aa r X i v : . [ m a t h . R T ] F e b SUPPORT τ -TILTING AND 2-TORSION PAIRS JORDAN MCMAHON
Abstract.
The theory of τ -tilting was introduced by Adachi–Iyama–Reiten;one of the main results is a bijection between support τ -tilting modules andtorsion classes. We are able to generalise this result in the context of the higherAuslander–Reiten theory of Iyama. For a finite-dimensional algebra A with2-cluster-tilting subcategory C ⊆ mod A , we are able to find a correspondencebetween support τ -tilting A -modules and torsion pairs in C satisfying an addi-tional functorial finiteness condition. Introduction
Support-tilting modules were first studied by Ringel [31] and connected to tor-sion classes and cluster algebras by Ingalls–Thomas [14]. Since support-tiltingmodules are able to capture the behaviour of clusters, this led to further study.Significantly, Adachi, Iyama and Reiten [1] introduced support τ -tilting modules,and were able to find a bijection, for an finite-dimensional algebra A , between sup-port τ -tilting A -modules and functorially-finite torsion classes in mod A . See also[18] for a survey of τ -tilting theory, which has seen much activity in recent years[4], [8], [9], [17], [19], [28], [29] as well as its generalisations such as in silting theory[2], [3], [7]. A natural question to ask is whether similar results are true in thecontext of higher Auslander–Reiten theory, as introduced by Iyama in [15], [16],and an active area of research [10], [20], [22], [23], [24], [30]. As the name suggests,higher Auslander–Reiten theory has connections to higher-dimensional geometryand topology [11], [12], [25], [27], [32] and is hence a natural generalisation. Theresult we find is the following: Theorem 1.1.
Let A be a finite-dimensional algebra with -cluster-tilting sub-category C ⊆ mod A . Then there is a correspondence between support τ -tilting A -modules and 2-functorially finite torsion pairs in C . Background and Notation
Consider a finite-dimensional algebra A over a field K , and fix a positive integer d . We will assume that A is of the form KQ/I , where KQ is the path algebraover some quiver Q and I is an admissible ideal of KQ . An A -module will mean afinitely-generated left A -module; by mod A we denote the category of A -modules.The functor D = Hom K ( − , K ) defines a duality. Let add M be the full subcategoryof mod A composed of all A -modules isomorphic to direct summands of finite direct sums of copies of M . For an A -module M , the right annihilator of M is thetwo-sided ideal ann M := { a ∈ A | aM = 0 } ; for a class of modules X we setann X := { a ∈ A | aX = 0 ∀ X ∈ X } . For an A -module M , define Sub M := { N ∈ mod A |∃ injection N ֒ → M } ; for a class of modules X we set Sub X := { N ∈X |∃ X ∈ X , injection N ֒ → X } . Define Fac M and Fac X dually.2.1. Higher cluster-tilting subcategories.
Define τ d := τ Ω d − to be the d -Auslander–Reiten translation and τ − d := τ − Ω − ( d − to be the inverse d -Auslander–Reiten translation . A subcategory C of mod A is precovering or contravariantlyfinite if for any M ∈ modΛ there is an object C M ∈ C and a morphism f : C M → M such that Hom( C, − ) is exact on the sequence C M M C ∈ C . The module C M is said to be a right C -approximation . The dualnotion of precovering is preenveloping or covariantly finite . A subcategory C thatis both precovering and preenveloping is called functorially finite . Definition 2.1. [16, Definition 2.2] For a finite-dimensional algebra A , a sub-category C ⊆ mod( A ) is a d -cluster-tilting subcategory if it satisfies the followingconditions: C = { X ∈ C | Ext iA ( M, X ) = 0 ∀ < i < d } . C = { X ∈ C | Ext iA ( X, M ) = 0 ∀ < i < d } . A right C -resolution is a sequence · · · C C M C i ∈ C for each i , and which becomes exact under Hom A ( C, − ) for each C ∈ C . Define a left C -resolution dually. Theorem 2.2. [16, Theorem 3.6.1]
Let
C ⊆ mod A be a d -cluster-tilting subcate-gory. Then(1) Any M ∈ mod A has a right C -resolution C d − · · · C C M . (2) Any M ∈ mod A has a left C -resolution M C C · · · C d − . Recall the stable module category mod A is full subcategory of mod A obtainedby factoring out all morphisms that factor through a projective module. Theorem 2.3. [16, Theorem 1.5]
Let A be a finite-dimensional algebra. Then: • If Ext iA ( M, A ) = 0 for all < i < d , then Ext iA ( M, N ) ∼ = D Ext d − iA ( N, τ d M ) and Hom A ( M, N ) ∼ = D Ext dA ( N, τ d M ) for all M ∈ mod A and all < i < d . • If Ext iA ( DA, N ) = 0 for all < i < d , then Ext iA ( M, N ) ∼ = D Ext d − iA ( τ − d N, M ) and Hom A ( M, N ) ∼ = D Ext dA ( τ − d N, M ) for all N ∈ mod A for all < i < d . We may now generalise a result of Auslander–Smalø, the proof of which is un-changed apart from indices.
Proposition 2.4 (c.f. Proposition 5.8 of [6]) . Let A be a finite-dimensional alge-bra. Then for X, Y ∈ mod A the following are equivalent:(i) Hom A ( τ − Y, X ) = 0 .(ii)
Hom A ( τ − Y, Sub X ) = 0 .(iii) Ext A (Sub X, Y ) = 0 .Proof.
Firstly, statements ( ii ) and ( iii ) are equivalent by Theorem 2.3 and state-ment ( i ) trivially implies ( ii ). It remains to show ( ii ) implies ( i ); we prove by con-tradiction. So suppose there is a morphism f : τ − Y → X is a non-zero morphismwith image Im f ∈ Sub X , and such that the induced morphism f ′ : τ − Y ։ Im f factors through a projective module P . Since f ′ is surjective, we may assume P is the projective cover of Im f . Let g : τ − M → P be any morphism; since τ − Y has no non-trivial projective summands, we have g ( τ − M ) ⊂ rad P , hence anycomposition τ − Y → P → Im f is not an epimorphism. Therefore the image of f ′ in Hom A ( τ − Y, Im f ) = 0 is not zero. Hence Hom A ( τ − Y, X ) = 0 if and only ifHom A ( τ − Y, Sub X ) = 0. (cid:3) Key homological tools for higher cluster-tilting categories are d -pushouts and d -pullbacks, constructed as follows. Proposition 2.5. [21, Proposition 3.8]
Let A be a finite-dimensional algebra and C ⊆ mod A . For any d -exact sequence in C → Y → Y → · · · → Y d +1 → and any morphism f : X d +1 → Y d +1 there exists a commutative diagram in C : Y X · · · X d X d +1 Y Y · · · Y d Y d +1 f such that there is an induced d -exact sequence → X → X ⊕ Y → · · · → X d +1 ⊕ Y d → Y d +1 → X d is defined to be the right C -approximation of the pullback of (( X d +1 ⊕ Y d ) → Y d +1 ). Subsequently X d − is defined to be the right C -approximation of the pull-back of (( X d ⊕ Y d − ) → Y d ). This continues until X = Y is reached. JORDAN MCMAHON
Remark . In this setting, it is actually immaterial whether Y d +1 is in C or not:Theorem 2.2 ensures that the so-called d -kernel of the morphism ( X d +1 ⊕ Y d ) → Y d +1 ) exists. Equally, it is not important whether Y ∈ C .A technical result is the following: Lemma 2.7.
Consider a module X ∈ C and two d -exact sequences: → L → M → N → X → , → L ′ → M ′ → N ′ → X → . Then there exist
P, Q ∈ C with no common non-zero summands and a commutativediagram with exact rows and columns:
L LQ Q ⊕ M M L ′ Q ⊕ M ′ P N L ′ M ′ N ′ X
00 0 0
Proof.
Just as in the above method for d -pullbacks, we may form the followingcommutative diagram with exact rows and columns by taking the appropriate right C -approximations of pullbacks: P, Q, R, S
L LQ S M L ′ R P N L ′ M ′ N ′ X
00 0 0It suffices to show the surjection R ։ M ′ is split (by symmetry, also S ։ M splits). By the pullback property of P , we find M ′ → N ′ factors through P . Butthe pullback property of R then implies both M ′ → P factors through R and that M ′ → R → M ′ ∼ = id. Hence R ∼ = M ′ ⊕ Q .Finally, suppose P and Q have a common summand Y . Then any morphism Y → N ′ must factor through M ′ . But as above, M ′ → N ′ factors through P .Hence Y must be a summand of M ′ . This is a contradiction, since M ′ is alreadyincluded a summand of Q ⊕ M ′ , and hence cannot be a summand of Q . Thisfinishes the proof. (cid:3) A second technical result will prove similarly useful.
Lemma 2.8.
In the setting of Lemma 2.7, in case there exists P ′ ∈ C such that P ∼ = P ′ ⊕ M ′ , then there is a commutative diagram with exact rows and columns JORDAN MCMAHON
L M L ′ Q P ′ N L ′ M ′ N ′ X Proof.
In this case we know that the morphism L ′ → M ′ factors through Q . Sincewe are dealing with a 2-pullback diagram, the 2-exact sequence:0 → L → M ⊕ Q → P ′ ⊕ M ′ → N ′ → (cid:3) Main results
In this section let
C ⊂ mod A be a fixed 2-cluster-tilting subcategory. Definition 3.1.
Define a subcategory
X ⊆ C to be 2 -contravariantly finite in C iffor any M ∈ C there exists a right X -approximation X → M and object X ∈ X and a sequence X → X → M on which Hom( C, − ) is exact for all C ∈ C .Dually, define a subcategory X ⊆ C to be 2 -covariantly finite in C if for any M ∈ C there exists a left X -approximation M → X and object X ∈ X anda sequence M → X → X on which Hom( − , C ) is exact for all C ∈ C . Asubcategory X ⊆ C will be said to be 2 -functorially finite in C if it is both 2-contravariantly finite in C and 2-covariantly finite in C . Lemma 3.2.
Let
X ⊆ C be functorially finite. Then X is 2-covariantly finite in C if and only if it is 2-contravariantly finite in C .Proof. It suffices to show any subcategory X C is also2-covariantly finite in C . So suppose X is 2-contravariantly finite in C . Let f : M → X be a left X approximation and let X ′ → X ′ → coker f be a right X -approximation. We have that X → coker f factors through X ′ , and since X ′ =coker f , this induces an additional non-zero morphism M → X ′ . The sequence M → X ′ → coker f is now zero, and hence induces a morphism g : M → X ′ .Since f : M → X is a left X -approximation, we have that g factors through X . But this means X → coker f factors through X ′ → X ′ and is hence zero, a contradiction. This is illustrated below: M X coker fX ′ X ′ coker f fg (cid:3) Recall that for a finite-dimensional algebra A , then a torsion pair in mod A consists of two subcategories ( T , F ) such that • Hom A ( T, F ) = 0 for all T ∈ T , F ∈ F . • T = { X ∈ mod A | Hom A ( X, F ) = 0 ∀ F ∈ F } . • F = { Y ∈ mod A | Hom A ( T, Y ) = 0 ∀ T ∈ T } .We may now define one of our primary objects of study. Definition 3.3. A C consists of two subcate-gories ( T , F ) each 2-functorially finite in C and such that • Hom A ( T, F ) = 0 for all T ∈ T , F ∈ F . • T = { X ∈ C| Hom A ( X, F ) = 0 ∀ F ∈ F } . • F = { Y ∈ C| Hom A ( T, Y ) = 0 ∀ T ∈ T } .For a 2-functorially-finite torsion pair ( T , F ) in C , we say T is a torsion class and F a torsion-free class . Lemma 3.4.
For any 2-functorially-finite torsion pair ( T , F ) in C then (Fac T , Sub F ) is a torsion pair in mod A .Proof. Suppose ( T , F ) is a 2-functorially-finite torsion pair in C . Clearly any X ∈ mod A satisfies that Hom A (Fac T , X ) = 0 if and only if Hom A ( T , X ) = 0. Let f : X → C X be a left C -approximation of X , moreover f must be injective. SoHom A ( T , X ) = 0 if and only if Hom A ( T , C X ) = 0 and hence whenever C X ∈ F and X ∈ Sub F . The result follows. (cid:3) Proposition 3.5.
Let A be a finite-dimensional algebra and C ⊆ mod A a 2-cluster-tilting subcategory. Let T , F ⊆ C be 2-functorially-finite subcategories in C . The following are equivalent.(i) ( T , F ) is a 2-functorially-finite torsion pair in C .(ii) For every M ∈ C there is an exact sequence T M → M → F M such that M → F M is a left F -approximation and T M → M is a right T -approximation.Proof. (i) = ⇒ (ii): Given M ∈ C , and a 2-functorially-finite torsion pair ( T , F )in C , if follows from Lemma 3.4 that (Fac T , Sub F ) is a torsion pair in mod A . Aclassical property of torsion pairs (see for example [5, Proposition V1.1.5]) there JORDAN MCMAHON is an exact sequence in mod A : 0 → T → M → F → T ∈ Fac T and F ∈ Sub F . By taking appropriate approximations we obtain the result.(ii) = ⇒ (i) This is trivial. (cid:3) We may characterise torsion classes in C . Proposition 3.6.
For a 2-functorially-finite subcategory
T ⊆ C , the following areequivalent:(i) There is an inclusion (Fac
T ∩ C ) ⊆ T and for every d -exact sequence in C with T , T ∈ T : → T → X → Y → T → there exists a d -pushout diagram with exact rows: T ′ T ′ T ′ T T X Y T such that T ′ , T ′ , T ′ ∈ T .(ii) T is the torsion part of a 2-functorially-finite torsion pair in C .Proof. (i) = ⇒ (ii) For an arbitrary M ∈ C , let g : T M → M be a right T -approximation. By Proposition 3.5 it suffices to show that Hom A ( T , coker g ) = 0.So assume there exists a morphism T → coker g for some T ∈ T . Then thereexist X, Y ∈ C and a pullback diagram by Remark 2.60 T X Y T T T M M coker g T ′ , T ′ , T ′ ∈ T making the following diagramcommute: 0 T ′ T ′ T ′ T T X Y T T T M M coker g T ′ ։ T → coker g is non-zero there exists a non-zeromorphism T ′ → M , which by assumption factors through T M . But this impliesthe above non-zero morphism T ′ → coker g factors through T M → M → coker g , acontradiction. (ii) = ⇒ (i) Clearly T is closed under factor modules in C . Now let( ∗ ) : 0 → T → X → Y → T → d -exact with T , T ∈ T . By Proposition 3.5 there exist T X , T Y ∈ T and F X , F Y ∈ F and exact sequences T X → X → F X ,T Y → Y → F Y . Applying Hom A ( − , F ) to ( ∗ ) implies an isomorphism Hom A ( Y, F ) ∼ = Hom A ( X, F )for any F ∈ Sub F ; this implies Im( X → F X ) ∼ = Im( Y → F Y ). We are in thesituation of Lemma 2.8, so there exist T ′ X , T ′ Y , T ′ ∈ T such that there there is apullback diagram 0 00 T ′ X T ′ Y T T X T Y T ′ T X Y T d -pushout diagram0 T ⊕ T ′ X T X ⊕ T ′ Y T Y T T X Y T → T ′ X → T X ⊕ T ′ Y → X ⊕ T Y → Y → (cid:3) Recall an object T ∈ C is τ -tilting [1] if • Hom A ( T, τ T ) = 0 • | T | = | A | , where | · | denotes the number of indecomposable summands.If T is τ -tilting as an A/ h e i -module for some idempotent e , then T is support τ -tilting . This can be generalised as follows, using the generalised tilting theory of[13], [26]: recall that an A -module T is a d -tilting module if:(1) proj . dim( T ) ≤ d . (2) Ext iA ( T, T ) = 0 for all 0 < i ≤ d .(3) there exists an exact sequence0 → A → T → T → · · · → T d → T , . . . , T d ∈ add T .We may now define the latter of our primary objects of study. Definition 3.7.
An object T ∈ C is τ -tilting if • Hom A ( T, τ T ) = 0 • There exists an exact sequence0 → A → T → T → T → T , T , T ∈ add T .If T is τ -tilting as an A/ h e i -module for some idempotent e , then we say that T is support τ -tilting .We now show that support τ -tilting modules are indeed 2-tilting. Recall thatan A -module T is faithful if its right annihilator ann T vanishes. Lemma 3.8 (c.f. Lemma IV.2.7 of [5]; c.f Lemma VII.5.1 of [5]; c.f. Proposition2.2(b) of [1]) . Let A be a finite-dimensional algebra with 2-cluster-tilting subcate-gory C ⊆ mod A . Then(i) For any T ∈ C , proj . dim T ≤ if and only if Hom A ( DA, τ T ) = 0 .(ii) Let T ∈ C be a faithful A -module. If Hom A ( T, τ T ) = 0 , then proj . dim T ≤ .(iii) Any τ -tilting A -module T is a tilting ( A/ ann T ) -module.Proof. ( i ): Apply the left exact functor ν − = Hom A ( DA, − ) to the exact sequence0 → τ M → νP → νP → νP → νM → → ν − τ M → ν − νP → ν − νP → ν − νP → ν − νM → A ( DA, τ M ) = ν − τ M vanishes if and only if proj . dim M ≤ ii ): It is known (see [5, V1.2.2]) that an A -module T is faithful if and only if DA is generated by T . So let T i ։ DA be a surjection. Applying the functorHom A ( − , τ T ) results in a monomorphism Hom A ( DA, τ T ) ֒ → Hom A ( T, τ T ). Theresult now follows from part (i) above.( iii ) Note that for any idempotent ideal h e i of A , and any M, N ∈ mod A thereis a natural inclusion Ext A/ h e i ( M, N ) ֒ → Ext A ( M, N ). In this case let h e i :=ann T . Since Hom A ( T, τ T ) = 0, Proposition 2.4 implies Ext A ( T, Fac T ) = 0, andit follows that Ext A/ h e i ( T, Fac T ) = 0. By Proposition 2.4 once more, we haveHom A/ h e i ( T, τ T ) = 0. Since T is faithful as an ( A/ ann T )-module, it follows frompart ( ii ) that proj . dim T ≤ d and that T is 2-tilting as an ( A/ ann T )-module. (cid:3) Lemma 3.9 (Happel [13]) . Let A be a finite-dimensional algebra and T be a d -tilting A -module. Assume that M ∈ mod( A ) satisfies Ext iA ( T, M ) = 0 for all i > . Then there exists an exact sequence → T m → · · · → T → T → M → such that T j ∈ add( T ) for all ≤ j ≤ m and m ≤ gl . dim A . We may now prove our main result.
Theorem 1.1.
An object T ∈ C is a support τ -tilting module if and only if Fac
T ∩ C is the torsion part of a 2-functorially-finite torsion pair in C .Proof. Let T be a torsion class in C , then by 2-functorial finiteness and closureunder factor modules, there exists an exact sequence A → T → T → T → f : A → T is a left T -approximation and T , T ∈ T . Clearly Im f ∼ = A/ Ann T .Now we need to prove that T i are Ext-projective in T for all 0 ≤ i ≤
2, i.e.that Ext A ( T i , T ) = 0 for all T ∈ T . First suppose Ext A ( T , T ) = 0. Then byProposition 3.6 there exists some T ′ ∈ T and a non-split surjection T ′ ։ T . Butthen f must factor through T ′ , a contradiction.We next claim that there is a surjection Ext A ( T , T ) ։ Ext A ( T , T ) for any T ∈ T . Let X := Im( T → T ). This induces exact sequencesExt A ( T , T ) → Ext A ( T , T ) → Ext A ( X, T )Ext A ( A/ ann T , T ) → Ext A ( X, T ) → Ext A ( T , T ) = 0 . Suppose Ext A ( A/ ann T , T ) = 0 and there exists a short exact sequence0 → T → E → A/ ann T → . But then E must be an A/ ann T -module, since both T and A/ ann T are, meaningthat this sequence splits, a contradiction. So Ext A ( A/ ann T , T ) = 0 = Ext A ( X, T ),and there is a surjection Ext A ( T , T ) ։ Ext A ( T , T ) as claimed.Now suppose Ext A ( T , T ′ ) = 0 for some T ′ ∈ T . By Proposition 3.6 there exists T ′ , T ′ , T ′ ∈ T and a d -exact sequence 0 → T ′ → T ′ → T ′ → T →
0. By Lemma
P, Q ∈ C and a commutative diagram with exact rows and columns:0 0 A/ ann T A/ ann T Q Q ⊕ T T T ′ Q ⊕ T ′ P T T ′ T ′ T ′ T
00 0Moreover
P, Q ∈ T by assumption, since they have no common non-zero sum-mands and we may replace T ′ if necessary. This implies a morphism A → Q ,which by assumption factors through T . But this implies the morphism T → T also factors through Q , and is hence the zero composition Q → P → T , a contra-diction.On the other hand, let T be a support τ -tilting module. We show that Fac T ∩ C satisfies the conditions of Proposition 3.6. Closure under factor modules is trivial,and 2-functorial finiteness in C follows from Lemma 3.9. So let0 → T → X → Y → T → d -exact sequence in C with T , T ∈ Fac T . By Lemma 2.4, we have thatHom A ( T, τ T ) = 0 ⇐⇒ Ext A ( T, Fac T ) = 0, so T is not in add T . Lemma 3.9implies there exists an exact sequence in Fac T : 0 → T ′ → T ′ → T ′ → T → . ByLemma 2.7 there exist
P, Q ∈ C to form a commutative diagram with exact rowsand columns: T ′ T ′ Q Q ⊕ T ′ T ′ T Q ⊕ X P T ′ T X Y T
00 0Applying Lemma 2.4 once more, we find Ext A ( T ′ , T ) = 0. This implies thediagram splits, Q ∼ = T , P ∼ = T ′ ⊕ X and that we are in the situation of Lemma2.8, with a commutative diagram with exact rows and columns:0 00 T ′ T ′ T T T ′ T ′ T X Y T d -pushout diagram0 T ⊕ T ′ T ⊕ T ′ T ′ T T X Y T → T ′ → T ⊕ T ′ → X ⊕ T ′ → Y → (cid:3) References
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