Cluster-tilting objects.
Let C be a k -linear Hom-finite Krull-Schmidt triangulated cate-gory. Assume that C is ( for short) i.e. there exists a functorial isomorphism D Ext C ( X, Y ) ≃ Ext C ( Y, X ). An important class of objects in these categories are the cluster-tilting objects. We recall the definition of these and related objects.
Definition 1.9. (a) We call T in C rigid if Hom C ( T, T [1]) = 0.(b) We call T in C cluster-tilting if add T = { X ∈ C | Hom C ( T, X [1]) = 0 } .(c) We call T in C maximal rigid if it is rigid and maximal with respect to this property, thatis, add T = { X ∈ C | Hom C ( T ⊕ X, ( T ⊕ X )[1]) = 0 } .We denote by c-tilt C the set of isomorphism classes of basic cluster-tilting objects in C . In thissetting, there are also mutations of cluster-tilting objects defined via approximations, which werecall [BMRRT, IY]. Definition-Proposition 1.10. [IY, Theorem 5.3] Let T = X ⊕ U be a basic cluster-tilting objectin C and X indecomposable in C . We consider the triangle X f U ′ Y X [1]with a minimal left ( add U )-approximation f of X . Let µ − X ( T ) := Y ⊕ U . Dually we define µ + X ( T ).A different feature in this case is that we have µ − X ( T ) ≃ µ + X ( T ). This is a basic cluster-tiltingobject which as before we call the mutation of T with respect to X .In this case we get just a graph rather than a quiver. We define the cluster-tilting graph G(c-tilt C ) of C as follows: • The set of vertices is c-tilt C . • We draw an edge between T and U if U is a mutation of T . These notations µ − and µ + are the opposite of those in [AI]. They are easy to remember since they are thesame direction as τ − and τ . TAKAHIDE ADACHI, OSAMU IYAMA AND IDUN REITEN
Note that U is a mutation of T if and only if T and U have all but one indecomposable directsummand in common [IY, Theorem 5.3] (see Corollary 4.5(a)).2. Support τ -tilting modules Our aim in this section is to develop a basic theory of support τ -tilting modules over any finitedimensional k -algebra. We start with discussing some basic properties of τ -rigid modules andconnections between τ -rigid modules and functorially finite torsion classes (Theorem 2.7). As anapplication, we introduce Bongartz completion of τ -rigid modules (Theorem 2.10). Then we givecharacterizations of τ -tilting modules (Theorem 2.12). We also give left-right duality of τ -rigidmodules (Theorem 2.14). Further we prove our main result which states that an almost completesupport τ -tilting module has exactly two complements (Theorem 2.18). As an application, weintroduce mutation of support τ -tilting modules. We show that mutation gives the Hasse quiverof the partially ordered set of support τ -tilting modules (Theorem 2.33).2.1. Basic properties of τ -rigid modules. When T is a Λ-module with I an ideal containedin ann T , we investigate the relationship between T being τ -rigid as a Λ-module and as a (Λ /I )-module. We have the following. Lemma 2.1.
Let Λ be a finite dimensional algebra, and I an ideal in Λ . Let M and N be (Λ /I ) -modules. Then we have the following. (a) If Hom Λ ( N, τ M ) = 0 , then
Hom Λ /I ( N, τ Λ /I M ) = 0 . (b) Assume I = h e i for an idempotent e in Λ . Then Hom Λ ( N, τ M ) = 0 if and only if
Hom Λ /I ( N, τ Λ /I M ) = 0 .Proof. Note that we have a natural inclusion Ext /I ( M, N ) → Ext ( M, N ). This is an isomor-phism if I = h e i for an idempotent e since mod (Λ / h e i ) is closed under extensions in mod Λ.(a) Assume Hom Λ ( N, τ M ) = 0. Then by Proposition 1.2, we have Ext ( M, Fac N ) = 0.By the above observation, we have Ext /I ( M, Fac N ) = 0. By Proposition 1.2 again, we haveHom Λ /I ( N, τ Λ /I M ) = 0.(b) Assume that I = h e i and Hom Λ /I ( N, τ Λ /I M ) = 0. By Proposition 1.2, we have Ext /I ( M, Fac N ) =0. By the above observation, we have Ext ( M, Fac N ) = 0. By Proposition 1.2 again, we haveHom Λ ( N, τ M ) = 0. (cid:3)
Recall that M in mod Λ is sincere if every simple Λ-module appears as a composition factor in M . This is equivalent to the fact that there does not exist a non-zero idempotent e of Λ whichannihilates M . Proposition 2.2. (a) τ -tilting modules are precisely sincere support τ -tilting modules. (b) Tilting modules are precisely faithful support τ -tilting modules. (c) Any τ -tilting (respectively, τ -rigid) Λ -module T is a tilting (respectively, partial tilting) (Λ / ann T ) -module.Proof. (a) Clearly sincere support τ -tilting modules are τ -tilting. Conversely, if a τ -tilting Λ-module T is not sincere, then there exists a non-zero idempotent e of Λ such that T is a (Λ / h e i )-module. Since T is τ -rigid as a (Λ / h e i )-module by Lemma 2.1(a), we have | T | = | Λ | > | Λ / h e i| , acontradiction to Proposition 1.3.(b) Clearly tilting modules are faithful τ -tilting. Conversely, any faithful support τ -tiltingmodule T is partial tilting by Proposition 1.4 and satisfies | T | = | Λ | . Thus T is tilting.(c) By Lemma 2.1(a), we know that T is a faithful τ -tilting (respectively, τ -rigid) (Λ / ann T )-module. Thus the assertion follows from (b) (respectively, Proposition 1.4). (cid:3) Immediately we have the following basic observation, which will be used frequently in this paper.
Proposition 2.3.
Let ( M, P ) be a pair with M ∈ mod Λ and P ∈ proj Λ . Let e be an idempotentof Λ such that add P = add Λ e . -TILTING THEORY 9 (a) ( M, P ) is a τ -rigid (respectively, support τ -tilting, almost complete support τ -tilting) pairfor Λ if and only if M is a τ -rigid (respectively, τ -tilting, almost complete τ -tilting) (Λ / h e i ) -module. (b) If ( M, P ) and ( M, Q ) are support τ -tilting pairs for Λ , then add P = add Q . In other words, M determines P and e uniquely.Proof. (a) The assertions follow from Lemma 2.1 and the equation | Λ / h e i| = | Λ | − | P | .(b) This is a consequence of Proposition 2.2(a). (cid:3) The following observations are useful.
Proposition 2.4.
Let X be in mod Λ with a minimal projective presentation P d −→ P d −→ X → . (a) For Y in mod Λ , we have an exact sequence → Hom Λ ( Y, τ X ) → D Hom Λ ( P , Y ) D ( d ,Y ) −−−−−→ D Hom Λ ( P , Y ) D ( d ,Y ) −−−−−→ D Hom Λ ( X, Y ) → . (b) Hom Λ ( Y, τ X ) = 0 if and only if the map
Hom Λ ( P , Y ) ( d ,Y ) −−−−→ Hom Λ ( P , Y ) is surjective. (c) X is τ -rigid if and only if the map Hom Λ ( P , X ) ( d ,X ) −−−−→ Hom Λ ( P , X ) is surjective.Proof. (a) We have an exact sequence 0 → τ X → νP νd −−→ νP . Applying Hom Λ ( Y, − ), we havea commutative diagram of exact sequences:0 −→ Hom Λ ( Y, τ X ) Hom Λ ( Y, νP ) ( Y,νd ) ≀ Hom Λ ( Y, νP ) ≀ D Hom Λ ( P , Y ) D ( d ,Y ) D Hom Λ ( P , Y ) D ( d ,Y ) D Hom Λ ( X, Y ) −→ . Thus the assertion follows.(b)(c) Immediate from (a). (cid:3)
We have the following standard observation (cf. [HU2, DK]).
Proposition 2.5.
Let X be in mod Λ with a minimal projective presentation P d −→ P d −→ X → .If X is τ -rigid, then P and P have no non-zero direct summands in common.Proof. We only have to show that any morphism s : P → P is in the radical. By Proposition2.4(c), there exists t : P → X such that d s = td . Since P is projective, there exists u : P → P such that t = d u . Since d ( s − ud ) = 0, there exists v : P → P such that s = ud + d v . P d sv P d tu X P d P d X d is in the radical, so is s . Thus the assertion is shown. (cid:3) The following analog of Wakamatsu’s lemma [AR4] will be useful.
Lemma 2.6.
Let η : 0 → Y → T ′ f −→ X be an exact sequence in mod Λ , where T is τ -rigid, and f : T ′ → X is a right ( add T ) -approximation. Then we have Y ∈ ⊥ ( τ T ) .Proof. Replacing X by Im f , we can assume that f is surjective. We apply Hom Λ ( − , τ T ) to η toget the exact sequence0 = Hom Λ ( T ′ , τ T ) → Hom Λ ( Y, τ T ) → Ext ( X, τ T ) Ext ( f,τT ) −−−−−−−→ Ext ( T ′ , τ T ) , where we have Hom Λ ( T ′ , τ T ) = 0 because T is τ -rigid. Since f : T ′ → X is a right ( add T )-approximation, the induced map ( T, f ) : Hom Λ ( T, T ′ ) → Hom Λ ( T, X ) is surjective. Then also the induced map Hom Λ ( T, T ′ ) → Hom Λ ( T, X ) of the maps modulo projectives is surjective, so bythe AR duality the map Ext ( f, τ T ) : Ext ( X, τ T ) → Ext ( T ′ , τ T ) is injective. It follows thatHom Λ ( Y, τ T ) = 0. (cid:3) τ -rigid modules and torsion classes. The following correspondence is basic in our paper,where we denote by f-torsΛ the set of functorially finite torsion classes in mod Λ. Theorem 2.7.
There is a bijection s τ -tiltΛ ←→ f-torsΛ given by s τ -tiltΛ ∋ T Fac T ∈ f-torsΛ and f-torsΛ ∋ T 7→ P ( T ) ∈ s τ -tiltΛ .Proof. Let first T be a functorially finite torsion class in mod Λ. Then we know that T = P ( T )is τ -rigid by Proposition 1.2(c). Let e ∈ Λ be a maximal idempotent such that
T ⊆ mod (Λ / h e i ).Then we have | Λ / h e i| = | Λ / ann T | , and | Λ / ann T | = | T | by Proposition 1.1(e). Hence ( T, Λ e ) isa support τ -tilting pair for Λ. Moreover we have T = Fac P ( T ) by Proposition 1.1(g).Assume conversely that T is a support τ -tilting Λ-module. Then T is a τ -tilting (Λ / h e i )-modulefor an idempotent e of Λ. Thus Fac T is a functorially finite torsion class in mod (Λ / h e i ) such that T ∈ add P ( Fac T ) by Proposition 1.2(b). Since | T | = | Λ / h e i| , we have add T = add P ( Fac T ) byProposition 1.3. Thus T ≃ P ( Fac T ). (cid:3) We denote by τ -tiltΛ (respectively, tiltΛ) the set of isomorphism classes of basic τ -tilting Λ-modules (respectively, tilting Λ-modules). On the other hand, we denote by sf-torsΛ (respectively,ff-torsΛ) the set of sincere (respectively, faithful) functorially finite torsion classes in mod Λ. Corollary 2.8.
The bijection in Theorem 2.7 induces bijections τ -tiltΛ ←→ sf-torsΛ and tiltΛ ←→ ff-torsΛ . Proof.
Let T be a support τ -tilting Λ-module. By Proposition 2.2, it follows that T is a τ -tiltingΛ-module (respectively, tilting Λ-module) if and only if T is sincere (respectively, faithful) if andonly if Fac T is sincere (respectively, faithful). (cid:3) We are interested in the torsion classes where our original module U is a direct summand of T = P ( T ), since we would like to complete U to a (support) τ -tilting module. The conditions forthis to be the case are the following. Proposition 2.9.
Let T be a functorially finite torsion class and U a τ -rigid Λ -module. Then U ∈ add P ( T ) if and only if Fac U ⊆ T ⊆ ⊥ ( τ U ) .Proof. We have T = Fac P ( T ) by Proposition 1.1(g).Assume Fac U ⊆ T ⊆ ⊥ ( τ U ). Then U is in T . We want to show that U is Ext-projective in T ,that is, Ext ( U, T ) = 0, or equivalently Hom Λ ( P ( T ) , τ U ) = 0, by Proposition 1.2(a). This followssince P ( T ) ∈ T ⊆ ⊥ ( τ U ). Hence U is a direct summand of P ( T ).Conversely, assume U ∈ add P ( T ). Then we must have U ∈ T , and hence Fac U ⊆ T . Since U is Ext-projective in T , we have Ext ( U, T ) = 0. Since T = Fac T , we have Hom Λ ( T , τ U ) = 0 byProposition 1.2(a). Hence we have T ⊆ ⊥ ( τ U ). (cid:3) We now prove the analog, for τ -tilting modules, of the Bongartz completion of classical tiltingmodules. Theorem 2.10.
Let U be a τ -rigid Λ -module. Then T := ⊥ ( τ U ) is a sincere functorially finitetorsion class and T := P ( T ) is a τ -tilting Λ -module satisfying U ∈ add T and ⊥ ( τ T ) = Fac T . We call P ( ⊥ ( τ U )) the Bongartz completion of U . Proof.
The first part follows from the following observation.
Lemma 2.11.
For any τ -rigid Λ -module U , we have a sincere functorially finite torsion class ⊥ ( τ U ) . -TILTING THEORY 11 Proof.
When U is τ -rigid, then Sub τ U is a torsionfree class by the dual of Proposition 1.2(b). Then( ⊥ ( τ U ) , Sub τ U ) is a torsion pair, and
Sub τ U and ⊥ ( τ U ) are functorially finite by Proposition 1.1.Assume that ⊥ ( τ U ) is not sincere. Then we have ⊥ ( τ U ) ⊆ mod (Λ / h e i ) for some primitiveidempotent e in Λ. The corresponding simple Λ-module S is not a composition factor of anymodule in ⊥ ( τ U ); in particular Hom( ⊥ ( τ U ) , D ( e Λ)) = 0. Then D ( e Λ) is in
Sub τ U . But this isa contradiction since τ U , and hence also any module in
Sub τ U , has no nonzero injective directsummands. (cid:3)
By Corollary 2.8, it follows that T is a τ -tilting Λ-module such that ⊥ ( τ U ) = Fac T . ByProposition 2.9, we have U ∈ add T . Clearly ⊥ ( τ U ) ⊇ ⊥ ( τ T ) since U is in add T . Hence we get Fac T = ⊥ ( τ U ) ⊇ ⊥ ( τ T ) ⊇ Fac T , and consequently ⊥ ( τ T ) = Fac T . (cid:3) We have the following characterizations of a τ -rigid module being τ -tilting. Theorem 2.12.
The following are equivalent for a τ -rigid Λ -module T . (a) T is τ -tilting. (b) T is maximal τ -rigid, i.e. if T ⊕ X is τ -rigid for some Λ -module X , then X ∈ add T . (c) ⊥ ( τ T ) = Fac T . (d) If Hom Λ ( T, τ X ) = 0 and
Hom Λ ( X, τ T ) = 0 , then X ∈ add T .Proof. (a) ⇒ (b): Immediate from Proposition 1.3.(b) ⇒ (c): Let U be the Bongartz completion of T . Since T is maximal τ -rigid, we have T ≃ U ,and hence ⊥ ( τ T ) = ⊥ ( τ U ) = Fac U = Fac T , using Theorem 2.10.(c) ⇒ (a): Let T be τ -rigid with ⊥ ( τ T ) = Fac T . Let U be the Bongartz completion of T . Thenwe have Fac T = ⊥ ( τ T ) ⊇ ⊥ ( τ U ) ⊇ Fac U ⊇ Fac T, and hence all inclusions are equalities. Since Fac U = Fac T , there exists an exact sequence0 Y T ′ f U f : T ′ → U is a right ( add T )-approximation. By the Wakamatsu-type Lemma 2.6 we haveHom Λ ( Y, τ T ) = 0, and hence Hom Λ ( Y, τ U ) = 0 since ⊥ ( τ T ) = ⊥ ( τ U ). By the AR duality we haveExt ( U, Y ) ≃ D Hom Λ ( Y, τ U ) = 0, and hence the sequence (1) splits. Then it follows that U is in add T . Thus T is a τ -tilting Λ-module.(a)+(c) ⇒ (d): Assume that (a) and (c) hold, and Hom Λ ( T, τ X ) = 0 and Hom Λ ( X, τ T ) =0. Then Ext ( X, Fac T ) = 0 by Proposition 1.2(a) and X is in ⊥ τ T = Fac T . Thus X is in add P ( Fac T ) = add T by Theorem 2.7.(d) ⇒ (b): This is clear. (cid:3) We note the following generalization.
Corollary 2.13.
The following are equivalent for a τ -rigid pair ( T, P ) for Λ . (a) ( T, P ) is a support τ -tilting pair for Λ . (b) If ( T ⊕ X, P ) is τ -rigid for some Λ -module X , then X ∈ add T . (c) ⊥ ( τ T ) ∩ P ⊥ = Fac T . (d) If Hom Λ ( T, τ X ) = 0 , Hom Λ ( X, τ T ) = 0 and
Hom Λ ( P, X ) = 0 , then X ∈ add T .Proof. In view of Lemma 2.1(b), the assertion follows immediately from Theorem 2.12 by replacingΛ by Λ / h e i for an idempotent e of Λ satisfying add P = add Λ e . (cid:3) In the rest of this subsection, we discuss the left-right symmetry of τ -rigid modules. It issomehow surprising that there exists a bijection between support τ -tilting Λ-modules and support τ -tilting Λ op -modules. We decompose M in mod Λ as M = M pr ⊕ M np where M pr is a maximalprojective direct summand of M . For a τ -rigid pair ( M, P ) for Λ, let(
M, P ) † := (Tr M np ⊕ P ∗ , M ∗ pr ) = (Tr M ⊕ P ∗ , M ∗ pr ) . We denote by τ -rigidΛ the set of isomorphism classes of basic τ -rigid pairs of Λ. Theorem 2.14. ( − ) † gives bijections τ -rigidΛ ←→ τ -rigidΛ op and s τ -tiltΛ ←→ s τ -tiltΛ op such that ( − ) †† = id . For a support τ -tilting Λ-module M , we simply write M † := Tr M np ⊕ P ∗ where ( M, P ) is asupport τ -tilting pair for Λ. Proof.
We only have to show that (
M, P ) † is a τ -rigid pair for Λ op since the correspondence( M, P ) ( M, P ) † is clearly an involution. We have0 = Hom Λ ( M np , τ M ) = Hom Λ op (Tr M, DM np ) = Hom Λ op (Tr M, τ Tr M ) . (2)Moreover we have0 = Hom Λ ( M pr , τ M ) = Hom Λ op (Tr M, DM pr ) = D Hom Λ op ( M ∗ pr , Tr M ) . (3)On the other hand, we have0 = Hom Λ ( P, M ) = Hom Λ ( P, M pr ) ⊕ Hom Λ ( P, M np ) . (4)Thus we have 0 = D ( P ∗ ⊗ Λ M np ) = Hom Λ op ( P ∗ , DM np ) = Hom Λ op ( P ∗ , τ Tr M ) . This together with (2) shows that Tr M ⊕ P ∗ is a τ -rigid Λ op -module. We have Hom Λ op ( M ∗ pr , P ∗ ) =0 by (4). This together with (3) shows that ( M, P ) † is a τ -rigid pair for Λ op . (cid:3) Now we discuss dual notions of τ -rigid and τ -tilting modules even though we do not use themin this paper. • We call M in mod Λ τ − -rigid if Hom Λ ( τ − M, M ) = 0. • We call M in mod Λ τ − -tilting if M is τ − -rigid and | M | = | Λ | . • We call M in mod Λ support τ − -tilting if M is a τ − -tilting (Λ / h e i )-module for some idem-potent e of Λ.Clearly M is τ − -rigid (respectively, τ − -tilting, support τ − -tilting) Λ-module if and only if DM is τ -rigid (respectively, τ -tilting, support τ -tilting) Λ op -module.We denote by cotiltΛ (respectively, τ − -tiltΛ, s τ − -tiltΛ) the set of isomorphism classes of basiccotilting (respectively, τ − -tilting, support τ − -tilting) Λ-modules. On the other hand, we denoteby f-torfΛ the set of functorially finite torsionfree classes in mod Λ, and by sf-torfΛ (respectively,ff-torfΛ) the set of sincere (respectively, faithful) functorially finite torsionfree classes in mod
Λ.We have the following results immediately from Theorem 2.7 and Corollary 2.8.
Theorem 2.15.
We have bijections s τ − -tiltΛ ←→ f-torfΛ , τ − -tiltΛ ←→ sf-torfΛ and cotiltΛ ←→ ff-torfΛ given by s τ − -tiltΛ ∋ T Sub T ∈ f-torfΛ and f-torfΛ ∋ F 7→ I ( F ) ∈ s τ − -tiltΛ . On the other hand, we have a bijections τ -tiltΛ ←→ s τ − -tiltΛgiven by ( M, P ) D (( M, P ) † ) = ( τ M ⊕ νP, νM pr ). Thus we have bijectionsf-torsΛ ←→ s τ -tiltΛ ←→ s τ − -tiltΛ ←→ f-torfΛby Theorems 2.7 and 2.15. We end this subsection with the following observation. Proposition 2.16. (a)
The above bijections send
T ∈ f-torsΛ to T ⊥ ∈ f-torfΛ . (b) For any support τ -tilting pair ( M, P ) for Λ , the torsion pairs ( Fac
M, M ⊥ ) and ( ⊥ ( τ M ⊕ νP ) , Sub ( τ M ⊕ νP )) in mod Λ coincide. -TILTING THEORY 13 Proof. (b) We only have to show
Fac M = ⊥ ( τ M ⊕ νP ). It follows from Proposition 1.2(b) andits dual that ( Fac
M, M ⊥ ) and ( ⊥ ( τ M ⊕ νP ) , Sub ( τ M ⊕ νP )) are torsion pairs in mod Λ. Theycoincide since
Fac M = ⊥ ( τ M ) ∩ P ⊥ = ⊥ ( τ M ⊕ νP ) holds by Corollary 2.13(c).(a) Let T ∈ f-torsΛ and (
M, P ) be the corresponding support τ -tilting pair for Λ. Since T ⊥ = M ⊥ and D ( M † ) = τ M ⊕ νP , the assertion follows from (b). (cid:3) Mutation of support τ -tilting modules. In this section we prove our main result oncomplements for almost complete support τ -tilting pairs. Let us start with the following result. Proposition 2.17.
Let T be a basic τ -rigid module which is not τ -tilting. Then there are at leasttwo basic support τ -tilting modules which have T as a direct summand.Proof. By Theorem 2.12, T = Fac T is properly contained in T = ⊥ ( τ T ). By Theorem 2.7 andLemma 2.11, we have two different support τ -tilting modules P ( T ) and P ( T ) up to isomorphism.By Proposition 2.9, they are extensions of T . (cid:3) Our aim is to prove the following result.
Theorem 2.18.
Let Λ be a finite dimensional k -algebra. Then any basic almost complete support τ -tilting pair ( U, Q ) for Λ is a direct summand of exactly two basic support τ -tilting pairs ( T, P ) and ( T ′ , P ′ ) for Λ . Moreover we have { Fac T, Fac T ′ } = { Fac U, ⊥ ( τ U ) ∩ Q ⊥ } . Before proving Theorem 2.18, we introduce a notion of mutation.
Definition 2.19.
Two basic support τ -tilting pairs ( T, P ) and ( T ′ , P ′ ) for Λ are said to be mu-tations of each other if there exists a basic almost complete support τ -tilting pair ( U, Q ) whichis a direct summand of (
T, P ) and ( T ′ , P ′ ). In this case we write ( T ′ , P ′ ) = µ X ( T, P ) or simply T ′ = µ X ( T ) if X is an indecomposable Λ-module satisfying either T = U ⊕ X or P = Q ⊕ X .We can also describe mutation as follows: Let ( T, P ) be a basic support τ -tilting pair for Λ, and X an indecomposable direct summand of either T or P .(a) If X is a direct summand of T , precisely one of the following holds. • There exists an indecomposable Λ-module Y such that X Y and µ X ( T, P ) :=(
T /X ⊕ Y, P ) is a basic support τ -tilting pair for Λ. • There exists an indecomposable projective Λ-module Y such that µ X ( T, P ) := (
T /X, P ⊕ Y ) is a basic support τ -tilting pair for Λ.(b) If X is a direct summand of P , there exists an indecomposable Λ-module Y such that µ X ( T, P ) := ( T ⊕ Y, P/X ) is a basic support τ -tilting pair for Λ.Moreover, such a module Y in each case is unique up to isomorphism.In the rest of this subsection, we give a proof of Theorem 2.18. The following is the first step. Lemma 2.20.
Let ( T, P ) be a τ -rigid pair for Λ . If U is a τ -rigid Λ -module satisfying ⊥ ( τ T ) ∩ P ⊥ ⊆ ⊥ ( τ U ) , then there is an exact sequence U f −→ T ′ → C → satisfying the following conditions. • f is a minimal left ( Fac T ) -approximation. • T ′ is in add T , C is in add P ( Fac T ) and add T ′ ∩ add C = 0 .Proof. Consider the exact sequence U f −→ T ′ g −→ C →
0, where f is a minimal left ( add T )-approximation. Then g ∈ rad( T ′ , C ).(i) f is a minimal left ( Fac T )-approximation: Take any X ∈ Fac T and s : U → X . By theWakamatsu-type Lemma 2.6, there exists an exact sequence0 → Y → T ′′ h −→ X → h is a right ( add T )-approximation and Y ∈ ⊥ ( τ T ). Moreover we have Y ∈ P ⊥ since T ′′ ∈ P ⊥ . By the assumption that ⊥ ( τ T ) ∩ P ⊥ ⊆ ⊥ ( τ U ), we have Hom Λ ( Y, τ U ) = 0, henceExt ( U, Y ) = 0. Then we have an exact sequenceHom Λ ( U, T ′′ ) → Hom Λ ( U, X ) → Ext ( U, Y ) = 0 . Thus there is some t : U → T ′′ such that s = ht . U t s f T ′ Y T ′′ h X T ′′ ∈ add T and f is a left ( add T )-approximation, there is some u : T ′ → T ′′ such that t = uf . Hence we have hu : T ′ → X such that ( hu ) f = ht = s , and the claim follows.(ii) C ∈ add P ( Fac T ): We have an exact sequence 0 → Im f i −→ T ′ → C →
0, which gives rise toan exact sequenceHom Λ ( T ′ , Fac T ) ( i, Fac T ) −−−−−→ Hom Λ (Im f, Fac T ) → Ext ( C, Fac T ) → Ext ( T ′ , Fac T ) . We know from (i) that ( f, Fac T ) : Hom Λ ( T ′ , Fac T ) → Hom Λ ( U, Fac T ) is surjective, and hence( i, Fac T ) is surjective. Further, Ext ( T ′ , Fac T ) = 0 by Proposition 1.2 since T ′ is in add T and T is τ -rigid. Then it follows that Ext ( C, Fac T ) = 0. Since C ∈ Fac T , this means that C isExt-projective in Fac T .(iii) add T ′ ∩ add C = 0: To show this, it is clearly sufficient to show Hom Λ ( T ′ , C ) ⊆ rad( T ′ , C ).Let s : T ′ → C be an arbitrary map. We have an exact sequence Hom Λ ( U, T ′ ) → Hom Λ ( U, C ) → Ext ( U, Im f ). Since Ext ( U, Im f ) = 0 because Im f is in Fac U , and U is τ -tilting, there is a map t : U → T ′ such that sf = gt . Since f is a left ( add T )-approximation, and T ′ is in add T , there is amap u : T ′ → T ′ such that t = uf . Then ( s − gu ) f = sf − gt = 0, hence there is some v : C → C such that s − gu = vg , and hence s = gu + vg . U ft T ′ gsu C v U f T ′ g C f Since g ∈ rad( T ′ , C ), it follows that s ∈ rad( T ′ , C ). Hence Hom Λ ( T ′ , C ) ⊆ rad( T ′ , C ), andconsequently add T ′ ∩ add C = 0. (cid:3) The following information on the previous lemma is useful.
Lemma 2.21.
In Lemma 2.20, assume C = 0 . Then f : U → T ′ induces an isomorphism U/ h e i U ≃ T ′ for a maximal idempotent e of Λ satisfying eT = 0 . In particular, if T is sincere,then U ≃ T ′ .Proof. By our assumption, we have an exact sequence0 Ker f U f T ′ . (5)Applying Hom Λ ( − , Fac T ), we have an exact sequenceHom Λ ( T ′ , Fac T ) ( f, Fac T ) −−−−−→ Hom Λ ( U, Fac T ) → Hom Λ (Ker f, Fac T ) → Ext ( T ′ , Fac T ) . We have Ext ( T ′ , Fac T ) = 0 because T ′ is in add T and T is τ -tilting. Since ( f, Fac T ) is surjective,it follows that Hom Λ (Ker f, Fac T ) = 0 and so Ker f ∈ ⊥ ( Fac T ). On the other hand, since T isa sincere (Λ / h e i )-module, mod (Λ / h e i ) is the smallest torsionfree class of mod Λ containing
Fac T .Thus we have a torsion pair ( ⊥ ( Fac T ) , mod (Λ / h e i )), and the canonical sequence for X associatedwith this torsion pair is given by0 h e i X X X/ h e i X . Since Ker f ∈ ⊥ ( Fac T ) and T ′ ∈ Fac T ⊆ mod (Λ / h e i ), the canonical sequence of U is given by (5).Thus we have U/ h e i U ≃ T ′ . (cid:3) -TILTING THEORY 15 In the next result we prove a useful restriction on X when T = X ⊕ U is τ -tilting and X isindecomposable. Proposition 2.22.
Let T = X ⊕ U be a basic τ -tilting Λ -module, with X indecomposable. Thenexactly one of ⊥ ( τ U ) ⊆ ⊥ ( τ X ) and X ∈ Fac U holds.Proof. First we assume that ⊥ ( τ U ) ⊆ ⊥ ( τ X ) and X ∈ Fac U both hold. Then we have Fac U = Fac T = ⊥ ( τ T ) = ⊥ ( τ U ) , which implies that U is τ -tilting by Theorem 2.12, a contradiction.Let Y ⊕ U be the Bongartz completion of U . Then we have ⊥ τ ( Y ⊕ U ) = ⊥ ( τ U ) ⊇ ⊥ τ T . Usingthe triple ( T, , Y ⊕ U ) instead of ( T, P, U ) in Lemma 2.20, there is an exact sequence Y ⊕ U ( f
00 1 ) T ′ ⊕ U T ′′ , where f : Y → T ′ and (cid:0) f
00 1 (cid:1) : Y ⊕ U → T ′ ⊕ U are minimal left ( Fac T )-approximations, T ′ and T ′′ are in add T and add ( T ′ ⊕ U ) ∩ add T ′′ = 0. Then we have T ′′ ∈ add X .Assume first T ′′ = 0. Then T ′′ ≃ X ℓ for some ℓ ≥
1, so we have T ′ ∈ add U . Since we have asurjective map T ′ → T ′′ , we have X ∈ Fac T ′ ⊆ Fac U .Assume now that T ′′ = 0. Applying Lemma 2.21, we have that (cid:0) f
00 1 (cid:1) : Y ⊕ U → T ′ ⊕ U isan isomorphism since T is sincere. Thus Y ∈ add T , and we must have Y ≃ X . Thus ⊥ ( τ X ) = ⊥ ( τ Y ) ⊇ ⊥ ( τ U ). (cid:3) Now we are ready to prove Theorem 2.18.(i) First we assume that Q = 0 (i.e. U is an almost complete τ -tilting module).In view of Proposition 2.17 it only remains to show that there are at most two extensions of U toa support τ -tilting module. Using the bijection in Theorem 2.7, we only have to show that for anysupport τ -tilting module X ⊕ U , the torsion class Fac ( X ⊕ U ) is either Fac U or ⊥ ( τ U ). If X = 0 (i.e. U is a support τ -tilting module), then this is clear. If X = 0, then X ⊕ U is a τ -tilting Λ-module.Moreover by Proposition 2.22 either X ∈ Fac U or ⊥ ( τ U ) ⊆ ⊥ ( τ X ) holds. If X ∈ Fac U , then wehave Fac ( X ⊕ U ) = Fac U . If ⊥ ( τ U ) ⊆ ⊥ ( τ X ), then we have Fac ( X ⊕ U ) = ⊥ ( τ ( X ⊕ U )) = ⊥ ( τ U ).Thus the assertion follows.(ii) Let ( U, Q ) be a basic almost complete support τ -tilting pair for Λ and e be an idempotent ofΛ such that add Q = add Λ e . Then U is an almost complete τ -tilting (Λ / h e i )-module by Proposition2.3(a). It follows from (i) that U is a direct summand of exactly two basic support τ -tilting (Λ / h e i )-modules. Thus the assertion follows since basic support τ -tilting (Λ / h e i )-modules which have U asa direct summand correspond bijectively to basic support τ -tilting pairs for Λ which have ( U, Q )as a direct summand. (cid:3)
The following special case of Lemma 2.20 is useful.
Proposition 2.23.
Let T be a support τ -tilting Λ -module. Assume that one of the followingconditions is satisfied. (i) U is a τ -rigid Λ -module such that Fac T ⊆ ⊥ ( τ U ) . (ii) U is a support τ -tilting Λ -module such that U ≥ T .Then there exists an exact sequence U f −→ T → T → such that f is a minimal left ( Fac T ) -approximation of U and T and T are in add T and satisfy add T ∩ add T = 0 .Proof. Let (
T, P ) be a support τ -tilting pair for Λ. Then ⊥ ( τ T ) ∩ P ⊥ = Fac T holds by Corollary2.13(c). Thus ⊥ ( τ T ) ∩ P ⊥ ⊆ ⊥ ( τ U ) holds for both cases. Hence the assertion is immediate fromLemma 2.20 since C is in add P ( Fac T ) = add T by Theorem 2.7. (cid:3) The following well-known result [HU1] can be shown as an application of our results.
Corollary 2.24.
Let Λ be a finite dimensional k -algebra and U a basic almost complete tilting Λ -module. Then U is faithful if and only if U is a direct summand of precisely two basic tilting Λ -modules.Proof. It follows from Theorem 2.18 that U is a direct summand of exactly two basic support τ -tilting Λ-modules T and T ′ such that Fac T = Fac U . If U is faithful, then T and T ′ are tiltingΛ-modules by Proposition 2.2(b). Thus the ‘only if’ part follows. If U is not faithful, then T is nota tilting Λ-module since it is not faithful because Fac T = Fac U . Thus the ‘if’ part follows. (cid:3) Partial order, exchange sequences and Hasse quiver.
In this section we investigatetwo quivers. One is defined by partial order, and the other one by mutation. We show that theycoincide.Since we have a bijection T Fac T between s τ -tiltΛ and f-torsΛ, then inclusion in f-torsΛ givesrise to a partial order on s τ -tiltΛ, and we have an associated Hasse quiver. Note that s τ -tiltΛ hasa unique maximal element Λ and a unique minimal element 0.The following description of when T ≥ U holds will be useful. Lemma 2.25.
Let ( T, P ) and ( U, Q ) be support τ -tilting pairs for Λ . Then the following conditionsare equivalent. (a) T ≥ U . (b) Hom Λ ( U, τ T ) = 0 and add P ⊆ add Q . (c) Hom Λ ( U np , τ T np ) = 0 , add T pr ⊇ add U pr and add P ⊆ add Q .Proof. (a) ⇒ (c) Since Fac T ⊇ Fac U , we have add T pr ⊇ add U pr and Hom Λ ( U, τ T ) = 0. Moreover add P ⊆ add Q holds by Proposition 2.2(a).(b) ⇒ (a) We have Fac T = ⊥ ( τ T ) ∩ P ⊥ by Corollary 2.13(c). Since add P ⊆ add Q , we have U ∈ Q ⊥ ⊆ P ⊥ . Since Hom Λ ( U, τ T ) = 0, we have U ∈ ⊥ ( τ T ) ∩ P ⊥ = Fac T , which implies Fac T ⊇ Fac U .(c) ⇒ (b) This is clear. (cid:3) Also we shall need the following.
Proposition 2.26.
Let
T, U, V ∈ s τ -tiltΛ such that T ≥ U ≥ V . Then add T ∩ add V ⊆ add U .Proof. Clearly we have P ( Fac T ) ∩ Fac U ⊆ P ( Fac U ) = add U . Thus we have add T ∩ add V ⊆ P ( Fac T ) ∩ Fac U ⊆ add U . (cid:3) The following observation is immediate.
Proposition 2.27. (a)
For any idempotent e of Λ , the inclusion s τ -tilt(Λ / h e i ) → s τ -tiltΛ preserves the partial order. (b) The bijection ( − ) † : s τ -tiltΛ → s τ -tiltΛ op in Theorem 2.14 reverses the partial order.Proof. (a) This is clear.(b) Let ( T, P ) and (
U, Q ) be support τ -tilting pairs of Λ. By Lemma 2.25, T ≥ U if andonly if Hom Λ ( U np , τ T np ) = 0, add T pr ⊇ add U pr and add P ⊆ add Q . This is equivalent toHom Λ op (Tr T np , τ Tr U np ) = 0, add T ∗ pr ⊇ add U ∗ pr and add P ∗ ⊆ add Q ∗ . By Lemma 2.25 again,this is equivalent to (Tr T np ⊕ P ∗ , T ∗ pr ) ≤ (Tr U np ⊕ Q ∗ , U ∗ pr ). (cid:3) In the rest of this section, we study a relationship between partial order and mutation.
Definition-Proposition 2.28.
Let T = X ⊕ U and T ′ be support τ -tilting Λ-modules such that T ′ = µ X ( T ) for some indecomposable Λ-module X . Then either T > T ′ or T < T ′ holds byTheorem 2.18. We say that T ′ is a left mutation (respectively, right mutation ) of T and we write T ′ = µ − X ( T ) (respectively, T ′ = µ + X ( T )) if the following equivalent conditions are satisfied.(a) T > T ′ (respectively, T < T ′ ).(b) X / ∈ Fac U (respectively, X ∈ Fac U ).(c) ⊥ ( τ X ) ⊇ ⊥ ( τ U ) (respectively, ⊥ ( τ X ) ⊥ ( τ U )). -TILTING THEORY 17 If T is a τ -tilting Λ-module, then the following condition is also equivalent to the above conditions.(d) T is a Bongartz completion of U (respectively, T is a non-Bongartz completion of U ). Proof.
This follows immediately from Theorem 2.18 and Proposition 2.22. (cid:3)
Definition 2.29.
We define the support τ -tilting quiver Q(s τ -tiltΛ) of Λ as follows: • The set of vertices is s τ -tiltΛ. • We draw an arrow from T to U if U is a left mutation of T .Next we show that one can calculate left mutation of support τ -tilting Λ-modules by exchangesequences which are constructed from left approximations. Theorem 2.30.
Let T = X ⊕ U be a basic τ -tilting module which is the Bongartz completion of U , where X is indecomposable. Let X f −→ U ′ g −→ Y → be an exact sequence, where f is a minimalleft ( add U ) -approximation. Then we have the following. (a) If U is not sincere, then Y = 0 . In this case U = µ − X ( T ) holds and this is a basic support τ -tilting Λ -module which is not τ -tilting. (b) If U is sincere, then Y is a direct sum of copies of an indecomposable Λ -module Y and isnot in add T . In this case Y ⊕ U = µ − X ( T ) holds and this is a basic τ -tilting Λ -module.Proof. We first make some preliminary observations. We have ⊥ ( τ U ) ⊆ ⊥ ( τ X ) because T is aBongartz completion of U . By Lemma 2.20, we have an exact sequence X f −→ U ′ g −→ Y → U ′ is in add U , Y is in add P ( Fac U ), add U ′ ∩ add Y = 0 and f is a left ( Fac U )-approximation. We have Ext ( Y, Fac U ) = 0 since Y ∈ add P ( Fac U ), and hence Hom Λ ( U, τ Y ) = 0by Proposition 1.2. We have an injective map Hom Λ ( Y, τ ( Y ⊕ U )) → Hom Λ ( U ′ , τ ( Y ⊕ U )). Since U is τ -rigid, we have that Hom Λ ( U ′ , τ ( Y ⊕ U )) = 0, and consequently Hom Λ ( Y, τ ( Y ⊕ U )) = 0. Itfollows that Y ⊕ U is τ -rigid.We show that g : U ′ → Y is a right ( add T )-approximation. To see this, consider the exactsequence Hom Λ ( T, U ′ ) → Hom Λ ( T, Y ) → Ext ( T, Im f ) . Since Im f ∈ Fac T , we have Ext ( T, Im f ) = 0, which proves the claim.We have that Y does not have any indecomposable direct summand from add T . For if T ′ in add T is an indecomposable direct summand of Y , then the natural inclusion T ′ → Y factorsthrough g : U ′ → Y . This contradicts the fact that f : X → U ′ is left minimal.Now we are ready to prove the claims (a) and (b).(a) Assume first that U is not sincere. Let e be a primitive idempotent with eU = 0. Then U is a τ -rigid (Λ / h e i )-module. Since | U | = | Λ |− | Λ / h e i| , we have that U is a τ -tilting (Λ / h e i )-module,and hence a support τ -tilting Λ-module which is not τ -tilting.(b) Next assume that U is sincere. Since we have already shown that Y ⊕ U is τ -rigid and Y / ∈ add T , it is enough to show Y = 0. Otherwise we have X ≃ U ′ by Lemma 2.21 since U issincere. This is not possible since U ′ is in add U , but X is not. Hence it follows that Y = 0. (cid:3) We do not know the answer to the following.
Question 2.31. Is Y always indecomposable in Theorem 2.30(b)? Note that right mutation can not be calculated as directly as left mutation.
Remark 2.32.
Let T and T ′ be support τ -tilting Λ-modules such that T ′ = µ X ( T ) for someindecomposable Λ-module X .(a) If T ′ = µ − X ( T ), then we can calculate T ′ by applying Theorem 2.30.(b) If T ′ = µ + X ( T ), then we can calculate T ′ using the following three steps: First calculate T † . Then calculate T ′† by applying Theorem 2.30 to T † . Finally calculate T ′ by applying( − ) † to T ′† . Our next main result is the following.
Theorem 2.33.
For
T, U ∈ s τ -tiltΛ , the following conditions are equivalent. (a) U is a left mutation of T . (b) T is a right mutation of U . (c) T > U and there is no V ∈ s τ -tiltΛ such that T > V > U . Before proving Theorem 2.33, we give the following result as a direct consequence.
Corollary 2.34.
The support τ -tilting quiver Q(s τ -tiltΛ) is the Hasse quiver of the partiallyordered set s τ -tiltΛ . The following analog of [AI, Proposition 2.36] is a main step to prove Theorem 2.33.
Theorem 2.35.
Let U and T be basic support τ -tilting Λ -modules such that U > T . Then: (a)
There exists a right mutation V of T such that U ≥ V . (b) There exists a left mutation V ′ of U such that V ′ ≥ T . Before proving Theorem 2.35, we finish the proof of Theorem 2.33 by using Theorem 2.35.(a) ⇔ (b) Immediate from the definitions.(a) ⇒ (c) Assume that V ∈ s τ -tiltΛ satisfies T > V ≥ U . Then we have add T ∩ add U ⊆ add V by Proposition 2.26. Thus T and V have an almost complete support τ -tilting pair for Λ as acommon direct summand. Hence we have V ≃ U by Theorem 2.18.(c) ⇒ (a) By Theorem 2.35, there exists a left mutation V of T such that T > V ≥ U . Then V ≃ U by our assumption. Thus U is a left mutation of T . (cid:3) To prove Theorem 2.35, we shall need the following results.
Lemma 2.36.
Let U and T be basic support τ -tilting Λ -modules such that U > T . Let U f −→ T → T → be an exact sequence as given in Proposition 2.23. If X is an indecomposable directsummand of T which does not belong to add T , then we have U ≥ µ X ( T ) > T .Proof. First we show µ X ( T ) > T . Since X is in Fac T ⊆ Fac U , there exists a surjective map a : U ℓ → X for some ℓ >
0. Since f ℓ : U ℓ → ( T ) ℓ is a left ( add T )-approximation, a factorsthrough f ℓ and we have X ∈ Fac T . It follows from X / ∈ add T that X ∈ Fac T ⊆ Fac µ X ( T ).Thus Fac T ⊆ Fac µ X ( T ) and we have µ X ( T ) > T .Next we show U ≥ µ X ( T ). Let ( U, Λ e ) and ( T, Λ e ′ ) be support τ -tilting pairs for Λ. ByProposition 2.27(b), we know that U † = Tr U ⊕ e Λ and T † = Tr T ⊕ e ′ Λ are support τ -tiltingΛ op -modules such that U † < T † . In particular, any minimal right ( add T † )-approximationTr T ⊕ P → U † (6)of U † with T ∈ add T np and P ∈ add e ′ Λ is surjective. The following observation shows T ∈ add T . Lemma 2.37.
Let X and Y be in mod Λ and P in proj Λ op . Let f : Y → X be a left ( add X ) -approximation of Y and g : Tr X ⊕ P → Tr Y be a minimal right ( add Tr X ⊕ P ) -approximationof Tr Y with X ∈ add X np and P ∈ add P . If g is surjective, then X is a direct summand of X .Proof. Assume that g is surjective and consider the exact sequence0 K h Tr X ⊕ P g Tr Y . Then h is in rad( K, Tr X ⊕ P ) since g is right minimal. It is easy to see that in the stable category mod Λ op , a pseudokernel of g is given by h , which is in the radical of mod Λ op . In particular, g isa minimal right ( add Tr X )-approximation in mod Λ op . Since Tr : mod Λ → mod Λ op is a duality,we have that Tr g : Tr Tr Y → Tr(Tr X ⊕ P ) = X is a minimal left ( add X )-approximation ofTr Tr Y in mod Λ. On the other hand, f : Y → X is clearly a left ( add X )-approximation of Y in mod Λ. Since Tr Tr Y is a direct summand of Y , we have that X is a direct summand of X in mod Λ. Thus the assertion follows. (cid:3) -TILTING THEORY 19
We now finish the proof of Lemma 2.36.Since T ∈ add T and X / ∈ add T , we have X / ∈ add T and hence U † ∈ Fac (Tr(
T /X ) ⊕ e ′ Λ) by(6). Hence we have U † ≤ µ X ( T ) † , which implies U ≥ µ X ( T ) by Proposition 2.27(b). (cid:3) Now we are ready to prove Theorem 2.35.We only prove (a) since (b) follows from (a) and Proposition 2.27(b).(i) Let ( U, Λ e ) and ( T, Λ e ′ ) be support τ -tilting pairs for Λ. Let U T T T / ∈ add T , then any indecomposable directsummand X of T which is not in add T satisfies U ≥ µ X ( T ) > T by Lemma 2.36. Thus weassume T ∈ add T in the rest of proof. Since add T ∩ add T = 0, we have T = 0 which implies T = U/ h e ′ i U by Lemma 2.21.(ii) By Proposition 2.27(b), we know that U † = Tr U ⊕ e Λ and T † = Tr T ⊕ e ′ Λ are support τ -tilting Λ op -modules such that U † < T † . Let T † f U † add T † )-approximation of U † . If e ′ Λ / ∈ add T † , then any indecomposable directsummand Q of e ′ Λ which is not in add T † satisfies U † ∈ Fac ( T † /Q ). Thus we have U † ≤ µ Q ( T † )and U ≥ µ Q ∗ ( T ) > T by Proposition 2.27. We assume e ′ Λ ∈ add T † in the rest of proof.(iii) We show that there exists an exact sequence P a Tr T ⊕ P Tr U mod Λ op such that P ∈ proj Λ op , P ∈ add e ′ Λ, a ∈ rad( P , Tr T ⊕ P ) and the map( a, U † ) : Hom Λ op (Tr T ⊕ P , U † ) Hom Λ op ( P , U † ) (9)is surjective.Let Q d −→ Q → U → U . Let d ′ : Q ′ → Q be aright ( add Λ e ′ )-approximation of Q . Since T = U/ h e ′ i U by (i), we have a projective presentation Q ′ ⊕ Q ( d ′ d ) −−→ Q → T → T . Thus we have an exact sequence Q ∗ d ′∗ d ∗ ) Q ′ ∗ ⊕ Q ∗ ( c ′ c ) Tr T ⊕ Q op -module Q . We have a commutative diagram Q ∗ d ∗ d ′∗ Q ∗ − c Tr U Q ′ ∗ c ′ Tr T ⊕ Q Tr U c ′ as c ′ = (cid:0) a
00 1 Q ′′ (cid:1) : Q ′ ∗ = P ⊕ Q ′′ Tr T ⊕ Q = Tr T ⊕ P ⊕ Q ′′ , where a is in the radical. Then we naturally have an exact sequence (8), and clearly we have P ∈ proj Λ op and P ∈ add e ′ Λ by our construction. It remains to show that (9) is surjective. Weonly have to show that the map( c ′ , U † ) : Hom Λ op (Tr T ⊕ Q, U † ) Hom Λ op ( Q ′ ∗ , U † )is surjective. Take any map s : Q ′ ∗ → U † . By Proposition 2.4(c), there exists t : Q ∗ → U † suchthat sd ′∗ = td ∗ . Thus there exists u : Tr T ⊕ Q → U † such that s = uc ′ and t = − uc , which showsthe assertion. (iv) First we assume P in (iii) is non-zero. Since e ′ Λ ∈ add T † by (ii) and P ∈ add e ′ Λ, wehave P ∈ add T † . Thus there exists a morphism s : P → T † which is not in the radical. Since(9) is surjective, there exists t : Tr T ⊕ P → U † such that ta = f s . Since f is a surjective right( add T † )-approximation and P is projective, there exists u : Tr T ⊕ P → T † such that t = f u . P as Tr T ⊕ P tu Tr U T † f U † f ( s − ua ) = 0 and f is right minimal, we have that s − ua is in the radical. Since a is in theradical, so is s , a contradiction.Consequently, we have P = 0. Thus Tr T ⊕ P ≃ Tr U and Tr T ≃ Tr U . Since T ∈ add T byour assumption, we have add T np = add U np . Since U > T , we have T pr ∈ add U pr . Thus U ≃ T ⊕ P for some projective Λ-module P .(v) It remains to consider the case U ≃ T ⊕ P for some projective Λ-module P .Since U > T , we have add Λ e ( add Λ e ′ . Take any indecomposable summand Λ e ′′ of Λ( e ′ − e )and let V := µ Λ e ′′ ( T, Λ e ′ ), which has a form ( T ⊕ X, Λ( e ′ − e ′′ )) with X indecomposable. Clearly V > T holds. Since τ U ∈ add τ ( T ⊕ X ) by our assumption and Λ e ∈ add Λ( e ′ − e ′′ ) by our choiceof e ′′ , we have Fac U = ⊥ ( τ U ) ∩ (Λ e ) ⊥ ⊇ ⊥ ( τ ( T ⊕ X )) ∩ (Λ( e ′ − e ′′ )) ⊥ = Fac V by Corollary 2.13(c). Thus U ≥ V holds. (cid:3) We end this section with the following application, which is an analog of [HU2, Corollary 2.2].
Corollary 2.38. If Q(s τ -tiltΛ) has a finite connected component C , then Q(s τ -tiltΛ) = C .Proof. Fix T in C . Applying Theorem 2.35(a) to Λ ≥ T , we have a sequence T = T < T
Throughout this section, let Λ be a finite dimensional algebra over a field k . Any almost completesilting complex has infinitely many complements. But if we restrict to two-term silting complexes,we get another class of objects extending the (classical) tilting modules and satisfying the twocomplement property (Corollary 3.8). Moreover we will show that there is a bijection betweensupport τ -tilting Λ-modules and two-term silting complexes for Λ, which is of independent interest(Theorem 3.2). The two-term silting complexes are defined as follows. Definition 3.1.
We call a complex P = ( P i , d i ) in K b ( proj Λ) two-term if P i = 0 for all i = 0 , − P ∈ K b ( proj Λ) is two-term if and only if Λ ≥ P ≥ Λ[1].We denote by 2-siltΛ (respectively, 2-presiltΛ) the set of isomorphism classes of basic two-termsilting (respectively, presilting) complexes for Λ.Clearly any two-term complex is isomorphic to a two-term complex P = ( P i , d i ) satisfying d − ∈ rad( P − , P ) in K b ( proj Λ). Moreover, for any two-term complexes P and Q , we haveHom K b ( proj Λ) ( P, Q [ i ]) = 0 for any i = − , , Theorem 3.2.
Let Λ be a finite dimensional k -algebra. Then there exists a bijection ←→ s τ -tiltΛ -TILTING THEORY 21 given by ∋ P H ( P ) ∈ s τ -tiltΛ and s τ -tiltΛ ∋ ( M, P ) ( P ⊕ P ( f −−−→ P ) ∈ where f : P → P is a minimal projective presentation of M . The following result is quite useful.
Proposition 3.3.
Let P be a two-term presilting complex for Λ . (a) P is a direct summand of a two-term silting complex for Λ . (b) P is a silting complex for Λ if and only if | P | = | Λ | .Proof. (a) This is shown in [Ai, Proposition 2.16].(b) The ‘only if’ part follows from Proposition 1.6(a). We will show the ‘if’ part. Let P bea two-term presilting complex for Λ with | P | = | Λ | . By (a), there exists a complex X such that P ⊕ X is silting. Then we have | P ⊕ X | = | Λ | = | P | by Proposition 1.6(a), so X is in add P . Thus P is silting. (cid:3) The following lemma is important.
Lemma 3.4.
Let
M, N ∈ mod Λ . Let P p → P p → M → and Q q → Q q → N → be minimalprojective presentations of M and N respectively. Let P = ( P p → P ) and Q = ( Q q → Q ) betwo-term complexes for Λ . Then the following conditions are equivalent: (a) Hom Λ ( N, τ M ) = 0 . (b) Hom K b ( proj Λ) ( P, Q [1]) = 0 .In particular, M is a τ -rigid Λ -module if and only if P is a presilting complex for Λ .Proof. The condition (a) is equivalent to the fact that ( p , N ) : Hom Λ ( P , N ) → Hom Λ ( P , N ) issurjective by Proposition 2.4(b).(a) ⇒ (b) Any morphism f ∈ Hom K b ( proj Λ) ( P, Q [1]) is given by some f ∈ Hom Λ ( P , Q ). Since( p , N ) is surjective, there exists g : P → N such that q f = gp . Moreover, since P is projective,there exists h : P → Q such that q h = g . Since q ( f − h p ) = 0, we have h : P → Q with f = q h + h p . 0 P p fh P p gh M Q q Q q N . Hence we have Hom K b ( proj Λ) ( P, Q [1]) = 0.(b) ⇒ (a) Take any f ∈ Hom Λ ( P , N ). Since P is projective, there exists g : P → Q such that q g = f . P p g f P Q q Q q N . Then g gives a morphism P → Q [1] in K b ( proj Λ). Since Hom K b ( proj Λ) ( P, Q [1]) = 0, there exist h : P → Q and h : P → Q such that g = q h + h p . Hence we have f = q ( q h + h p ) = q h p . Therefore ( p , N ) is surjective. (cid:3) We also need the following observation.
Lemma 3.5.
Let P p → P p → M → be a minimal projective presentation of M in mod Λ and P := ( P p → P ) be a two-term complex for Λ . Then for any Q in proj Λ , the following conditionsare equivalent. (a) Hom Λ ( Q, M ) = 0 . (b) Hom K b ( proj Λ) ( Q, P ) = 0 . Proof.
The proof is left to the reader since it is straightforward. (cid:3)
The following result shows that silting complexes for Λ give support τ -tilting modules. Proposition 3.6.
Let P = ( P d → P ) be a two-term complex for Λ and M := Cok d . (a) If P is a silting complex for Λ and d is right minimal, then M is a τ -tilting Λ -module. (b) If P is a silting complex for Λ , then M is a support τ -tilting Λ -module.Proof. (b) We write d = ( d ′
0) : P = P ′ ⊕ P ′′ → P , where d ′ is right minimal. Then the sequence P ′ d ′ → P → M → M . We show that ( M, P ′′ ) is a support τ -tilting pair for Λ. Since P is silting, M is a τ -rigid Λ-module by Lemma 3.4. On the other hand,since P is silting, we have Hom K b ( proj Λ) ( P ′′ , P ) = 0. By Lemma 3.5, we have Hom Λ ( P ′′ , M ) = 0.Thus ( M, P ′′ ) is a τ -rigid pair for Λ. Since d ′ is a minimal projective presentation of M , we have | M | = | P ′ d ′ −→ P | . Thus we have | M | + | P ′′ | = | P ′ d ′ −→ P | + | P ′′ | = | P | , which is equal to | Λ | by Proposition 1.6(a). Hence ( M, P ′′ ) is a support τ -tilting pair for Λ.(a) This is the case P ′′ = 0 in (b). (cid:3) The following result shows that support τ -tilting Λ-modules give silting complexes for Λ. Proposition 3.7.
Let P d −→ P d −→ M → be a minimal projective presentation of M in mod Λ . (a) If M is a τ -tilting Λ -module, then ( P d −→ P ) is a silting complex for Λ . (b) If ( M, Q ) is a support τ -tilting pair for Λ , then P ⊕ Q ( d −−−−→ P is a silting complex for Λ .Proof. (b) We know that ( P d −→ P ) is a presilting complex for Λ by Lemma 3.4. Let P :=( P ⊕ Q ( d −−−−→ P ). By Lemmas 3.4 and 3.5, we have that P is a presilting complex for Λ. Since d is a minimal projective presentation, we have | P d −→ P | = | M | . Moreover, since ( M, Q ) is asupport τ -tilting pair for Λ, we have | M | + | Q | = | Λ | . Thus we have | P | = | P d −→ P | + | Q | = | M | + | Q | = | Λ | . Hence P is a silting complex for Λ by Proposition 3.3(b).(a) This is the case Q = 0 in (b). (cid:3) Now Theorem 3.2 follows from Propositions 3.6 and 3.7. (cid:3)
We give some applications of Theorem 3.2.
Corollary 3.8.
Let Λ be a finite dimensional k -algebra. (a) Any basic two-term presilting complex P for Λ with | P | = | Λ | − is a direct summand ofexactly two basic two-term silting complexes for Λ . (b) Let
P, Q ∈ . Then P and Q have all but one indecomposable direct summand incommon if and only if P is a left or right mutation of Q .Proof. (a) This follows from Theorems 2.18 and 3.2.(b) This is immediate from (a). (cid:3) Now we define Q(2-siltΛ) as the full subquiver of Q(siltΛ) with vertices corresponding to two-term silting complexes for Λ.
Corollary 3.9.
The bijection in Theorem 3.2 is an isomorphism of the partially ordered sets.In particular, it induces an isomorphism between the two-term silting quiver
Q(2-siltΛ) and thesupport τ -tilting quiver Q(s τ -tiltΛ) . -TILTING THEORY 23 Proof.
Let ( M, Λ e ) and ( N, Λ f ) be support τ -tilting pairs for Λ. Let P := ( P → P ) and Q :=( Q → Q ) be minimal projective presentations of M and N respectively. We only have to showthat M ≥ N if and only if Hom K b ( proj Λ) ( P ⊕ Λ e [1] , ( Q ⊕ Λ f [1])[1]) = 0.We know that M ≥ N if and only if Hom Λ ( N, τ M ) = 0 and Λ e ∈ add Λ f by Lemma 2.25.Moreover Hom Λ ( N, τ M ) = 0 if and only if Hom K b ( proj Λ) ( P, Q [1]) = 0 by by Lemma 3.4. On theother hand Λ e ∈ add Λ f if and only if Hom Λ (Λ e, N ) = 0 since N is a sincere (Λ / h f i )-module. ThusΛ e ∈ add Λ f is equivalent to Hom K b ( proj Λ) (Λ e, Q ) = 0 by Lemma 3.5. Consequently M ≥ N if andonly if Hom K b ( proj Λ) ( P ⊕ Λ e [1] , Q [1]) = 0, and this is equivalent to Hom K b ( proj Λ) ( P ⊕ Λ e [1] , ( Q ⊕ Λ f [1])[1]) = 0 since Hom K b ( proj Λ) ( P ⊕ Λ e [1] , Λ f [2]) = 0 is automatic. Thus the assertion follows. (cid:3) Immediately we have the following application.
Corollary 3.10. If Q(2-siltΛ) has a finite connected component C , then Q(2-siltΛ) = C .Proof. This is immediate from Corollaries 2.38 and 3.9. (cid:3)
Note also that Theorem 3.2 and Corollary 3.9 give an alternative proof of Theorem 2.35 sincethe corresponding property for two-term silting complexes holds by [AI, Proposition 2.36].4.
Connection with cluster-tilting theory
Let C be a Hom-finite Krull-Schmidt 2-Calabi-Yau (2-CY for short) triangulated category (forexample, the cluster category C Q associated with a finite acyclic quiver Q [BMRRT]). We shallassume that our category C has a cluster-tilting object T . Associated with T , we have by definitionthe 2-CY-tilted algebra Λ = End C ( T ) op , whose module category is closely connected with the 2-CY-category C . In particular, there is an equivalence of categories [BMR1, KR]:( − ) := Hom C ( T, − ) : C / [ T [1]] → mod Λ . (10)In this section we investigate this relationship more closely by giving a bijection between cluster-tilting objects in C and support τ -tilting Λ-modules (Theorem 4.1). This was the starting pointfor the theory of τ -rigid and τ -tilting modules. As an application, we give a proof of some knownresults for cluster-tilting objects (Corollary 4.5). Also we give a direct connection between cluster-tilting objects in C and two-term silting complexes for Λ (Theorem 4.7). There is an inducedisomorphism between the associated graphs (Corollary 4.8).4.1. Support τ -tilting modules and cluster-tilting objects. In this subsection we show thatthere is a close relationship between the cluster-tilting objects in C and support τ -tilting Λ-modules.We use this to apply our main Theorem 0.4 to get a new proof of the fact that almost completecluster-tilting objects have exactly two complements, and of the fact that all maximal rigid objectsare cluster-tilting, as first proved in [IY] and [ZZ], respectively.We denote by iso C the set of isomorphism classes of objects in a category C . From our equivalence(10), we have a bijection g ( − ) : iso C ←→ iso( mod Λ) × iso( proj Λ)given by X = X ′ ⊕ X ′′ e X := ( X ′ , X ′′ [ − X ′′ is a maximal direct summand of X whichbelongs to add T [1]. We denote by rigid C (respectively, m-rigid C ) the set of isomorphism classesof basic rigid (respectively, maximal rigid) objects in C , and by c-tilt T C the set of isomorphismclasses of basic cluster-tilting objects in C which do not have non-zero direct summands in add T [1].Our main result in this section is the following. Theorem 4.1.
The bijection g ( − ) induces bijections rigid C ←→ τ -rigidΛ , c-tilt C ←→ s τ -tiltΛ and c-tilt T C ←→ τ -tiltΛ . Moreover we have c-tilt C = m-rigid C = { U ∈ rigid C | | U | = | T |} . We start with the following easy observation (see [KR]).
Lemma 4.2.
The functor ( − ) induces an equivalence of categories between add T (respectively, add T [2] ) and proj Λ (respectively, inj Λ ). Moreover we have an isomorphism ( − ) ◦ [2] ≃ ν ◦ ( − ) : add T → inj Λ of functors. Now we express Ext C ( X, Y ) in terms of the images X and Y in our fixed 2-CY tilted algebraΛ. We let h X, Y i Λ = h X, Y i := dim k Hom Λ ( X, Y ) . Proposition 4.3.
Let X and Y be objects in C . Assume that there are no nonzero indecomposabledirect summands of T [1] for X and Y . (a) We have X [1] ≃ τ X and Y [1] ≃ τ Y as Λ -modules. (b) We have an exact sequence → D Hom Λ ( Y , τ X ) → Ext C ( X, Y ) → Hom Λ ( X, τ Y ) → . (c) dim Ext C ( X, Y ) = h X, τ Y i Λ + h Y , τ X i Λ .Proof. (a) This can be shown as in the proof of [BMR1, Proposition 3.2]. Here we give a directproof. Take a triangle T g T f X T [1] (11)with a minimal right ( add T )-approximation f and T , T ∈ add T . Applying ( ) to (11), we havean exact sequence T g T f X . (12)This gives a minimal projective presentation of X since X has no nonzero indecomposable directsummands of T [1]. Applying the Nakayama functor to (12) and Hom C ( T, − ) to (11) and comparingthem by Lemma 4.2, we have the following commutative diagram of exact sequences:0 τ X νT νg ≀ νT ≀ T [1] X [1] T [2] g [2] T [2] . Thus we have τ X ≃ X [1].(b) We have an exact sequence0 → [ T [1]]( X, Y [1]) → Hom C ( X, Y [1]) → Hom C / [ T [1]] ( X, Y [1]) → , where [ T [1]] is the ideal of C consisting of morphisms which factor through add T [1]. We have afunctorial isomorphismHom C / [ T [1]] ( X, Y [1]) ≃ Hom Λ ( X, Y [1]) (a) ≃ Hom Λ ( X, τ Y ) . (13)On the other hand, the first of following functorial isomorphism was given in [P, 3.3].[ T [1]]( X, Y [1]) ≃ D Hom C / [ T [1]] ( Y, X [1]) (13) ≃ D Hom Λ ( Y , τ X ) . Thus the assertion follows.(c) This is immediate from (b). (cid:3)
We now consider the general case, where we allow indecomposable direct summands from T [1]in X or Y . Proposition 4.4.
Let X = X ′ ⊕ X ′′ and Y = Y ′ ⊕ Y ′′ be objects in C such that X ′′ and Y ′′ arethe maximal direct summands of X and Y respectively, which belong to add T [1] . Then dim Ext C ( X, Y ) = h X ′ , τ Y ′ i Λ + h Y ′ , τ X ′ i Λ + h X ′′ [ − , Y ′ i Λ + h Y ′′ [ − , X ′ i Λ . -TILTING THEORY 25 Proof.
Since Ext C ( X ′′ , Y ′′ ) = 0, we havedim Ext C ( X, Y ) = dim Ext C ( X ′ , Y ′ ) + dim Ext C ( X ′′ , Y ′ ) + dim Ext C ( X ′ , Y ′′ ) . By Proposition 4.3, the first term equals h X ′ , τ Y ′ i Λ + h Y ′ , τ X ′ i Λ . Clearly the second term equals h X ′′ [ − , Y ′ i Λ , and the third term equals h Y ′′ [ − , X ′ i Λ . (cid:3) Now we are ready to prove Theorem 4.1.By Proposition 4.4, we have that X is rigid if and only if e X is a τ -rigid pair for Λ. Thus wehave bijections rigid C ↔ τ -rigidΛ, which induces a bijection m-rigid C ↔ s τ -tiltΛ by Corollary2.13(a) ⇔ (b).On the other hand we show that a bijection c-tilt C ↔ s τ -tiltΛ is induced. Since c-tilt C ⊆ m-rigid C , we only have to show that any X ∈ rigid C satisfying that e X is a support τ -tiltingpair for Λ is a cluster-tilting object in C . Assume that Y ∈ C satisfies Ext C ( X, Y ) = 0. ByProposition 4.4, we have Hom Λ ( X ′ , τ Y ′ ) = 0, Hom Λ ( Y ′ , τ X ′ ) = 0, Hom Λ ( X ′′ [ − , Y ′ ) = 0 andHom Λ ( Y ′′ [ − , X ′ ) = 0. By the first 3 equalities, we have Y ′ ∈ add X ′ by Corollary 2.13(a) ⇔ (d).By the last equality we have Y ′′ [ − ∈ add X ′′ [ − Y ∈ add X holds, which shows that X isa cluster-tilting object in C .The remaining statements follow immediately. (cid:3) Now we recover the following results in [IY] and [ZZ].
Corollary 4.5.
Let C be a 2-CY triangulated category with a cluster-tilting object T . (a) [IY] Any basic almost complete cluster-tilting object is a direct summand of exactly twobasic cluster-tilting objects. In particular, T is a mutation of V if and only if T and V have all but one indecomposable direct summand in common. (b) [ZZ] An object X in C is cluster-tilting if and only if it is maximal rigid if and only if it isrigid and | X | = | T | .Proof. (a) This is immediate from the bijections given in Theorem 4.1 and the corresponding resultfor support τ -tilting pairs given in Theorem 2.18.(b) This is the last equality in Theorem 4.1. (cid:3) Connections between cluster-tilting objects in C and tilting Λ-modules have been investigatedin [Smi, FL]. It was shown that a tilting Λ-module always comes from a cluster-tilting object in C , but the image of a cluster-tilting object is not always a tilting Λ-module. This is explained byTheorem 4.1 asserting that the Λ-modules corresponding to the cluster-tilting objects of C are thesupport τ -tilting Λ-modules, which are not necessarily tilting Λ-modules.4.2. Two-term silting complexes and cluster-tilting objects.
Throughout this section, let C be a 2-CY category with a cluster-tilting object T . Fix a cluster-tilting object T ∈ C . Let Λ :=End C ( T ) op and let K b ( proj Λ) be the homotopy category of bounded complexes of finitely generatedprojective Λ-modules. In this section, we shall show that there is a bijection between cluster-tiltingobjects in C and two-term silting complexes for Λ and that the mutations are compatible with eachother.The following result will be useful, where we denote by K ( proj Λ) the full subcategory of K b ( proj Λ) consisting of two-term complexes for Λ.
Proposition 4.6.
There exists a bijection iso
C ←→ iso( K ( proj Λ)) which preserves the number of non-isomorphic indecomposable direct summands.Proof.
For any object U ∈ C , there exists a triangle T g T f U T [1] where T , T ∈ add T and f is a minimal right ( add T )-approximation. By Lemma 4.2, we have atwo-term complex T g −→ T in K b ( proj Λ).Conversely, let P d → P be a two-term complex for Λ. By Lemma 4.2, there exists a morphism g : T → T in add T such that g = d . Taking the cone of g , we have an object U in C . Then we caneasily check that the correspondence gives a bijection and preserves the number of non-isomorphicindecomposable direct summands. (cid:3) Using this, we get the desired correspondence.
Theorem 4.7.
The bijection in Proposition 4.6 induces bijections rigid
C ←→ and c-tilt
C ←→ . Proof. (i) For any rigid object U ∈ C , we have a triangle T g T f U h T [1]where T , T ∈ add T and f is a minimal right ( add T )-approximation. Let a : T → T be anarbitrary morphism in C . Since U is rigid, we have f ah [ −
1] = 0. Thus we have a commutativediagram U [ − h [ − T ga T fb UT g T f U h T [1]of triangles in C . Since hb = 0, there exists k : T → T such that b = f k . Since f ( a − k g ) = 0,there exists k : T → T such that gk = a − k g . Therefore we haveHom K b ( proj Λ) (( T g −→ T ) , ( T g −→ T )[1]) = 0 . Thus T g −→ T is a presilting complex for Λ.(ii) Let P := ( P d → P ) be a two-term presilting complex for Λ. There exists a unique g : T → T in add T such that g = d . We consider a triangle T g T f U h T [1]in C . We take a morphism a : U → U [1] in C . Then we have the commutative diagram T g T h [1] af T [2] g [2] T [2] . Applying ( − ), we have a commutative diagram P d P h [1] af νP νd νP . Thus we have a morphism P → νP [ −
1] in K b ( proj Λ). Since P is a presilting complex for Λ, wehave Hom K b ( proj Λ) ( P, νP [ − ≃ D Hom K b ( proj Λ) ( P [ − , P ) = 0 . -TILTING THEORY 27 Therefore h [1] af = 0, and the morphism h [1] af factors through add T [1]. Hence we have h [1] af = 0.Thus we have a commutative diagram T g T fa U ha T [1] T [1] g [1] T [1] f [1] U [1] h [1] T [2] . Since T ∈ add T , we have a = 0. Thus af = 0, so there exists ϕ : T [1] → U [1] such that a = ϕh .Since T ∈ add T , we have h [1] ϕ = 0. Thus there exists b : T [1] → T [1] such that ϕ = f [1] b .Consequently, we have commutative diagrams0 T gb [ − T T g T P db [ − P P d P P is a presilting complex for Λ, there exist s : T [1] → T [1] and t : T [1] → T [1] such that b = sg [1] + g [1] t . Therefore we have a = ϕh = f [1] bh = f [1] sg [1] h + f [1] g [1] th = 0 . Hence Hom C ( U, U [1]) = 0, that is, U is rigid, and the claim follows. (cid:3) Corollary 4.8.
The bijections in Theorems 3.2 and 4.7 induce isomorphisms of the followinggraphs. (a)
The underlying graph of the support τ -tilting quiver Q(s τ -tiltΛ) of Λ . (b) The underlying graph of the two-term silting quiver
Q(2-siltΛ) of Λ . (c) The cluster-tilting graph
G(c-tilt C ) of C .Proof. (a) and (b) are the same by Corollary 3.9.We show that (b) and (c) are the same. Let U and V be cluster-tilting objects in C . Let P and Q be the two-term silting complexes for Λ corresponding respectively to U and V by Theorem 4.7.By Corollary 4.5(a) the following conditions are equivalent:(a) There exists an edge between U and V in the exchange graph.(b) U and V have all but one indecomposable direct summand in common.Clearly (b) is equivalent to the following condition:(c) P and Q have all but one indecomposable direct summand in common.Now (c) is equivalent to the following condition by Corollary 3.8(b).(d) There exists an edge between P and Q in the underlying graph of the silting quiver.Therefore the exchange graph of C and the underlying graph of the silting full subquiver consistingof two-term complexes for Λ coincide. (cid:3) We end this section with the following application.
Corollary 4.9. If G(c-tilt C ) has a finite connected component C , then G(c-tilt C ) = C .Proof. This is immediate from Corollaries 2.38 and 4.8. (cid:3) Numerical invariants
In this section, we introduce g -vectors following [AR3] and [DK]. We show that g -vectors of inde-composable direct summands of support τ -tilting modules form a basis of the Grothendieck group(Theorem 5.1). Moreover we observe that non-isomorphic τ -rigid pairs have different g -vectors(Theorem 5.5). In [DWZ] the authors defined what they called E -invariants of finite dimensionaldecorated representations of Jacobian algebras, and used this to solve several conjectures from [FZ]. In the case of finite dimensional Jacobian algebras they showed that the E -invariants weregiven by formulas which we were led to in section 4.1, by considering dim k Ext C ( T, T ) for a cluster-tilting object T in C . We here consider E -invariants for any finite dimensional algebra, using thesame formula, and show that they can be expressed in terms of homomorphism spaces, dimensionvectors and g -vectors. We give some further results on the case of 2-CY tilted algebras, includinga comparison for neighbouring 2-CY tilted algebras (Theorem 5.7).In the rest of this paper we assume that our base field k is algebraically closed. Let Λ be a finitedimensional k -algebra.5.1. g -vectors and E -invariants for finite dimensional algebras. Recall from [DK] that the g -vectors are defined as follows: Let K ( proj Λ) be the Grothendieck group of the additive category proj
Λ. Then the isomorphism classes P (1) , . . . , P ( n ) of indecomposable projective Λ-modules forma basis of K ( proj Λ). Consider M in mod Λ and let P P M mod Λ. Then we write P − P = n X i =1 g Mi P ( i ) , where by definition g M = ( g M , . . . , g Mn ) is the g -vector of M . The element P − P is also calledan index of M , which was investigated in [AR3], in connection with studying modules determinedby their composition factors, and in [DK].Another useful vector associated with M is the dimension vector c M = ( c M , . . . , c Mn ). Denoteby S ( i ) the simple top of P ( i ). Then c Mi is by definition the multiplicity of the simple module S ( i ) as composition factor of M . This vector has played an important role in cluster theory forthe acyclic case, since the denominators of cluster variables are determined by dimension vectorsof indecomposable rigid modules over path algebras [BMRT, CK]. Now this result is not true ingeneral [BMR2].We have the following result on g -vectors of support τ -tilting modules. Theorem 5.1.
Let ( M, P ) be a support τ -tilting pair for Λ with M = L ℓi =1 M i and P = L ni = ℓ +1 P i with M i and P i indecomposable. Then g M , · · · , g M ℓ , g P ℓ +1 , · · · , g P n form a basis ofthe Grothendieck group K ( proj Λ) .Proof. By Theorem 3.2, we have a corresponding silting complex Q = L ni =1 Q i for Λ with indecom-posable Q i , where the vectors g M , · · · , g M ℓ , g P ℓ +1 , · · · , g P n are exactly the classes of Q , · · · , Q n in the Grothendieck group K ( K b ( proj Λ)) = K ( proj Λ). By Proposition 1.6(b), we have the asser-tion. (cid:3)
This gives a result below due to Dehy-Keller. Recall that for a cluster-tilting object T ∈ C andan object X ∈ C , there exists a triangle T ′′ → T ′ → X → T ′′ [1]in C with T ′ , T ′′ ∈ add T . We call ind T ( X ) := T ′ − T ′′ ∈ K ( add T ) the index of X . Corollary 5.2. [DK, Theorem 2.4]
Let C be a 2-CY triangulated category, and T and U = L ni =1 U i be basic cluster-tilting objects with U i indecomposable. Then the indices ind T ( U ) , · · · , ind T ( U n ) form a basis of the Grothendieck group K ( add T ) of the additive category add T .Proof. We can assume that U i / ∈ add T [1] for 1 ≤ i ≤ ℓ , and U i ∈ add T [1] for ℓ + 1 ≤ i ≤ n .Then ( L ℓi =1 U i , L ni = ℓ +1 U i [ − τ -tilting pair for Λ by Theorem 4.1. The equivalenceHom C ( T, − ) : add T → proj Λ gives an isomorphism K ( add T ) ≃ K ( proj Λ). This sends ind T ( U i )to g U i for 1 ≤ i ≤ ℓ , and to − g U i [ − for ℓ + 1 ≤ i ≤ n . Thus the assertion follows from Theorem5.1. (cid:3) -TILTING THEORY 29 Now we consider a pair M = ( X, P ) of a Λ-module X and a projective Λ-module P . We regarda Λ-module X as a pair ( X, M = ( X, P ) and N = ( Y, Q ), let g M := g X − g P ,E ′ Λ ( M, N ) := h X, τ Y i + h P, Y i ,E Λ ( M, N ) := E ′ Λ ( M, N ) + E ′ Λ ( N, M ) ,E Λ ( M ) := E Λ ( M, M ) . We call g M the g -vector of M , and E Λ ( M, N ) the E -invariant of M and N . Clearly a pair ( M, τ -rigid if and only if E Λ ( M ) = 0.There is the following relationship between E -invariants and g -vectors, where we denote by a · b the standard inner product P ni =1 a i b i for vectors a = ( a , · · · , a n ) and b = ( b , · · · , b n ). Proposition 5.3.
Let Λ be a finite dimensional algebra, and let X and Y be in mod Λ . Then wehave the following. E ′ Λ ( X, Y ) = h Y, X i − g Y · c X ,E Λ ( X, Y ) = h Y, X i + h X, Y i − g Y · c X − g X · c Y ,E Λ ( X ) = 2( h X, X i − g X · c X ) . Proof.
We only have to show the first equality. Since P − P = P ni =1 g Yi P ( i ), then h P , X i −h P , X i = g Y · c X . By Proposition (2.4)(a), we have E ′ Λ ( X, Y ) = h X, τ Y i = h Y, X i + h P , X i − h P , X i = h Y, X i − g Y · c X . (cid:3) The following more general description of E -invariants is also clear. Proposition 5.4.
For any pair M = ( X, P ) and N = ( Y, Q ) , we have E Λ ( M, N ) = h Y, X i + h X, Y i − g M · c Y − g N · c X . We end this subsection with the following analog of [DK, Theorem 2.3], which was also observedby Plamondon.
Theorem 5.5.
The map M g M gives an injection from the set of isomorphism classes of τ -rigidpairs for Λ to K ( proj Λ) .Proof. The proof is based on Propositions 2.4(c) and 2.5, and is the same as that of [DK, Theorem2.3]. (cid:3) E -invariants for -CY tilted algebras. In the rest of this section, let C be a 2-CY trian-gulated k -category and let T be a cluster-tilting object in C . Let Λ := End C ( T ) op . For any object X ∈ C , we take a decomposition X = X ′ ⊕ X ′′ where X ′′ is a maximal direct summand of X whichbelongs to add T [1] and define a pair by e X Λ := ( X ′ , X ′′ [ − , where ( − ) is an equivalence Hom C ( T, − ) : C / [ T [1]] → mod Λ given in (10).We have the following interpretation of E -invariants. Proposition 5.6.
We have E Λ ( e X Λ , e Y Λ ) = dim k Ext C ( X, Y ) for any X, Y ∈ C .Proof.
This is immediate from Proposition 4.4 and our definition of E -invariants. (cid:3) Now let T ′ be a cluster-tilting mutation of T . Then we refer to the 2-CY-tilted algebras Λ =End C ( T ) op and Λ ′ = End C ( T ′ ) op as neighbouring e X Λ ′ forΛ ′ in a similar way to e X Λ by using the equivalence Hom C ( T ′ , − ) : C / [ T ′ [1]] → mod Λ ′ .By our approach to the E -invariant, the following is now a direct consequence. Theorem 5.7.
With the above notation, let M and N be objects in C . Then E Λ ( f M Λ , e N Λ ) = E Λ ′ ( f M Λ ′ , e N Λ ′ ) .Proof. This is clear from Proposition 5.6 since both sides are equal to dim k Ext C ( M, N ). (cid:3) In particular, f M Λ is τ -rigid if and only if f M Λ ′ is τ -rigid.This result is analogous to the corresponding result for (neighbouring) Jacobian algebras provedin [DWZ], in a larger generality. It is however not clear whether the two concepts of neighbouringalgebras coincide for finite dimensional neighbouring Jacobian algebras. See [BIRS] for moreinformation. 6. Examples
In this section we illustrate some of our work with easy examples.
Example 6.1.
Let Λ be a local finite dimensional k -algebra. Then we have s τ -tiltΛ = { Λ , } sincethe condition Hom Λ ( M, τ M ) = 0 implies either M = 0 or τ M = 0 (i.e. M is projective). We haveQ(s τ -tiltΛ) = ( Λ 0 ), Q(f-torsΛ) = ( mod Λ 0 ) and Q(2-siltΛ) = ( Λ Λ[1] ).
Example 6.2.
Let Λ be a finite dimensional k -algebra given by the quiver 1 a a with relations a = 0. Then Q(s τ -tiltΛ), Q(f-torsΛ) and Q(2-siltΛ) are the following: ⊕
21 12 ⊕ ⊕ mod Λ add ( ⊕ add add (2 ⊕ ) add (cid:20)
21 [ a −−−→ ⊕ (cid:21) (cid:20) ⊕
21 [ a −−−→ (cid:21)(cid:20)
12 [ a −−−→ ⊕ (cid:21) (cid:20) ⊕
12 [ a −−−→ (cid:21) Λ[1]
Example 6.3.
Let Λ be a finite dimensional k -algebra given by the quiver 2 a a a with relations a = 0. Then Λ is a cluster-tilted algebra of type A , and there are 14 elements in c-tilt C for thecluster category C of type A . By our bijections, we know that there are 14 elements in each sets τ -tiltΛ, f-torsΛ and 2-siltΛ. -TILTING THEORY 3112 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕
31 12 ⊕ ⊕
31 12 ⊕ ⊕ ⊕ ⊕
31 31 ⊕ ⊕ Example 6.4.
Let Λ = kQ/ h βα i , where Q is the quiver 1 α −→ β −→
3. Then T = S ⊕ P ⊕ P isa τ -tilting module which is not a tilting module. Here S i denotes the simple Λ-module associatedwith the vertex i , and P i denotes the corresponding indecomposable projective Λ-module.In this case there are 12 basic support τ -tilting Λ-modules, and Q(s τ -tiltΛ) is the following. ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ τ -tilting modules. ndex ( − ) ⊥ , ( − ) ⊥ , ⊥ ( − ), ⊥ ( − ), 4( − ) ∗ , 5( M, P ) † , 11 M † , 12( − ), 23 ≥ , 3 h− , −i , 24 g ( − ), 232-Calabi-Yau category, 72-presiltΛ, 202-siltΛ, 3, 20 add M , 2 c M , 28cotiltΛ, 12c-tilt C , 3, 7c-tilt T C , 23 D , 5 E ′ Λ ( M, N ), E Λ ( M, N ), E Λ ( M ), 29 Fac M , 2ff-torfΛ, 12ff-torsΛ, 10f-torfΛ, 12f-torsΛ, 3, 10 g M , 28, 29G(c-tilt C ), 7 I ( F ), 5ind T ( X ), 28 inj Λ, 2iso C , 23 K ( proj Λ), 25 K b ( proj Λ), 6 mod
Λ, 2 mod Λ, mod Λ, 6m-rigid C , 23 µ , 13 µ + , µ − , 7, 16 ν , ν − , 5 P ( T ), 5 proj Λ, 2Q(2-siltΛ), 22Q(siltΛ), 7Q(s τ -tiltΛ), 17rigid C , 23sf-torfΛ, 12sf-torsΛ, 10siltΛ, 6s τ -tiltΛ, 3s τ − -tiltΛ, 12 Sub M , 2 τ , τ − , 5, 6 τ -rigidΛ, 11 τ -tiltΛ, 10 τ − -tiltΛ, 12tiltΛ, 10Tr, 5almost complete silting complex, 7almost complete support τ -tilting pair, 3almost complete τ -tilting module, 2Auslander-Bridger transpose duality, 6 AR duality, 6AR translation, 6basic pair, 3Bongartz completion τ -tilting module, 10tilting module, 5cluster-tilting graph, 7cluster-tilting object, 7costable category, 6cotilting module, 5direct summand of pair, 3 E -invariant, 29Ext-injective module, 5Ext-projective module, 5 g -vectorof a module, 28of a pair, 29indexof a module, 28of an object, 28left mutationsilting complex, 7support τ -tilting module, 16maximal rigid object, 7mutationcluster-tilting object, 7support τ -tilting pair, 3, 13Nakayama functor, 5neighbouring 2-CY-tilted algebras, 29partial cotilting module, 5partial tilting module, 4presilting complex, 6right mutationsilting complex, 7support τ -tilting module, 16rigid object, 7silting complex, 6silting quiver, 7sincere module, 8stable category, 6support τ -tilting module, 2support τ -tilting pair, 3support τ -tilting quiver, 17support τ − -tilting module, 12support tilting module, 2 τ -rigid module, 2 τ -rigid pair, 3 τ -tilting module, 2 τ − -rigid module, 12 τ − -tilting module, 12tilting module, 4 -TILTING THEORY 33 torsion class, 5torsion pair, 5torsionfree class, 5two-term complex, 20 References [Ab] H. Abe,
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Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya. 464-8602, Japan
E-mail address : [email protected] Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya. 464-8602, Japan
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