aa r X i v : . [ m a t h . R T ] J a n ON HIGHER TORSION CLASSES
J. ASADOLLAHI, P. JØRGENSEN, S. SCHROLL, AND H. TREFFINGER
Abstract.
Building on the embedding of an n -abelian category M into an abelian category A as an n -cluster-tilting subcategory of A , in this paper we relate the n -torsion classes of M with the torsion classes of A . Indeed, we show that every n -torsion class in M is given by theintersection of a torsion class in A with M . Moreover, we show that every chain of n -torsionclasses in the n -abelian category M induces a Harder-Narasimhan filtration for every objectof M . We use the relation between M and A to show that every Harder-Narasimhan filtrationinduced by a chain of n -torsion classes in M can be induced by a chain of torsion classes in A .Furthermore, we show that n -torsion classes are preserved by Galois covering functors, thuswe provide a way to systematically construct new (chains of) n -torsion classes. Introduction
Higher homological algebra has its origin in the study of n -cluster-tilting subcategories ofabelian and triangulated categories in [16, 17]. The subject has greatly developed since itsintroduction with more and more of the classical notions emerging in the higher setting. Thekey idea of higher homological algebra is the study of categories where the shortest non-splitexact sequences are composed of n + 2 objects, for a fixed positive integer n . In particular,1-homological algebra corresponds to the classical theory of abelian, exact and triangulatedcategories and their classical generalisations such as quasi-abelian and extriangulated categories.In recent years the importance of higher homological algebra is starting to emerge through ar-ticles showing connections between this subject and other branches of the mathematical sciences,such as combinatorics and homological mirror symmetry [10, 15, 19, 27, 34].Since its inception, it has been shown that many of the fundamental homological conceptsin the classical theory have an analogue in higher homological algebra. Classical homologicalalgebra is a by now well-developed subject and many of the fundamental concepts have severalequivalent definitions characterising different properties and aspects of the various concepts.However, this poses a difficulty in generalising these ideas into the setting of higher homologicalalgebra, since the classically equivalent definitions might lift to non-equivalent concepts in thehigher setting. Therefore, even if n -exact sequences are easy to identify, the search for the bestdefinition for higher analogues of classical notions is not an easy task. A breakthrough in thisdirection was achieved in [20], where the definitions of n -abelian and n -exact categories wereintroduced and in [11, 24] where it is shown that any n -abelian category arises as an n -clustertilting subcategory of an abelian category.A key notion in representation theory and homological algebra is the concept of torsiontheories, introduced by Dickson in [9]. Their natural relevance, for example, in relation to derivedcategories and tilting theory, have led to the use of homological algebra in many branches ofmathematics, including algebraic geometry and mathematical physics. Torsion theory is builton the notion of torsion pairs, where a torsion pair is a pair of full subcategories ( T , F ) with nonon-zero morphism from the torsion class T to the torsionfree class F . A definition of torsion classes in higher homological algebra has recently been given by the second author in [22] basedon the classical characterisation of the existence of a unique torsion subobject and a uniquetorsion free quotient for every object in the category.Part of our motivation for writing this paper is to show that when considering an n -abeliancategory in the context of its ambient abelian category, that is viewed as an n -cluster tiltingsubcategory of an abelian category, then the definition of higher torsion classes in [22] seemsto encode all the relevant properties one would expect. More precisely, one of the main ideasof the paper is built on the comparison of n -torsion classes in an n -abelian category and thecorresponding minimal torsion classes generated by these n -torsion classes in an abelian categorywhich contains the n -abelian category as a n -cluster-tilting subcategory. In particular, we dothis in the context of Harder-Narasimhan filtrations and n -Harder Narasimhan filtrations whichwe define and which we also study in the context of Galois coverings.Our first result characterises the minimal torsion classes in an abelian category containing a n -torsion class. Note that for every n ≥
1, the set of all n -torsion classes in an n -abelian category M forms a poset under the natural order given by the inclusion. Theorem 1.1 (Corollary 3.3, Theorem 3.6) . Let M be an n -cluster-tilting subcategory of askeletally small abelian length category A . Then there is an injective morphism of posets T : { n -torsion classes in M } → { torsion classes in A} given by sending an n -torsion class U in M to the minimal torsion class in A containing U .Moreover, a torsion class T in A is of the form T ( U ) for some n -torsion class U in M if andonly if the following hold:(1) tM ∈ U for all M ∈ M ; where t = t T is the torsion functor associated to the torsionclass T ;(2) T is the minimal torsion class in A containing { tM : M ∈ M } ;(3) Ext n − A ( X, Y ) = 0 , for all X ∈ { tM : M ∈ M } and Y ∈ { coker( tM ֒ → M ) | M ∈ M } .In this case, U = T ∩ M = { tM : M ∈ M } . Stability conditions were introduced in [26] to attack algebro-geometric problems. Given theirsimplicity and effectiveness, their definition was later adapted to other contexts, such as quiverrepresentations [23, 31], abelian categories [30] and triangulated categories [5].The success of stability conditions relies on the fact that every stability condition determinesfor each non-zero object in a category a stratification by more well-behaved objects. Thisstratification, usually known as the
Harder-Narasimhan filtration , has been used to make possiblecalculations that otherwise would be highly complicated or even impossible. Applications of thiscan be found, for example, in the study of Donaldson-Thomas invariants and in the mirrorsymmetry program [6, 28].In a recent paper [33], the fourth author has introduced an axiomatic approach to Harder-Narasimhan filtrations for abelian categories, by showing that the existence of such filtration forevery object in an abelian category is equivalent to the existence of a chain of torsion classes inthe category. Since this construction of Harder-Narasimhan filtrations does not depend on theexistence of a stability condition, it allows the introduction of Harder-Narasimhan filtrations tonon-abelian settings such as quasi-abelian categories [32]. In the second main result of this paperwe push this idea further by showing that chains of n -torsion classes induce Harder-Narasimhanfiltrations in n -abelian categories. Moreover, we show that the Harder-Narasimhan filtrationsobtained in this way coincide with the Harder-Narasimhan filtrations in the ambient abeliancategory which contains the n -abelian category as an n -cluster tilting subcategory. IGHER TORSION CLASSES 3
Theorem 1.2 (Theorem 4.6 and Theorem 5.2) . Let M be an n -abelian category and let δ be achain of n -torsion classes in M . Then for M a non-zero object in M the following hold:(1) δ induces an n -Harder-Narasimhan filtration of M which is unique up to isomorphism.(2) If M is the n -cluster-tilting subcategory of a skeletally small abelian length category A thenthe n -Harder-Narasimhan filtration of M in M induced by δ is equal to the Harder-Narasimhanfiltration of M in A induced by T ( δ ) , where T ( δ ) = { T ( U ) | U ∈ δ } . Another important concept in representation theory is the notion of Galois coverings, intro-duced by Gabriel in [12, 13] and studied further by many authors since. The initial aim wasto reduce a problem for modules over an algebra A to that of a category C with an action ofa group G such that A is equivalent to the orbit category C /G . The theory has much evolvedsince its inception leading to a vast body of literature on the subject [1, 3, 4, 7, 14, 25, 29]. Inparticular, it has been shown that several nice properties, such as local finiteness or Cohen-Maculay finiteness, are preserved by Galois coverings [2, 12]. Recently, Darp¨o and Iyama showin [7] that n -cluster-tilting subcategories are, under certain conditions, preserved by Galois cov-erings. Their construction is based on the fact that under certain technical conditions, whichare described in Theorem 1.3, given a Galois covering functor P : C −→ C /G , there exists aGalois precovering functor P • : mod- C −→ mod-( C /G ), called push-down functor, between thecategories of finitely presented functors over C and C /G such that P • ( M ) is an n -cluster-tiltingsubcategory in mod- C /G , where M is a certain n -cluster tilting subcategory of mod- C . We addto this by showing that under similar conditions as in [7], n -torsion classes, chains of n -torsionclasses, and n -Harder-Narasimhan filtrations are preserved by Galois coverings. Note that someauthors use ‘G-covering’ instead of ‘Galois covering’ for this generalized version of the classicalGalois covering theory, see e.g. [1]. More precisely, we show the following. Theorem 1.3 (Theorem 6.1, Proposition 6.3) . Let C be a small locally bounded Krull-Schmidt k -category with an admissible action of a group G on C inducing an admissible action on mod - C .Suppose that M is an n -cluster tilting G -equivariant full subcategory of mod - C such that P • ( M ) is functorially finite in mod - ( C /G ) . If U is a G -equivariant n -torsion class of M then P • ( U ) is an n -torsion class of P • ( M ) .Moreover, if U is a G -equivariant n -torsion class in M then the following statements holdfor M ∈ M .(1) An object U M in U is the torsion object of M with respect to U if and only if P • ( U M ) is the torsion object of P • ( M ) with respect to P • ( U ) .(2) If δ = { U s : s ∈ [0 , } is a chain of G -equivariant n -torsion classes in M then M ( M ( · · · ( M r − ( M r = M is the n -Harder-Narasimhan filtration of M with respect to δ in M if and only if P • ( M ) ( P • ( M ) ( · · · ( P • ( M r − ) ( P • ( M r ) = P • ( M ) is the n -Harder-Narasimhan filtration of P • ( M ) with respect to the chain of n -torsionclasses P • ( δ ) in P • ( M ) .(3) If T ( U ) is G -equivariant then T ( P • ( U )) = P • ( T ( U )) , that is the following diagram iscommutative ASADOLLAHI, JØRGENSEN, SCHROLL, AND TREFFINGER { G -equivariant n -torsion classes in M } T ( − ) / / P • ( − ) (cid:15) (cid:15) { G -equivariant torsion classes in mod - C} P • ( − ) (cid:15) (cid:15) { n -torsion classes in P • ( M ) } T ( − ) / / { torsion classes in mod - C /G } Acknowledgements:
Part of this work was done while the first author visited the Univer-sity of Leicester. He would like to thank the third and fourth authors for their warm hospitalityand also excellent mathematical discussions during his visit. The second author was supportedby a DNRF Chair from the Danish National Research Foundation (grant number DNRF156), byAarhus University Research Foundation (grant no. AUFF-F-2020-7-16), and by the Engineer-ing and Physical Sciences Research Council (grant number EP/P016014/1). The third and thefourth author are supported by the EPSRC through the Early Career Fellowship, EP/P016294/1.The fourth author is also funded by the Deutsche Forschungsgemeinschaft (DFG, German Re-search Foundation) under Germany’s Excellence - EXC-2047/1-390685813. The fourth authoralso thanks the Hausdorff Center for Mathematics, Bonn, where some of the work for this paperwas carried out. 2.
Background
An abelian category A is said to be a length category if every object of A is of finite length.A category A is called skeletally small if the class of all isomorphism classes of objects in A isa set. In this paper, whenever we say that A is an abelian category, we assume that A is askeletally small abelian length category.Given a full subcategory X of A which is closed under direct sums, we define the subcategoryFac ( X ) of A to be the full subcategory of A whose objects are quotients of objects in X .Fac ( X ) = { Y ∈ A : ∃ exact sequence X → Y → , for some X ∈ X } Similarly, the category Sub ( X ) is the full subcategory of A whose objects are subobjects ofobjects in X .Sub ( X ) = { Y ∈ A : ∃ exact sequence 0 → Y → X, for some X ∈ X } We say that X is a generating subcategory of A if Fac ( X ) = A . Dually, we say that X is cogenerating if Sub ( X ) = A .We say that an object M of A is filtered by X if there exists a finite sequence of subobjects M ⊂ M ⊂ · · · ⊂ M n such that M = 0, M n = M and M i /M i − ∈ X for all 1 ≤ i ≤ n . We denote by Filt X the fullsubcategory of all objects filtered by X . Note that X is a full subcategory of Fac ( X ), Sub ( X )and Filt ( X ).2.1. n -cluster-tilting subcategories and n -abelian categories. Let n be an integer greateror equal to 1. The theory of higher homological algebra started in [16, 17] with the study ofthe so-called n -cluster-tilting subcategories of module categories. Their definition for arbitraryabelian categories is the following.Let us preface the definition by recalling some notions. Let X be a full subcategory of A . Wesay that X is a contravariantly finite subcategory of A if every object A ∈ A , admits a right IGHER TORSION CLASSES 5 X -approximation, that is, for every A ∈ A there exists a morphism π : M → A with M ∈ X such that any other morphism π ′ : M ′ → A , with M ′ ∈ X , factors through π . Dually, thenotion covariantly finite subcategories is defined. A functorially finite subcategory of A , is asubcategory which is both contravariantly and covariantly finite. Definition 2.1. [20, Definition 3.14] Let A be an abelian category. A functorially finitegenerating-cogenerating subcategory M of A is n -cluster-tilting if M = { X ∈ A : Ext i A ( X, M ) = 0 for all M ∈ M and all 1 ≤ i ≤ n − } = { Y ∈ A : Ext i A ( M, Y ) = 0 for all M ∈ M and all 1 ≤ i ≤ n − } . The concept of n -abelian category was introduced in [20] as a generalisation of the classicalconcept of abelian categories, to formalize the homological structure of n -cluster-tilting subcat-egories. The formal definition uses the notions of n -kernel and n -cokernel of a morphism, thatwe now recall. Let f : X → X be a morphism in an additive category M . A sequence ofmorphisms X f −→ X f −→ . . . f n − −−−→ X n f n −−→ X n +1 is called an n -cokernel of f if, for every M ∈ M , the following sequence0 → M ( X n +1 , M ) → M ( X n , M ) → · · · → M ( X , M ) → M ( X , M )of abelian groups is exact. An n -cokernel of f is denoted by ( f , . . . , f n ). The notion of n -kernelof a morphism is defined dually. The sequence X f −→ X f −→ . . . f n − −−−→ X n f n −−→ X n +1 . is called n -exac t if ( f , . . . , f n ) is an n -cokernel of f and ( f , . . . , f n − ) is an n -kernel of f n . Definition 2.2. [20, Definition 3.1] Let n be a positive integer. An additive category M is n -abelian if the following axioms hold:(A0) M has split idempotents;(A1) Every morphism in M has an n -kernel and an n -cokernel;(A2) For every monomorphism f : X → X and any n -cokernel ( f , . . . , f n ) of f , thefollowing sequence is n -exact; X f −→ X f −→ . . . f n − −−−→ X n f n −−→ X n +1 . (A2 op ) For every epimorphism f n : X n → X n +1 and any n -kernel ( f , . . . , f n − ) of f n , thefollowing sequence is n -exact; X f −→ X f −→ . . . f n − −−−→ X n f n −−→ X n +1 The motivating example for n -abelian categories are n -cluster-tilting subcategories and indeedas stated below it is now known that these are the only small n -abelian categories. Theorem 2.3. [20, Theorem 3.16] Let M be an n -cluster-tilting subcategory of an abeliancategory A . Then M is n -abelian. It is worth noticing that all n -exact sequences in M are the n -extensions of A where all termsof the extensions are in M . The converse of the previous result also holds. Theorem 2.4. [11, Theorem 4.3] [24, Theorem 7.3] Let M be an n -abelian category. Thenthere exists an abelian category A and a fully faithful functor F : M → A such that F ( M ) isan n -cluster-tilting subcategory of A . ASADOLLAHI, JØRGENSEN, SCHROLL, AND TREFFINGER
Torsion and n -torsion classes. Generalising the classical properties of abelian groups,Dickson introduced in [9] the notion of torsion pair as follows.
Definition 2.5.
Let A be an abelian category. Then the pair ( T , F ) of full subcategories of A is a torsion pair if the following conditions are satisfied: • Hom A ( X, Y ) = 0 for all X ∈ T and Y ∈ F . • For every module M in A there exists a short exact sequence0 → tM ι M −−→ M π M −−→ f M → tM ∈ T and f M ∈ F .This short exact sequence is unique up to isomorphisms and is known as the canonical shortexact sequence of M with respect to ( T , F ). Moreover we say that T is a torsion class and F isa torsion free class .In the same paper [9] where he introduced the concept of torsion pair, Dickson gave an usefulcharacterisation of torsion and torsion free classes. Theorem 2.6. [9, Theorem 2.3] A full subcategory T of an abelian category A is a torsion classif and only if T is closed under factors and extensions. Dually, a full subcategory F of an abeliancategory A is a torsion free class if and only if F is closed under subobjects and extensions. We denote by tors( A ) the set of all torsion classes in A . It is clear that the natural inclusionof sets induces a natural partial order in tors( A ).Given a subcategory X of A , we denote by T ( X ) the minimal torsion class of A containing X . It is well-known that T ( X ) coincides with all the objects of A filtered by elements in thecategory Fac ( X ), that is, T ( X ) = Filt (Fac ( X )).With the development of higher homological algebra, it is natural to consider higher analoguesof torsion classes in this framework. The first such notion is introduced in [22]. The formaldefinition is as follows. Definition 2.7. [22, Definition 1.1] Let M be an n -abelian category. A full subcategory U of M is an n -torsion class if for every M ∈ M there exists an n -exact sequence(1) 0 −→ U M −→ M −→ V v −→ · · · v n − −−−→ V n −→ , where U M is an object of U and the sequence(2) 0 −→ Hom M ( U, V ) −→ Hom M ( U, V ) −→ · · · −→ Hom M ( U, V n ) −→ U in U . U M is called the n -torsion subobject of M with respect to U .2.3. Harder-Narasimhan filtrations in abelian categories.
Inspired by the relation be-tween stability conditions and torsion classes, in [33] the relation between Harder-Narasimhanfiltrations and torsion classes was studied. This was done through the introduction of chains oftorsion classes as follows.
Definition 2.8. [33, Definition 2.1] A chain of torsion classes η in an abelian category A is aset of torsion classes η := {T s : s ∈ [0 , T = A , T = { } and T s ⊆ T r if r ≤ s } . We denote by T ( A ) the set of all chains of torsion classes of A . IGHER TORSION CLASSES 7
Associated to every chain of torsion classes η ∈ T ( A ) there is a set P η = {P t : t ∈ [0 , } of full subcategories of A where each P t is defined as follows. Note that in the following definitionwe assume T s< T s = A and S s> T s = { } . Definition 2.9.
Consider a chain of torsion classes η ∈ T ( A ). For every t ∈ [0 ,
1] we have thesubcategory S t = T s
1] and include a short proof that the two definitionsare equivalent: P t = T s> F s if t = 0 (cid:18) T s
1] and X ∈ (cid:0)T s
Galois coverings.
Let C be a skeletally small Krull-Schmidt k -category, where k is afield. Given a group G , we say that there is a G -action over C or simply C is a G -category ifthere is a group homomorphism A : G −→ Aut( C ), where Aut( C ) denotes the group of k -linearautomorphisms of C . We usually write A g instead of A ( g ) and gX for A g ( X ), for each g ∈ G and X ∈ C . The action of G on C is called admissible if gX ≇ X for each indecomposable object X in C and each g = 1 in G .Let C be a G -category. The orbit category C /G of C by G is a category whose objects are theobjects of C and for every X, Y ∈ C /G , the morphism set C /G ( X, Y ) is given by ( f h,g ) ( g,h ) ∈ Y ( g,h ) ∈ G × G C ( gX, hY ) | ( f h,g ) ( g,h ) is rcf and f g ′ h,g ′ g = g ′ ( f h,g ) , ∀ g ′ ∈ G where rcf denotes ( f h,g ) ( g,h ) being row and column finite, that is, for every g ∈ G there arefinitely many h ∈ G such that f h,g = 0 and, dually, for every h ′ ∈ G there are finitely many g ′ ∈ G such that f h ′ ,g ′ = 0. For two composable morphisms X f −→ Y f ′ −→ Z in C /G , we define f ′ f := X g ′ ∈ G f ′ h,g ′ f g ′ ,g ( g,h ) ∈ G × G . There is a canonical functor P : C −→ C /G which is given by P ( X ) = X and P ( f ) =( δ g,h gf ) ( g,h ) , for every X, Y ∈ C and for every f ∈ C ( X, Y ).Recall that a pair (
F, ϕ ) of a functor F : C −→ C ′ and a family ϕ := ( ϕ g ) g ∈ G of naturalisomorphisms ϕ g : F −→ F A g is called a G -invariant functor if, for every g, h ∈ G , the followingdiagram is commutative F ϕ g / / ϕ hg ' ' ◆◆◆◆◆◆◆◆◆◆ F A gϕ h A g (cid:15) (cid:15) F A hg = F A h A g . The family ϕ := ( ϕ g ) g ∈ G is called an invariant adjuster of F . Definition 2.12. [1, Definition 1.7] Let F : C −→ C ′ be a G -invariant functor. Then F is calleda G -precovering functor , if for every X, Y ∈ C the k -homomorphisms F (1) X,Y : M g ∈ G C ( gX, Y ) −→ C ′ ( F X, F Y ) , ( f g ) g ∈ G X g ∈ G F ( f g ) ϕ g,X ; F (2) X,Y : M g ∈ G C ( X, gY ) −→ C ′ ( F X, F Y ) , ( f g ) g ∈ G X g ∈ G ϕ g − ,gY F ( f g ) , are isomorphisms. If F is also dense, then it is called a G -covering functor .It is shown in [1, Proposition 2.6] that P : C −→ C /G is a G -covering functor which is universalamong G -invariant functors starting from C .We say that the k -category C is locally bounded if for each indecomposable X ∈ C , we havethat X Y ∈ ind - C (dim k (Hom C ( X, Y ) + dim k (Hom C ( Y, X )) < ∞ . From now on, assume moreover that C is a locally bounded k -category. IGHER TORSION CLASSES 9
Let C be a G -category. The G -action on C induces a G -action on Mod- C , where Mod- C denotesthe category of contravariant functors from C to Mod- k . In fact, for each g ∈ G , we can definean automorphism A g : Mod- C −→
Mod- C by A g ( M ) = g M := M ◦ A − g , for all M ∈ Mod- C . It follows from the definitions that for every M ∈ C , g C ( − , M ) = C ( g − ( − ) , M ) ∼ = C ( − , gM ).The canonical functor P : C −→ C /G induces a functor P • : Mod-( C /G ) −→ Mod- C , givenby P • ( M ) = M ◦ P for every M ∈ Mod-( C /G ). This functor is called the pull-up of P . It iswell known that P • possesses a left adjoint P • : Mod- C −→
Mod-( C /G ), which is called the push-down functor . For more details see [1]. It follows that the push-down functor P • is exact.A functor M ∈ Mod- C is called finitely presented if there exists an exact sequence ( − , Y ) −→ ( − , X ) −→ M −→ C . Let mod- C be the full subcategory of Mod- C consisting of allfinitely presented modules. It is known [1, Theorem 4.3] that the restriction of the push-downfunctor to mod- C induces a functor P • : mod- C −→ mod-( C /G )again denoted by P • , which is a G -precovering functor.The central result relating higher homological algebra and Galois coverings was proved byDarp¨o and Iyama and it reads as follows. Recall that a subcategory X of mod- C is said to be G -equivariant if g X = X , for all g ∈ G . Theorem 2.13. [7, Theorem 2.14] Let C be a locally bounded Krull-Schmidt G -category with anadmissible action of G on C inducing an admissible action on mod - C . If M is a G -equivariantfull subcategory of mod - C such that P • ( M ) is functorially finite in mod - ( C /G ) , then M is an n -cluster tilting subcategory of mod - C if and only if P • ( M ) is an n -cluster tilting subcategory of mod - ( C /G ) . For more details on the covering theory we refer the reader to [1, 3, 7].3.
Minimal torsion classes containing n -torsion classes Let M be an n -abelian category. By [24] , there exists an abelian category A and a fully faith-ful functor F : M → A such that F ( M ) is an n -cluster-tilting subcategory of A . Throughoutthe section we fix an n -abelian category M and consider it as the n -cluster tilting subcategoryof the abelian category A .For an n -torsion class U ⊆ M , we set T = T ( U ) to be the smallest torsion class of A containing U and for any M ∈ M , denote by U M the n -torsion object of M with respect to the n -torsion class U . For a torsion class T in A and for M ∈ A denote by tM the torsion objectof M with respect to T . Lemma 3.1.
Let U be an n -torsion class in M and let M ∈ M . Then tM ∼ = U M where tM is the torsion object of M with respect to T ( U ) . In other words, for all M ∈ M , thefundamental n -exact sequence (3) 0 −→ U M u −→ M −→ V v −→ · · · v n − −−−→ V n −→ of M with respect to U is isomorphic to (4) 0 −→ tM ι M −−→ M −→ V v −→ · · · v n − −−−→ V n −→ . Proof.
Let M ∈ M and take the canonical n -exact sequence of M with respect to U .(5) 0 −→ U M u −→ M −→ V v −→ V −→ · · · −→ V n −→ A .(6) 0 −→ coker u −→ V v −→ V v −→ · · · v n − −−−→ V n −→ M ( U, − ) with U ∈ U we obtain the following exact sequence.(7) 0 −→ M ( U, coker u ) −→ M ( U, V ) v ∗ −→ M ( U, V )But from the definition of n -torsion pair we have that(8) 0 −→ M ( U, V ) v ∗ −→ · · · v n − ∗ −−−→ M ( U, V n ) −→ U ∈ U . In particular,the exactness of the sequence (8) implies that v ∗ : M ( U, V ) → M ( U, V ) is injective. Thus M ( U, coker u ) = 0 for every U ∈ U . This impliesthat 0 → U M u −→ M −→ coker u → U M ∈ T ( U ) and coker u ∈ ( T ( U )) ⊥ , where for a class T of objects of A , T ⊥ = { Y ∈ A | Hom A ( X, Y ) = 0 , for all X ∈ T } . Since the canonical short exact sequence of any object with respect to a torsion pair is uniqueup-to-isomorphism, we conclude that tM ∼ = U M . (cid:3) As a direct consequence of Lemma 3.1 we obtain the following result.
Proposition 3.2.
Let U , U be two n -torsion classes in M . Then T ( U ) = T ( U ) if and onlyif U = U .Proof. The sufficiency is clear, so we show the necessity. Let M ∈ U \ U . Then the torsionobject U M of M with respect to U is not isomorphic to U M = M . By Lemma 3.1 we havethat the torsion object t M of M with respect to T ( U ) is not isomorphic to the torsion object t M = M of M with respect to T ( U ). Hence T ( U ) is different from T ( U ). (cid:3) In recent years there has been a great deal of interest regarding the poset of torsion classesin the module category of an algebra ordered by inclusion. As a consequence of our previousresult, we have the following.
Corollary 3.3.
The map T ( − ) : n -tors ( M ) → tors ( A ) from the set of n -torsion classes n -tors ( M ) of the n -cluster-tilting subcategory M to the set of torsion classes tors ( A ) of theabelian category A is injective and respects the order given by the inclusion.Proof. It follows from Proposition 3.2 that if U and U are two distinct n -torsion classes in M ,then T ( U ) and T ( U ) are two distinct torsion classes in A . The fact that T ( U ) ⊂ T ( U ) if U ⊂ U is immediate. (cid:3) Lemma 3.4.
Let U be an n -torsion class of M ⊂ A . If T = T ( U ) is the minimal torsionclass of A containing U , then U = t M , where t M = { tM : M ∈ M } . In particular we have T = T ( t M ) . IGHER TORSION CLASSES 11
Proof.
The fact that U ⊇ { tM : M ∈ M } follows directly from Lemma 3.1. Let U ∈ U ⊂ M .Then U ∈ T = T ( U ). Hence tU = U . Thus U ∈ { tM : M ∈ M } . (cid:3) Lemma 3.5.
Let M ⊆ A be an n -cluster tilting subcategory and T ⊆ A be a torsion classsatisfying t M ⊆ M . Then t M is an n -torsion class of M if and only if Ext n − A ( X, Y ) = 0 forall X ∈ t M and all Y ∈ f M , where f M = { coker ( tM ֒ → M ) | M ∈ M } .Proof. Assume first that t M ⊆ M is an n -torsion class. Let X ∈ t M and Y ∈ f M be arbitraryelements. So Y = f M , for some M ∈ M and there exists a short exact sequence(9) 0 −→ tM ι M −−→ M −→ f M −→ , where the monomorphism tM ι M ֒ → M sits in the canonical n -exact sequence(10) 0 −→ tM ι M −−→ M −→ V v −→ · · · v n − −−−→ V n −→ , in which the sequence(11) 0 −→ V v −→ · · · v n − −−−→ V n −→ t M -exact.The sequence (9) induces a long exact sequence which containsExt n − A ( X, M ) −→ Ext n − A ( X, f M ) −→ Ext n A ( X, tM ) Ext n A ( X,ι M ) −−−−−−−−→ Ext n A ( X, M ) . Since X ∈ t M ⊆ M while M is n -cluster tilting, the first term is zero and so the sequencereads as follows(12) 0 −→ Ext n − A ( X, f M ) −→ Ext n A ( X, tM ) Ext n A ( X,ι M ) −−−−−−−−→ Ext n A ( X, M ) . Moreover, X ∈ M implies that the sequence (10) induces a long exact Hom-Ext n -sequencewhich contains the following, see [21, Prop. 2.2].(13) A ( X, V n − ) v n − ∗ −−−→ A ( X, V n ) −→ Ext n A ( X, tM ) Ext n A ( X,ι M ) −−−−−−−−→ Ext n A ( X, M )Since the sequence (11) is t M -exact, we deduce that v n − ∗ is surjective and so the morphismExt n A ( X, ι M ) is injective. This in view of the sequence (12) implies that Ext n − A ( X, Y ) = 0For the converse, assume that Ext n − A ( X, Y ) = 0, for all X ∈ t M and all Y ∈ f M . Consider M ∈ M . Since t M ⊆ M , we have an n -exact sequence(14) 0 −→ tM ι M −−→ M −→ V v −→ · · · v n − −−−→ V n −→ . in M , where coker( tM ι M ֒ → M ) = f M ∈ f M . To conclude the result we should show that thesequence 0 −→ V v −→ · · · v n − −−−→ V n −→ . is t M -exact. But this follows easily from the assumption, in view of the facts that A ( t M , f M ) =0 and the sequence 14 is n -exact. (cid:3) We are now in place to give a characterisation of the torsion classes T in A which are of theform T = T ( U ) for some n -torsion class U of M . Theorem 3.6.
Let M ⊆ A be an n -cluster tilting subcategory and T be a torsion class of A .Then T is of the form T = T ( U ) for some n -torsion class U of M if and only if the followingholds:(1) t M ⊆ M ;(2) T = T ( t M ) ;(3) Ext n − A ( X, Y ) = 0 for all X ∈ t M and Y ∈ f M .Moreover, in this case U = T ∩ M = { tM : M ∈ M } .Proof. Necessity . Suppose that T = T ( U ) for an n -torsion class U ⊆ M . We must show that T has all three characteristics as in the statement. Parts 1. and 2. follow from Lemma 3.4. Inparticular we have U = t M ⊆ M , so Lemma 3.5 applies to complete the proof of this part. Sufficiency . Suppose that T is a torsion class in A satisfying 1.-3. We must show T = T ( U )for an n -torsion class U ⊆ M , and in view of 2. this holds if t M is an n -torsion class in M . But t M ⊆ M holds by 1, and so by 3. in view of Lemma 3.5, the sequence (10) is is a fundamental n -exact sequence for M with respect to t M .Now we show the moreover part of the statement. It is already proven in Lemma 3.4 that U = { tM : M ∈ M } . So we only need to prove that U = T ∩ M . We do it by doubleinclusion. The fact that U ⊂ M ∩ T follows immediately from the fact that T = T ( U ). Now,if X ∈ M ∩ T we have that X = tX . Then X ∈ U by Lemma 3.1. (cid:3) Harder-Narasimhan filtrations in n -abelian categories We start this section by introducing chains of n -torsion classes. Definition 4.1.
A chain of n -torsion classes δ in an n -abelian category M is a set of n -torsionclasses δ := { U s : s ∈ [0 , , U = M , U = { } and U s ⊆ U r if r ≤ s } . We denote by T ( M ) the set of all chains of n -torsion classes in M .In this section we show that every chain of n -torsion classes δ in T ( M ) induces an n -Harder-Narasimhan filtration for every non-zero object M ∈ M . We first need some preliminary results. Lemma 4.2.
Let U ⊂ U be two n -torsion classes in an n -abelian category M and let M be an object of M . Take the n -torsion subobjects U M , U M of M with respect to U and U ,respectively. Then the following hold:(1) U M is a subobject of U M .(2) The torsion object U U M of U M with respect to U is isomorphic to U M .Proof.
1. This follows directly from the definition of n -torsion classes and the fact that U M isan object of U .2. Since U M is a subobject of M , we have that U U M is a subobject of M which belongs to U . Then, by the universal properties of n -torsion objects we have that U U M is a subobject of U M . On the other hand, we have by 1 . that U M is a subobject of U M . Hence, the universalproperties of torsion objects imply that U M is a subobject of U U M . We can then conclude that U M is isomorphic to U U M . (cid:3) Proposition 4.3.
Let M be an n -abelian category and let δ be a chain of n -torsion classes in M . Then S r>s U r is an n -torsion class in M for all s ∈ [0 , and T t
Proof.
Let δ ∈ T ( M ) and M ∈ M . We first show that S r>s U r is an n -torsion class for all s ∈ [0 , t > s the n -torsion subobject U Mt of M with respect to U t . ThenLemma 4.2 implies that we have an ascending chain of subobjects of M as follows0 = U M ⊂ · · · ⊂ U Mt ⊂ · · · ⊂ M. Recall that A is a length category, in particular A is an Artinian category. This implies that theabove ascending chain of subobjects of M stabilises. In other words, this implies the existenceof a t M > s such that U Mt M = U Mt for all s < t < t M .Given that U Mt M is a subobject of M , there is a monomorphism α : U Mt M → M . Then we obtainan n -exact sequence in M (15) 0 −→ U Mt M α −→ M −→ V v −→ · · · v n − −−−→ V n −→ n -cokernel of α . We claim that(16) 0 −→ M ( X, V ) −→ M ( X, V ) −→ · · · −→ M ( X, V n ) −→ . is exact for all X ∈ S r>s U r . Indeed, for each X ∈ S r>s U r there exists a real number t ∈ ( s, X ∈ U t . If t M ≤ t , we have that X ∈ U Mt M . Then (16) is exact because U Mt M is thetorsion object of M in U t M . Otherwise, let s < t < t M . Then we have that U Mt ∼ = U Mt M byconstruction and (16) is exact for all X ∈ S r>s U r . Hence, we have shown that for each M ∈ M there exists a subobject U M>s := U Mt M in S r>s U r such that (16) is exact for all X ∈ S r>s U r .Thus S r>s U r is an n -torsion class.To show that T t
1) we denote by U M>s and U Ms U r and T t
Let δ be a chain of n -torsion classes. For every t ∈ [0 ,
1] we have the subcategory S t = T s
Let M be an n -abelian category and let δ be a chain of n -torsion classes in M .Then δ induces an n -Harder-Narasimhan filtration for every M ∈ M . That is a filtration M ( M ( · · · ( M r − ( M r = M such that there exists a finite ordered set s > s > · · · > s r such that s i ∈ [0 , and the n -cokernel of the inclusion M k − → M k is in Q s k for every ≤ k ≤ r . Moreover this filtration isunique up-to-isomorphism.In particular, we have that M i − is the torsion subobject of M i with respect to S r>s i U r forall ≤ i ≤ r .Proof. Given M ∈ M , we begin by showing the existence of a filtration with the desired proper-ties. For this we need to show the existence of s r ∈ [0 ,
1] such that M ∈ T ss r U t .Clearly M ∈ T = M and M
6∈ T = { } . Moreover either M ∈ U s or M U s for all s ∈ [0 , s r = inf { t ∈ [0 ,
1] : M U t } = sup { s ∈ [0 ,
1] : M ∈ U s } . is well-defined and uniquely determined by M . Now, consider the n -torsion subobject U M>s r of M with respect to the n -torsion class S t>s r U t . Note that U M>s r is a proper subobject of M since M S t>s r U t . Hence the n -cokernel of the natural inclusion U M>s t → M is in Q s r .Set M r := M and M r − := U M>s r . Applying the above argument to M r − , there is a unique s r − ∈ [0 ,
1] such that M r − ∈ T ss r − U t . Moreover, s r − = inf { t ∈ [0 ,
1] : M r − U t } = sup { s ∈ [0 ,
1] : M r − ∈ U s } . Note that M r − ∈ S t>s r U t ⊂ U s r . In particular, this implies that s r − > s r .Applying this process inductively, we get an ascending sequence s r < s r − < . . . correspond-ing to a descending chain of subobjects of M · · · ⊂ M r − i ⊂ M r − i +1 ⊂ · · · ⊂ M r = M. Recall that, by Theorem 2.4 M is a full subcategory of an abelian category A , which is assumedto be a length category. Hence M is of finite length in A . Thus there is a positive integer k that M r − k = 0. Without loss of generality we can suppose that k = r . This shows the existence of afiltration with the desired properties.We now show the uniqueness of this filtration. Suppose that there exists a second filtration0 = M ′ ( M ′ ( · · · ( M ′ t − ( M ′ t = M as in the statement of the theorem. Then M ′ t − is the torsion object U M>s t of M with respect tothe n -torsion class S x>s t U x for some s t in [0 ,
1] such that M ∈ T ss t U t . However,we have shown that there exists a unique s t ∈ [0 ,
1] such that M ∈ T ss t U t . Thisimplies that s t = s r . Moreover, we have that M r − ∼ = M ′ t − since the torsion objects are uniqueup to isomorphism.Repeating this process we show that s t − i = s r − i and that M r − i ∼ = M ′ t − i for all positive integer i . In particular we have that 0 = M ∼ = M ′ r − t +1 and 0 = M ∼ = M ′ r − t , implying that r = t andthe proof is complete. (cid:3) IGHER TORSION CLASSES 15 Embedding of n -Harder-Narasimhan filtrations In Section 3 we have shown that the map T ( − ) : n -tors( M ) → tors( A ) embeds the posetof n -torsion classes n -tors( M ) in an n -cluster tilting subcategory M of an abelian category A into the poset tors( A ) of torsion classes in A . This implies, in particular, that every chain of n -torsion classes δ in M induces naturally a chain of torsion classes T ( δ ) in A by setting T ( δ ) := { T ( U s ) : U s ∈ δ for all s ∈ [0 , } . In order to construct ( n -)Harder-Narasimhan filtrations and show that they are unique, weuse infinite unions and intersections of ( n -)torsion classes. We now show that the map T ( − ) : n -tors( M ) → tors( A ) commutes with infinite unions and intersections. Proposition 5.1.
Let δ = { U s : s ∈ [0 , } be a chain of n -torsion classes in M . Then T [ r>s U r ! = [ r>s T ( U r ) and T \ rs T ( U r ) , \ r>s F ( U r ) ! and \ rs U r (cid:19) = S r>s T ( U r ).Clearly U r ⊂ T ( U r ) for all r > s . Then S r>s U r ⊂ S r>s T ( U r ). We can then apply T ( − ) toboth sets to obtain T (cid:18) S r>s U r (cid:19) ⊂ T (cid:18) S r>s T ( U r ) (cid:19) . Now, we have that S r>s T ( U r ) is a torsionclass by [33, Proposition 2.3]. So, T (cid:18) S r>s U r (cid:19) ⊂ T (cid:18) S r>s T ( U r ) (cid:19) = S r>s T ( U r ).In the other direction, recall that T ( X ) = Filt (Fac ( X )) (cf. [8]). Then we have the followinginclusions. U r ⊂ [ r>s U r for all r > s Filt (Fac ( U r )) ⊂ Filt
Fac [ r>s U r !! for all r > sT ( U r ) ⊂ T [ r>s U r ! for all r > s [ r>s T ( U r ) ⊂ T [ r>s U r ! The moreover part of the statement follows directly from [33, Proposition 2.7] (cid:3)
By now, we have seen that for every non-zero object M ∈ M ⊂ A we have the n -Harder-Narasimhan filtration induced by δ given by Theorem 4.6 and the Harder-Narasimhan filtrationinduced by T ( δ ) given by Theorem 2.11. We show that both filtrations coincide. Theorem 5.2.
Let δ be a chain of n -torsion classes in M , T ( δ ) be the chain of torsion classesin A induced by δ and M be an object in M . Consider the n -Harder Narasimhan filtration M ( M ( . . . M t − ( M t = M of M induced by δ and the Harder-Narasimhan filtration N ( N ( . . . M t ′ − ( M t ′ = M of M induced by T ( δ ) . Then we have that t = t ′ and M i ∼ = N i for all ≤ i ≤ t .Proof. Let M be an object of M and δ a chain of n -torsion classes. Consider the n -HarderNarasimhan filtration 0 = M ( M ( . . . M t − ( M t = M of M induced by δ . By Theorem 4.6, we have that M i − is the n -torsion subobject of M i withrespect to the n -torsion class S r>s i U r . Applying Lemma 3.1 and Proposition 5.1 we obtain that M i − is the torsion object of M i with respect of S r>s i T ( U r ). In other words,0 −→ M i − α −→ M i −→ coker( α ) −→ M i with respect to the torsion pair [ r>s i T ( U r ) , \ r>s i F ( U r ) ! , where F ( U r ) is the torsion free class in A such that ( T ( U r ) , F ( U r )) is a torsion pair (see [33,Proposition 2.7]). Thus coker( α ) belongs to T r>s i F ( U r ).On the other hand, it follows from Theorem 4.6 that M i ∈ U s for all s < s i . Then M i ∈ T ( U s )for all s < s i . Hence M i ∈ T ss i F ( U s ) for all 1 ≤ i ≤ t .This means that 0 = M ( M ( . . . M t − ( M t = M is a filtration of M such that:(1) 0 = M and M n = M ;(2) there exists s k ∈ [0 ,
1] such that M k /M k − ∈ P s k , for all 1 ≤ k ≤ t ;(3) s > s > · · · > s t .Then this is the Harder-Narasimhan filtration of M with respect to the chain of torsion classes T ( δ ). (cid:3) As a consequence of the last theorem, we have the following.
Corollary 5.3.
Let δ be a chain of n -torsion classes of M ⊂ A and consider the family ofsubcategories {P s : s ∈ [0 , } be the slicing associated to T ( δ ) as defined in Definition 2.9. Ifthere exists a non-zero object in M then there exists a s ∈ [0 , such that P s ∩ M contains anon-zero object. IGHER TORSION CLASSES 17
Proof.
Let δ be a chain of n -torsion classes in M and M be a non-zero object of M . Then,Theorem 4.6 gives us the n -Harder-Narasimhan filtration of M .0 = M ( M ( . . . M t − ( M t = M Moreover, Theorem 5.2 implies that M i /M i − ∈ P s i for all 1 ≤ i ≤ t . In particular, M /M ∼ = M ∈ P s . (cid:3) Galois coverings of n -torsion classes Assume that C is a locally bounded Krull-Schmidt k -category, where k is a field, with anadmissible G -action on C inducing an admissible action on mod- C . Then, as in Subsection 2.4,the functor P : C → C /G induces a functor P • : mod- C −→ mod-( C /G ) which is a G -precoveringmap. Moreover, we know by Theorem 2.13 that, under these assumptions, a G -equivariantsubcategory M of mod- C such that P • ( M ) is functorially finite in mod-( C /G ) is an n -cluster-tilting subcategory of mod- C if and only if P • ( M ) is an n -cluster-tilting subcategory of mod- C /G .For more details see Subsection 2.4.We start this section by showing that G -equivariant n -torsion classes behave well under thepush-down functor P • : mod- C −→ mod-( C /G ). Theorem 6.1.
Let C be a locally bounded Krull-Schmidt k -category with an admissible action ofa group G on C inducing an admissible action on mod - C . Suppose that M is an n -cluster-tilting G -equivariant full subcategory of mod - C such that P • ( M ) is functorially finite in mod - ( C /G ) .Let U be a G -equivariant full subcategory of M . If U is an n -torsion class of M then P • ( U ) is an n -torsion class of P • ( M ) .Proof. First, note that by Theorem 2.13, P • ( M ) is an n -cluster-tilting subcategory of mod-( C /G ).Let U be an n -torsion class of M and P • ( M ) be an object in P • ( M ). By definition, there is an n -exact sequence(19) 0 −→ U M θ −→ M ϕ −→ V −→ · · · ϕ n −→ V n −→ M , where U ∈ U and(20) 0 → M ( U, V ) → M ( U, V ) → · · · → M ( U, V n ) → U in U .By applying the exact functor P • on the sequence (19), we get the following exact sequence(21) 0 −→ P • ( U M ) P • ( θ ) −→ P • ( M ) P • ( ϕ ) −→ P • ( V ) −→ · · · P • ( ϕ n ) −→ P • ( V n ) −→ . in P • ( M ). To show that P • ( U ) is an n -torsion class of P • ( M ), it is enough to show that thisis the canonical n -exact sequence of P • ( M ) with respect to P • ( U ).Since the sequence (21) is exact, we may deduce from [18, Lemma 3.5], that it is an n -exactsequence in P • ( M ).So it is enough to show that the sequence(22) 0 −→ P • ( M )( P • ( U ) , P • ( V )) −→ · · · −→ P • ( M )( P • ( U ) , P • ( V n )) −→ P • ( U ) of P • ( U ). Since the push-down functor P • : mod- C −→ mod-( C /G ) is G -precovering, there exists thefollowing commutative diagram / / ( P • ( U ) , P • ( V )) P • ( θ ) ∗ / / ( P • ( U ) , P • ( V )) / / · · · P • ( ϕ n ) ∗ / / ( P • ( U ) , P • ( V n )) / / / / ⊕ g ∈ G M ( g U, V ) P • (1) U,V O O ⊕ θ ∗ / / ⊕ g ∈ G M ( g U, V ) P • (1) U,V O O / / · · · ⊕ ϕ n ∗ / / ⊕ g ∈ G M ( g U, V n ) P • (1) U,V n O O / / , where the vertical maps are k -isomorphisms, see Definition 2.12.Since U is G -equivariant, g U belongs to U for all g ∈ G . Now, since the sequence (20) isexact, for all U ∈ U , the bottom row of the above diagram is exact. This implies the exactnessof the top row, as desired.This completes the proof of the theorem. (cid:3) In Theorem 3.6, given an n -cluster-tilting subcategory M of an abelian category A , wecharacterise the minimal torsion class T ( U ) of A containing the n -torsion class U ⊂ M . Inthe following corollary we compare the minimal torsion class T ( P • ( U )) of mod- C /G containing P • ( U ) with the torsion class P • ( T ( U )). Recall that for a subcategory X of an abelian category A , the minimal torsion class of A containing X is denoted by T ( X ) and is equal to Filt (Fac ( X )). Corollary 6.2.
Let the situation be as in the Theorem 6.1. Let U be an n -torsion class of M and suppose that T ( U ) is G -equivariant. Then P • ( T ( U )) = T ( P • ( U )) .Proof. By definition, T ( U ) = Filt (Fac ( U )) is the minimal torsion class of mod- C that contains U . Let M ∈ T ( U ). So there exists a filtration0 = M ⊂ M ⊂ · · · ⊂ M t = M of M such that M i /M i − ∈ Fac ( U ) for all 1 ≤ i ≤ t . Since the push-down functor P • is exact, weeasily deduce that P • ( M ) ∈ Filt (Fac ( P • ( U ))) = T ( P • ( U )). Therefore P • ( T ( U )) ⊆ T ( P • ( U )).For the reverse inclusion, note that the inclusion U ⊆ T ( U ) implies that P • ( U ) ⊆ P • ( T ( U )).Hence T ( P • ( U )) ⊆ T ( P • ( T ( U ))) . But T ( P • ( T ( U ))) = P • ( T ( U )), because by Theorem 6.1 we have that the functor P • preservestorsion classes. The proof is hence complete. (cid:3) Note that Theorem 6.1 implies that the functor P • : mod- C → mod- C /G induces a map nP • : G - n -tors( M ) → n -tors( P • ( M ))from the set G - n -tors( M ) of G -equivariant n -torsion classes of M ⊂ mod- C to the set n -tors( P • ( M ))of n -torsion classes of P • ( M ) ⊂ mod- C /G . Likewise, P • : mod- C → mod- C /G induces a map P • : G -tors(mod- C ) → tors(mod- C /G )from the set G -tors( M ) of G -equivariant torsion classes of mod- C to the set tors( P • ( M )) oftorsion classes of mod- C /G . Using this notation, Corollary 6.2 can be restated as follows. IGHER TORSION CLASSES 19 G - n -tors( M ) T ( − ) / / nP • ( − ) (cid:15) (cid:15) G -tors(mod- C ) P • ( − ) (cid:15) (cid:15) n -tors( P • ( M )) T ( − ) / / tors(mod- C /G )Suppose that δ = { U s : s ∈ [0 , } is a chain of G -equivariant n -torsion classes in M ⊂ mod- C .Then Theorem 6.1 implies that P • ( δ ) := { P • ( U s ) : s ∈ [0 , } is a chain of n -torsion classes in P • ( M ) ⊂ mod- C /G .Now, Theorem 4.6 implies that for every non-zero object M ∈ M , the chain of G -equivariant n -torsion classes δ induces an n -Harder-Narasimhan filtration, while P • ( δ ) induces an n -Harder-Narasimhan filtration of P • ( M ). In the following result we compare both filtrations. Proposition 6.3.
Let C be a locally bounded Krull-Schmidt k -category with an admissible actionof a group G on C inducing an admissible action on mod - C . Let M be a G -equivariant n -cluster-tilting subcategory of mod - C such that P • ( M ) is functorially finite in mod - C /G . Let δ = { U s : s ∈ [0 , } be a chain of G -equivariant n -torsion classes in M and M be a non-zeroobject of M . Then a filtration M ( M ( · · · ( M r − ( M r = M is the n -Harder-Narasimhan filtration of M with respect to δ in M if and only if P • ( M ) ( P • ( M ) ( · · · ( P • ( M r − ) ( P • ( M r ) = P • ( M ) is the n -Harder-Narasimhan filtration of P • ( M ) with respect to the chain of n -torsion classes P • ( δ ) in P • ( M ) .Proof. Clearly, the union and intersection of G -equivariant sets is itself G -equivariant. Thisfact together with Proposition 4.3 imply that S r>s U r and T ts U r ) and P • ( T ts U r ) = S r>s P • ( U r ) and P • ( T ts U r for all s < r ≤
1. Then P • ( U r ) ⊂ P • ( S r>s U r ) for all s < r ≤ S r>s P • ( U r ) ⊂ P • ( S r>s U r ). For the reverse inclusion, let X ∈ P • ( S r>s U r ). Then X = P • ( Y ) for some Y ∈ S r>s U r . This implies the existence of a r ∈ ( s,
1] such that Y ∈ U r .Thus X = P • ( Y ) ∈ P • ( U r ) ⊂ S r>s U r and our claim follows.Let M be a non-zero object of M and let0 = M ( M ( · · · ( M r − ( M r = M be the n -Harder-Narasimhan filtration of M with respect to δ in M , it follows from Definition 4.4that M r − is the n -torsion object of M with respect to the n -torsion class S r>s r U r , where s r = sup { t ∈ [0 ,
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