Local Coherence of Hearts Associated with Thomason filtrations
aa r X i v : . [ m a t h . R T ] J a n LOCAL COHERENCE OF HEARTS ASSOCIATED WITHTHOMASON FILTRATIONS
LORENZO MARTINI AND CARLOS E. PARRA
Abstract.
Prompted by [SˇS20], in which it is proved that the heart of acompactly generated t-structure in a triangulated category with coproduct isa locally finitely presented Grothendieck category, and inspired by [Hrb18], westudy the local coherence of the hearts associated with Thomason filtrationsof the prime spectrum of a commutative ring, achieving a useful recursivecharacterisation in case of finite length filtrations. Low length cases involvehereditary torsion classes of finite type of the ring, and even their Happel–Reiten–Smalø hearts; in these cases, the relevant characterisations are givenby few module-theoretic conditions.
Introduction
The main way to study an arbitrary abelian category is to provide good enoughcategorical correspondences to a category of modules over an arbitrary ring or, ifthis is not manageable, to define directly on the abelian category some homolog-ical properties that generalise the corresponding module-theoretic ones. Possibly,categories of modules over an associative ring are the “nicest” abelian categoriesone can work with, since they are Grothendieck categories with a finitely generatedprojective generator and carrying an additional finiteness condition, namely theyare locally finitely presented. However, there are other fundamental homologicalconditions that are not shared by all the module categories. In this sense, we areinterested in providing necessary and sufficient conditions for certain locally finitelypresented Grothendieck categories to be locally coherent. Such finiteness conditionif formulated just by miming the behaviour of the category of modules over a co-herent ring, namely by asking for the finitely presented objects to form an abeliancategory. Locally coherent Grothendieck categories constitute the abelian settingin which it is possible to perform a fruitful purity theory strictly related to the gen-eral purity theory for triangulated categories (see [Kra00]; in turn, these theoriesare a generalisation of the classical one for modules). Purity is a central topic inrepresentation theory and it is in fact interwoven with other powerful homologi-cal and categorical tools, such as localisation, tilting theory, cotorsion theory, andderivators (see [AHMV17, Lak18, SˇSV17, SˇS20]).The Grothendieck categories we want to examine come from the world of trian-gulated categories, more precisely they are the hearts of certain t-structures (theheart of any t-structure is an abelian category, see [BBD82]). In the literature,the hearts of two families of t-structures have been intensively studied in the sensementioned above, namely the Happel–Reiten–Smalø t-structures and the compactlygenerated ones.
Mathematics Subject Classification.
Key words and phrases. torsion pairs, TTF triples, t-structures, compactly generated,Thomason filtrations, heart, Happel–Reiten–Smalø, locally finitely presented, locally coherent,Grothendieck categories.The first named author is supported by a grant from University of Trento.The second named author is supported by grant CONICYT/FONDECYT/REGULAR/1200090.
HRS t-structures were introduced in [HRS96] and they are defined in the de-rived category of an abelian category by means of a torsion pair of this latter.Many authors (see e.g. [CGM07, CMT11, PS15, PS16, SˇSV17]) have investigatedon the module theoretic properties of their hearts, in fact establishing necessary andsufficient conditions for these latter to be equivalent to module categories. Whenthe underlying abelian category is Grothendieck, then the Grothendieck conditionfor the heart has been completely characterised in [PS16a]: it occurs if and onlyif the torsion pair is of finite type. On the other hand, crucial results concerningthe finiteness conditions have been achieved e.g. in [Sao17, PSV19]. An exhaustivesurvey devoted to the study of HRS hearts and related topics is [PS20].Compactly generated t-structures are defined in any triangulated category withcoproducts. Very recent works [SˇSV17, Bon19, SˇS20] show that their hearts arelocally finitely presented Grothendieck categories; thus, it is natural to ask whetherand when they are locally coherent.In the present paper we concentrate on the derived category of a commutativering; its compactly generated t-structures have been classified formerly in [AJS10]when the ring is noetherian, then in [Hrb18] without this latter condition, by meansof the Thomason filtrations of the Zariski spectrum of the ring. In details, we studythe local coherence of the hearts associated with arbitrary Thomason filtrations,achieving a characterisation for filtrations of finite length (Theorem 6.11). Suchresult is in turn a very special case of a more general characterisation, i.e. thelocal coherence of a Grothendieck category endowed with a TTF triple of finitetype (Theorem 2.1); furthermore, it is formulated recursively and constructively,meaning that it provides an algorithm to build a Thomason filtration of finite lengthwith a locally coherent Grothendieck heart out of filtrations of finite fewer lengthwith locally coherent hearts.The paper is organised as follows.Section 1 contains the notations and all the preliminary definitions and resultswe need, concerning abelian and Grothendieck categories (in particular their tor-sion theories), triangulated categories (in particular their t-structures), and a briefsurvey on prederivators.In section 2, Theorem 2.1 shows a general criterion for the local coherence of aGrothendieck category equipped with a TTF theory of finite type; this is the resultwe want to specialise in the body of the paper.Section 3 contains the definition of Thomason filtration, its t-structure, and someresults concerning both its heart and certain subcategories involved in the study ofthe finitely presented complexes of the heart, which will be useful in the sequel.Sections 4, 5 and 6 are the central part of the paper, since they provide the ma-chinery to generalise Theorem 2.1, hence to achieve the local coherence of the heartof filtrations of finite length. In more details, Section 4 is devoted to the boundedabove Thomason filtrations. Such filtrations contain those of finite length, andas particular case we deduce two crucial results: we can realise both a hereditarytorsion class of finite type and the HRS heart it gives rise as the heart of a suit-able Thomason filtration of finite length (respectively 0 and 1). We completelycharacterise the local coherence of the former category in Theorem 4.7; moreover,in Corollary 4.8 we extend from the noetherian to the coherent ones the class ofcommutative rings whose torsion classes of finite type are always locally coherentGrothendieck categories.Section 5 is devoted to detecting within the heart of an arbitrary Thomasonfiltration a TTF triple of finite type, in order to let Theorem 2.1 apply. By Propo-sition 5.4, such TTF classes are indeed the hearts of bounded below Thomason
OCAL COHERENCE OF THOMASON HEARTS 3 filtrations naturally associated with the given one; in particular, these hearts areall locally coherent Grothendieck categories in case the given heart is so.Section 6 is the core of the paper and contains the main results concerning theThomason filtrations of finite length. By the results of the previous section, suchlength is taken in the vein of providing a recursive argument for the characterisationof the local coherence. Theorem 6.11 builds the local coherence in such recursiveway, by means of five conditions. In this vein, the conditions are the most eligibleones, in the sense that for the crucial cases of length 0 , , Preliminaries
We will refer to [Pop73, Ste75] for the basics on abelian categories, and to [Nee01,Mil] for what does concern triangulated categories. Throughout the present paper, R will denote a commutative ring, while R -Mod, C h ( R ), K ( R ) and D ( R ) will denote,respectively, its module category, its category of cochain complexes, its homotopycategory, and its unbounded derived category. Given a preadditive category A anda set S of objects of A , we will denote, for short, S ⊥ = { M ∈ A | Hom A ( S, M ) = 0 ∀ S ∈ S} = Ker Hom A ( S , − ) , ⊥ S = { M ∈ A | Hom A ( M, S ) = 0 ∀ S ∈ S} = Ker Hom A ( − , S ) . Moreover, in case A is an abelian category with coproducts, Gen S (gen S ) andAdd S (add S ) will denote, respectively, the full subcategories formed by the objectsadmitting an epimorphism, resp. a split epimorphism, originating in a coproduct of(finitely many) objects of S . If Gen S = A , then S is said to be a set of generators for A .1.1. Abelian categories.
Let A be an additive category and let I be a smallcategory (i.e. its objects form a set). A functor F : I → A will be also called a diagram of shape I on A and denoted by ( F i ) i ∈ I , where F i = F ( i ) for all i ∈ I .The category having such diagrams as objects and the natural transformationsbetween them as morphisms will be denoted by A I . When any I -shaped diagramhas colimit (resp. limit) in A , then A is said to admit I -colimits (resp. I -limits). Inthis case, we have two adjoint pairscolim i ∈ I : A I −−→←−− A : ∆ I and ∆ I : A −−→←−− A I : lim i ∈ I where ∆ I is the constant functor. If A admits I -colimits (resp. I -limits) for everysmall category I , then A is said to be cocomplete (resp. complete ). A very importantcase occurs when I is a directed poset, hence regarded as a small category in theusual way: the I -shaped diagrams are called direct systems , while the I op -shapeddiagrams are called inverse systems , so that the corresponding I -colimit and I -limit functors are then called respectively the direct limit and the inverse limit ,and denoted by lim −→ i ∈ I F i and lim ←− i ∈ I op F i for every F = ( F i ) i ∈ I ∈ A I .Following the celebrated Tˆohoku paper [Gro57] by Grothendieck, an abeliancategory A is said to be L. MARTINI AND C. E. PARRA
AB-3 if it admits coproducts or, equivalently, if it cocomplete;AB-4 if it is AB-3 and the coproduct functor ` i ∈ I : A I → A is exact for every smallcategory I ;AB-5 if it is AB-3 and the direct colimit functor lim −→ i ∈ I : A I → A is exact for everydirected poset I .The most important and studied AB-5 abelian categories are the Grothendieckcategories , namely those having a set of generators (equivalently, a generator).For instance, it is well-known that Grothendieck categories are also complete, pro-vide injective envelopes for their objects and have an injective cogenerator ([Ste75,Corollary X.4.4], [Gro57]).We want to study certain Grothendieck categories having some additional finite-ness conditions which we now recall explicitly in a more general context (cf. [CB94,Kra97]). Let A be an additive category with direct limits (that is, A admits I -colimits for every directed poset I ); then • an object A ∈ A is called finitely generated if for every direct system ofmonomorphisms ( M i ) i ∈ I of A (i.e. each connection map M i → M j is amonomorphism), the natural group homomorphismlim −→ i ∈ I Hom A ( A, M i ) −→ Hom A ( A, lim −→ i ∈ I M i )is bijective. • An object B ∈ A is called finitely presented if for every direct system( M i ) i ∈ I of A , the natural group homomorphismlim −→ i ∈ I Hom A ( B, M i ) −→ Hom A ( B, lim −→ i ∈ I M i )is bijective; that is, the functor Hom A ( B, − ) : A → Ab commutes withdirect limits .These definitions in turn provide the aforementioned finiteness conditions for anadditive category with direct limits: A is called • locally finitely presented if the full subcategory fp( A ) of the finitely pre-sented objects is skeletally small and A = lim −→ fp( A ), meaning that eachobject of A is isomorphic to a direct limit of a direct system of A ; • locally coherent if it is locally finitely presented and fp( A ) is abelian (when A has kernels and cokernels, this latter condition is equivalent to fp( A )being closed under taking kernels).When A is an abelian category, then it is locally finitely presented if and only if itis a Grothendieck category with a generating set of finitely presented objects (see[Kra97]). In this case, the finitely generated objects are precisely the quotient ofthe finitely presented ones.Throughout this paper, we will set fg( R -Mod) = gen R and fp( R -Mod) = R -mod.1.2. Torsion theories.
Let A be an abelian category. A torsion pair in A is apair ( T , F ) of full subcategories such that Hom A ( T , F ) = 0 and for every M ∈ A there exists a (functorial) short exact sequence 0 → X → M → Y → X ∈ T and Y ∈ F . Generally, such approximating exact sequence will be expressedby means of two functors, say x and y , involved in the following adjoint pairs T ֒ −→←−− x A y −−→←− ֓ F . In view of this display, the induced endofunctors of A , denoted by x and y again,are called respectively the torsion radical and the torsion coradical . T is calledthe torsion class of the torsion pair and its objects are the torsion objects of A OCAL COHERENCE OF THOMASON HEARTS 5 (w.r.t. the torsion pair), whereas F is the torsionfree class and its objects are the torsionfree objects of A .When C is a Grothendieck category, then two full subcategories T and F forma torsion pair ( T , F ) in C if and only if T ⊥ = F and T = ⊥ F . Yet, a non-emptyclass T of objects of C is a torsion class iff it is closed under quotient objects, exten-sions, and coproducts; dually, a non-empty class F of objects of C is a torsionfreeclass iff it is closed under subobjects, extensions and products.A torsion pair ( T , F ) in C is said to be: • hereditary if T is closed under taking subobjects (equivalently, if F is closedunder taking injective envelopes); • stable if it is hereditary and also T is closed under injective envelopes; • of finite type if F is closed under taking direct limits.Moreover, we say that the torsion pair ( T , F ) restricts to fp( C ) if for every B ∈ fp( C )we have x ( B ) , y ( B ) ∈ fp( C ). Remark 1.1.
Let C be a Grothendieck category.(1) By the properties of a torsion pair, it is clear that any torsion class of C isan additive category with direct limits (thus, it makes sense to ask whetheror when it fulfils some finiteness condition), moreover it has cokernels andkernels, the former computed as in C , the latter by taking the torsion radicalof the kernels of C .However, a torsion class needs not to be an abelian category. For in-stance, given a tilting (non projective) object V ∈ C (see e.g. [PS20]),the induced tilting torsion class Gen V = Ker Ext C ( V, − ) is not abelian.Indeed, since C provides injective envelopes, then the tilting class is cogen-erating, meaning that for any M ∈ C there exists a short exact sequence0 → M → T q → T ′ → T, T ′ ∈ Gen V . Now, we can choose0 = M ∈ Ker Hom C ( V, − ), and if Gen V would be abelian, then q would bean isomorphism in Gen V , in particular a split epimorphism in C , contra-diction.(2) When ( T , F ) is a hereditary torsion pair of C such that T = Gen M forsome object M ∈ T (e.g. when the torsion pair is of finite type, see [PS15,Lemma 4.6]), then T is a Grothendieck category with the same exact struc-ture of C .We are particularly interested in (hereditary) torsion pairs of finite type (mostlyby Theorem 2.1).In the case of R -Mod (see [Ste75, Theorem VI.5.1] and [GP08, Appendix]), ahereditary torsion pair (of finite type) corresponds bijectively to a Gabriel filters (of finite type) of R ; that is, to a set G of ideals of R fulfilling the following axioms:(i) for any I, J ∈ G , I ∩ J ∈ G ;(ii) if I ∈ G and J is an ideal such that J ⊇ I , then J ∈ G ;(iii) if I ∈ G and r ∈ R , then ( I : r ) = { γ ∈ R | γr ∈ I } ∈ G ;(iv) for any ideal J , if there exists an ideal I ∈ G such that ( J : a ) ∈ G for all a ∈ I , then J ∈ G .Recall that a Gabriel filter G is of finite type if it has a basis of finitely generatedideals; that is, if every ideal in G contains a finitely generated ideal in G . Thebijective correspondence between hereditary torsion pairs of finite type ( T , F ) in R -Mod and Gabriel filters of finite type G of R is given by the mutually inverseassignments T 7−→ G T = { I ≤ R | R/I ∈ T } and
L. MARTINI AND C. E. PARRA
G 7−→ T G = { M ∈ R -Mod | Ann R ( x ) ∈ G ∀ x ∈ M } . Another particular case of torsion theories in a Grothendieck category C we areinterested in is given by the TTF triples , namely triples ( E , T , F ) such that both( E , T ) and ( T , F ) are torsion pairs of C (see [BR07] for a detailed reference). Themiddle term T is called TTF class of the triple; by the closure properties of torsionand torsionfree classes, it follows that a full subcategory T of C is a TTF classif and only if T is closed under subobjects, quotients, coproducts, products andextensions. In this case, since the right constituent ( T , F ) is hereditary, as well asthe left constituent ( E , T ) is of finite type, a TTF triple is hereditary resp. of finitetype in case its left, resp. right, constituent is so.TTF triples over a commutative ring R are well-understood (see [Ste75, VI.8]):they are in bijection with idempotent ideals of R , and T is a TTF class in R -Mod if,and only if, there is an idempotent ideal J ≤ R such that T consists of the modulesannihilated by J , i.e. T = R/J -Mod, so that in the left constituent ( ⊥ T , T )of the triple the torsion modules are precisely the J -divisible modules, i.e. those M ∈ R -Mod such that JM = M .1.3. t-structures. The corresponding notion of torsion pair for a triangulated cat-egory is the one of t-structure, introduced in the celebrated work [BBD82], to whichwe will refer to. t-structures provide a useful approximation theory in their ambienttriangulated category, as well as torsion pairs do in their ambient abelian category.The most powerful feature of such approximation theory is that each t-structuremakes a “homological algebra” avalaible within its triangulated category, and therelevant cohomologies belong to a suitable abelian category naturally associatedwith the t-structure.Let ( D , ( − )[1]) be a triangulated category. A t-structure in D is a pair ( U , V )of full subcategories closed under direct summands and satisfying the followingconditions:(i) U [1] ⊆ U ; that is, U is closed under positive shiftings;(ii) Hom D ( U , V [ − M ∈ D , there exists an exact triangle U → M → V + → with U ∈ U and V ∈ V [ − M U and M V provided by axiom (iii) well-define the so-called truncation functors τ ≤U and τ > U , which are adjoint to the relevant inclusions: U ֒ −→←−− D : τ ≤U and τ > U : D −−→←− ֓ V [ − . By the axioms of a triangulated category, it is readily seen that any t-structure( U , V ) can be expressed by means of the first component U via the equality V = U ⊥ [1]. U is called the aisle and U ⊥ is the coaisle of the t-structure. We recall that( U , V ) is a t-structure if, and only if, ( U [ n ] , V [ n ]) is a t-structure for every n ∈ Z .Let us recall the well-known and most important results from [BBD82] on at-structure ( U , V ) we are going to use in the sequel. The main one is that theintersection H := U ∩ V turns out to be an abelian category, called the heart of thet-structure. The “homological algebra” we referred to is provided by the naturallyisomorphic cohomological functors
D → H defined as H H := τ > U [1] ◦ τ ≤U ∼ = τ ≤U ◦ τ > U [1] . The short exact sequences of H are precisely the exact triangles of D whose verticesbelong to H . Consequently, we have the following crucial correspondences, valid OCAL COHERENCE OF THOMASON HEARTS 7 for all
M, N ∈ H : Ext H ( M, N ) ∼ = −→ Hom D ( M, N [1])Ext H ( M, N ) ֒ −→ Hom D ( M, N [2]) . Example 1.2.
Let A be an abelian category. Denote simply by D ≤ = { M ∈ D ( A ) | H k ( M ) = 0 ∀ k > } , D ≥ = { M ∈ D ( A ) | H k ( M ) = 0 ∀ k < } , the subcategories of bounded below resp. above complexes over A . Then ( D ≤ , D ≥ )is a t-structure of D ( A ), called the standard t-structure , and its heart is equivalentto A . Example 1.3.
Let C be a Grothendieck category and ( T , F ) be a torsion pairin C . The Happel-Reiten-Smalø t-structure associated with ( T , F ) (introduced in[HRS96]) is the t-structure of the bounded derived category D ( C ), whose membersare defined respectively as U ( T , F ) = { M ∈ D ≤ | H ( M ) ∈ T } and V ( T , F ) = { M ∈ D ≥− | H − ( M ) ∈ F} . Therefore, the associated
HRS heart H ( T , F ) consists of the cochain complexes0 → Y d → X → C concentrated in degrees − d ∈ F andCoker d ∈ T . Yet, such heart admits ( F [1] , T [0]) as torsion pair. We recall that in[PS15, PS16a] it is proved that H ( T , F ) is a Grothendieck category if and only if( T , F ) is of finite type.Let us recall some basic informations concerning the abelian structure of theheart H of a t-structure ( U , V ) in D . Let us start by computing kernels, imagesand cokernels. Given a morphism f : M → N in H , embed it in an exact triangleof D by means of a cone Z . Consider the approximation of Z [ −
1] within ( U , V ),then the following commutative diagram provided by the octahedral axiom τ ≤U ( Z [ − / / Z [ − / / (cid:15) (cid:15) τ > U ( Z [ − + / / (cid:15) (cid:15) τ ≤U ( Z [ − / / (cid:15) (cid:15) M / / f (cid:15) (cid:15) W + / / (cid:15) (cid:15) / / N (cid:15) (cid:15) N (cid:15) (cid:15) Z / / τ > U ( Z [ − W is a cone for the morphism τ ≤U ( Z [ − → M . We have:Ker H ( f ) := τ ≤U ( Z [ − H ( f ) := W Coker H ( f ) := τ > U ( Z [ − H H ( Z ) . If the ambient triangulated category D admits coproducts, say them denoted bythe symbol ` , then the heart H has coproducts as well, generally distinct to those L. MARTINI AND C. E. PARRA of D ; indeed, given a family ( M i ) i ∈ I of objects of H , it is not difficult to see thatthe objects M i ∈ I M i := H H (cid:16)a i ∈ I M i (cid:17) is the coproduct of the family in H . It is now clear how to compute direct limits,which we will denote by lim −→ H when necessary. The dual notion of products andinverse limits are also available in case D admits products. Remark 1.4.
We will deal with (hearts of) certain compactly generated t-structuresin the derived category of a commutative ring R . In relation to our instance ofproviding finiteness conditions (in particular, the local coherence) on a given abeliancategory, the interest in compactly generated t-structures of D ( R ) is motivated bythe recent paper [SˇS20], in which it is proved that the heart of a compactly generatedt-structure in a triangulated category with coproducts is a locally finitely presentedGrothendieck category (see Theorem 8.20 therein).We recall that D ( R ) admits coproducts (and products) and it is compactly gen-erated , meaning that there exists a set S of complexes such that, for every S ∈ S ,(i) the functor Hom D ( R ) ( S, − ) : D ( R ) → Ab commutes with coproducts;(ii) given M ∈ D ( R ), M = 0 if and only if Hom D ( R ) ( S [ k ] , M ) = 0 for all k ∈ Z .In other words, S is a set of compact generators of D ( R ). The full subcategoryof D ( R ) formed by the compact objects , i.e. those satisying (i), will be denoted by D c ( R ). A t-structure ( U , V ) of D ( R ) is compactly generated if there is a set S ofcompact generators for the aisle or, equivalently, V = T k ≥ Ker Hom D ( R ) ( S [ k ] , − ).1.4. Derivators.
We briefly recall some terminology and basic facts concerningGrothendieck prederivators, more precisely the strong and stable derivators, fol-lowing [Gro13, ˇSto14, Lak18, SˇSV17]. The aim is to remind that to any suchderivator it is naturally associated a triangulated category, called its base, in whichhomotopy limits and colimits are defined; furthermore, there is a strong and stablederivator whose base is equivalent to the derived category of a fixed ring, so thatthe homotopy colimits of this latter may be managed (and understood) in the baseinstead.Let
Cat be the 2-category of all categories, cat be the 2-category of smallcategories, and cat op be the 2-category obtained by cat reversing the arrows ofthe 1-cells and letting the 2-cells unchanged. A prederivator is a strict 2-functor D : cat op → Cat . Let be the discrete small category consisting of one object; D ( ) is called the base of the prederivator D . Since is a terminal object of cat ,for every small category I ∈ cat there is a unique functor pt I : I → . The ho-motopy colimit (resp. limit ) functor is the left (resp. right) adjoint to the functor D (pt I ) : D ( ) → D ( I ): hocolim i ∈ I : D ( I ) −−→←−− D ( ) : D (pt I )and D (pt I ) : D ( ) −−→←−− D ( I ) : holim i ∈ I . In general, a prederivator needs not to admit homotopy (co)limits; in fact, deriva-tors are axiomatised in order to guarantee (also) their existence for all I ∈ cat .Moreover, the axioms of strong and stable derivators provide the conditions in or-der to equip each of their images with a triangulated structure. More precisely, theintroduction of these latter derivators is motivated since, given a derivator D and OCAL COHERENCE OF THOMASON HEARTS 9 any small category I , the shifted derivator defined by setting D I : cat op −→ Cat J D ( I × J ) , has the base D I ( ) equivalent to the category D ( I ) of coherent diagrams of shape I ,though this is not true in general for the category D ( ) I of the incoherent diagram of shape I . In other words, not every incoherent diagram of shape I lifts to a coherent diagram of shape I , yet the diagram functor associated with D ,diag I : D ( I ) −→ D ( ) I X 7−→ ( i
7→ X i ) , is far from being an equivalence of categories, unless a strong and stable derivator isinvolved. If this is the case, each category D ( I ) carries a triangulated structure suchthat the homotopy (co)limits are triangulated functors (see [Gro13, Theorem 4.16,Corollary 4.19]); moreover, by [SˇSV17, Theorem A], as soon as a t-structure ( U , V )with heart H is considered in the base D ( ), then U I = {X ∈ D ( I ) | X i ∈ U , ∀ i ∈ I } and V I = {Y ∈ D ( I ) | Y i ∈ V , ∀ i ∈ I } form a t-structure with heart H I in the category D ( I ), and the diagram functorinduces an equivalence of abelian categories H I ∼ = H I . Example 1.5.
Let R be a ring. For any small category I ∈ cat there is a naturalequivalence of Grothendieck categories C h ( R -Mod I ) ∼ = C h ( R -Mod) I , which extendsto the relevant derived categories: D ( R -Mod I ) ∼ = D ( R ) I . The assignment D R : cat op −→ Cat I −→ D ( R -Mod I )( u : J → I ) −→ (cid:0) D ( R -Mod I ) u ∗ → D ( R -Mod J ) (cid:1) , where D R ( u ) = u ∗ is induced by the exact functor u : R -Mod I → R -Mod J , well-defines a strong and stable derivator, called the standard derivator of R . The base D R ( ) is then equivalent to the derived category of the ring. In particular, thehomotopy (co)limits of D R are naturally isomorphic to the total right (resp. left)derived functors of the ordinary (co)limits of D ( R ): for every I ∈ cat and X =( X i ) i ∈ I in D (Mod- R I ),holim i ∈ I X i = R lim i ∈ I X i and hocolim i ∈ I X i = L colim i ∈ I X i . We are mostly interested in the case of filtered homotopy colimits, namely when I is a directed poset. In this case, the ordinary colimit functor C h ( R ) I → C h ( R ) isexact, hence for any X as above we have a natural isomorphismholim −−−→ i ∈ I X i ∼ = lim −→ i ∈ I X i . A Criterion for the Local Coherence
We prove a general result characterising the local coherence of a Grothendieckcategories equipped with a TTF triple of finite type. The body of the presentpaper will be prominently focused in specialising this result to the hearts of certaint-structures, as announced in Remark 1.4 (cf. Remark 3.5(1)).
Theorem 2.1.
Let C be a Grothendieck category equipped with a TTF triple offinite type ( E , T , F ) . Consider the following three statements: (a) C is locally coherent; (b) The following conditions are satisfied: (i) E and T are locally coherent; (ii) For every P ∈ fp( E ) , the functor Ext C ( P, − ) commutes with directlimits of direct systems of T ; (iii) For every Q ∈ fp( T ) , the functor Ext C ( Q, − ) commutes with directlimits of direct systems of E . (c) The conditions (i) , (ii) of part (b) hold true, and moreover (iii)’ The torsion pair ( E , T ) restricts to fp( C ) .Then “(a) ⇔ (b) ⇒ (c)” , and the statements are all equivalent in case C is locallyfinitely presented.Proof. We will denote by E ֒ −→←−− x C y −−→←− ֓ T the adjunction provided by the left constituent of the TTF triple. Notice that thehypotheses on the TTF triple imply, by [PSV19, Lemma 1.11], that fp( E ) , fp( T ) ⊆ fp( C ). In turn, the local coherence of C always implies conditions (ii), (iii) and (iii)’;to see this latter, for every B ∈ fp( C ) we have x ( B ) ∈ fp( C ), since such object occursas the kernel of the epimorphism B → y ( B ) in fp( C ).Let us prove “(a) ⇒ (b)”. By what we just observed, we only have to checkcondition (i). T is a locally coherent Grothendieck category thanks to [Her97,Theorem 2.16] and [Kra97, Theorem 2.6]. Now, let us show that E is locally finitelypresented. Let X ∈ E and ( B i ) i ∈ I be a direct system in fp( C ) such that X =lim −→ i ∈ I B i . Since ( E , T ) is of finite type, we have X = x ( X ) = lim −→ i ∈ I x ( B i ), thus E is locally finitely presented since each x ( B i ) belongs to fp( E ). It remains to showthat E is locally coherent. By the previous part, it suffices to check that the kernelin E of an epimorphism f : P → P ′ in fp( E ) is finitely presented as well. Noticethat f is an epimorphism also in C , therefore Ker f ∈ fp( C ) by the local coherencehypothesis. Our claim then follows since Ker E ( f ) = x (Ker f ) and ( E , T ) restrictsto fp( C ).Now, let us now show that if C is locally finitely presented, then “(c) ⇒ (a)”.We have to prove that the kernel of any epimorphism f : B → B ′ in fp( C ) is finitelypresented as well. Since the torsion pair ( E , T ) restricts to fp( C ) by (iii)’, thefollowing commutative diagram with exact rows0 / / x ( B ) / / p (cid:15) (cid:15) B f (cid:15) (cid:15) (cid:15) (cid:15) / / y ( B ) q (cid:15) (cid:15) (cid:15) (cid:15) / / / / x ( B ′ ) / / B ′ / / y ( B ′ ) / / C ). Besides q , also p is an epimorphism, being Coker p ∈ E ∩ T = 0.Therefore, p and q are epimorphisms in fp( E ) and fp( T ) respectively, hence byhypothesis (i) we obtain that Ker E ( p ) and Ker T ( q ) = Ker q are finitely presentedobjects of C . Thus, once we prove that Ker p ∈ fp( C ), we infer that Ker f isfinitely presented by extension-closure again, applied on the short exact sequence0 → Ker p → Ker f → Ker q → −→ Ker E ( p ) −→ Ker p −→ y (Ker p ) −→ OCAL COHERENCE OF THOMASON HEARTS 11 of the relevant kernel within ( E , T ), and let us prove that the third term is finitelypresented in C . We have the following pushout diagram:Ker E ( p ) (cid:15) (cid:15) (cid:15) (cid:15) Ker E ( p ) (cid:15) (cid:15) (cid:15) (cid:15) Ker p (cid:15) (cid:15) (cid:15) (cid:15) / / / / P . O . x ( B ) (cid:15) (cid:15) (cid:15) (cid:15) p / / / / x ( B ′ ) y (Ker p ) / / / / C / / / / x ( B ′ )whose second column tells us that the pushout C is finitely presented as well.Eventually, given a direct system ( M i ) i ∈ I of objects of T , applying the functorslim −→ i ∈ I Ext r C ( − , M i ) and Ext r C ( − , lim −→ i ∈ I M i ) ( r ∈ N ∪ { } )on the second exact row, thanks to hypothesis (ii), by the Five Lemma we getthat Hom C ( y (Ker p ) , − ) preserves direct limits of T ; that is, y (Ker p ) is a finitelypresented object of T , hence of C , as desired.In order to conclude the proof, we now show that condition (b) implies that C is locally finitely presented and the condition (c). For the first claim we will followthe pattern of the proof of [PSV19, Lemma 1.12]. Let M be an arbitrary objectof C and consider its approximation 0 → x ( M ) → M → y ( M ) → E , T ).Since T is locally finitely presented by (i), there exists a direct system ( Q i ) i ∈ I infp( T ) such that y ( M ) = lim −→ i ∈ I Q i . We have the pullback diagram0 / / x ( M ) / / M i (cid:15) (cid:15) / / P . B . Q i (cid:15) (cid:15) / / / / x ( M ) / / M / / y ( M ) / / M i ’s form a direct system in C whose direct limit is M . Once we showthat M i ∈ Gen[fp( C )] for all i ∈ I , then we conclude our first claim (see the proofof [PSV19, Lemma 1.12]). Consider the extension ξ i : 0 → x ( M ) → M i → Q i → E is locally finitely presented, there existsa direct system ( P λ ) λ ∈ Λ ⊆ fp( E ) such that x ( M ) = lim −→ λ ∈ Λ P λ . By hypothesis (iii), weobtain ξ i ∈ Ext C ( Q i , lim −→ λ ∈ Λ P λ ) ∼ = lim −→ λ ∈ Λ Ext C ( Q i , P λ ) , i.e., by definition of Yoneda ext-group, there is an index γ ∈ Λ such that ξ i factorsas the pushout diagram (see again [PSV19] for details)0 / / P γ / / (cid:15) (cid:15) P . O . N γ (cid:15) (cid:15) / / Q i / / / / x ( M ) / / M i / / Q i / / N γ is a finitely presented object of C by [PSV19, Corollary 1.4]. Moreover,it is M i = lim −→ λ ≥ γ N λ so that our first claim is proved. Let us check that condition (iii)’ holds true. Let B ∈ fp( C ) and let us consider its approximation 0 → x ( B ) → B → y ( B ) → E , T ). We only have to show that x ( B ) ∈ fp( E ) ⊆ fp( C ), since y ( B ) ∈ fp( T ) ⊆ fp( C ). The approximation yields the following long exact sequence of covariantfunctors:0 → Hom C ( y ( B ) , − ) → Hom C ( B, − ) → Hom C ( x ( B ) , − ) − · · ·· · · → Ext C ( y ( B ) , − ) → Ext C ( B, − )which, when restricted to E , by hypothesis (iii), [PSV19, Lemma 1.3] and theFive Lemma, gives that x ( B ) ∈ fp( E ). (cid:3) Thomason Filtrations and Hearts
Let R be a commutative ring and Spec R be its prime spectrum i.e. the set ofall the prime idels of the ring. Let us recall that for every p ∈ Spec R one canconsider the localisation φ : R → R p of R at p and set M p = M ⊗ R R p for every M ∈ R -Mod. This assignment well-defines the so-called extension of scalars functor φ ∗ = − ⊗ R R p , which is left adjoint to the scalar restriction φ ∗ : R p -Mod → R -Modinduced by φ . Given M ∈ R -Mod, define its support by settingSupp M = { p ∈ Spec R | M ⊗ R R p = 0 } . Yet, recall that Spec R is a topological space whose closed subsets are of the form V ( J ) = { p ∈ Spec R | p ⊇ J } = Supp R/J for all ideals J ≤ R . Definition 3.1.
A subset X of Spec R is said to be Thomason if there exists afamily B X of finitely generated ideals of R such that X = S J ∈B X V ( J ).Notice that Spec R is itself Thomason, for one chooses B X as the family ofprincipal ideals generated by the elements of R , each of which is contained in somemaximal ideal.By [GP08, Theorem 2.2], a Thomason subset X corresponds bijectively to ahereditary torsion pair of finite type ( T X , F X ) in R -Mod, where T X = { M ∈ R -Mod | Supp M ⊆ X } , thus in turn it corresponds bijectively to a Gabriel filter of finite type on R definedby G X = { J ≤ R | V ( J ) ⊆ X } . Proposition 3.2.
Let X , T X and G X be as above. Then (i) T X is a Grothendieck category, and fp( T X ) = T X ∩ R -mod ; (ii) T X = Gen( R/J | J ∈ G X ∩ gen R ) .Proof. (i) It is well-known that T X is a Grothendieck category (we deduce it in Propo-sition 4.6). Let us show the equality in the second part of the statement. Theinclusion “ ⊇ ” is clear, while “ ⊆ ” follows by [PSV19, Lemma 1.11] since ( T X , F X )is a torsion pair of finite type.(ii) The inclusion “ ⊇ ” is clear from the properties of a torsion class. Conversely,since T X is a hereditary torsion class of R -Mod, every torsion object is the directlimit of a direct system in T X ∩ gen R , hence it suffices to show that each module M in the latter category is the homomorphic image of the direct sum of some R/J ’s, where each J is a finitely generated ideal in G X . Since M is a finitelygenerated module, then Supp M = V (Ann R ( M )) (see e.g. [Lam99, Exercise 23,p. 58]). Therefore, V (Ann R ( M )) ⊆ X , and since G X is a Gabriel filter of finitetype, Ann R ( M ) contains a finitely generated ideal J of the filter. This means JM = 0 i.e. M is a R/J -module, in fact finitely generated over
R/J as well, sothat there exists an epimorphism (
R/J ) n → M for some positive integer n . (cid:3) OCAL COHERENCE OF THOMASON HEARTS 13
Corollary 3.3.
Let X = S J ∈B X V ( J ) be a Thomason set, let G X be the associatedGabriel filter and set J X = G X ∩ gen R . Then X = S J ∈J X V ( J ) .Proof. The right-ward inclusion X ⊆ S J ∈J X V ( J ) is clear (notice that B X ⊆ J X ).Conversely, let p be a prime ideal containing some finitely generated ideal J in G X ,and let us prove that p contains an ideal in B X . The module R/p is a torsionby Proposition 3.2, whence Supp
R/p = V ( p ) ⊆ X , so we are done since clearly p ∈ V ( p ). (cid:3) Henceforth, we will always identify a Thomason subset X = S J ∈B X V ( J ) bysetting B X as the the family J X of all finitely generated ideals in the Gabriel filterassociated with X . Definition 3.4. A Thomason filtration of Spec R is a decreasing map Φ : ( Z , ≤ ) → (2 Spec R , ⊆ ) such that Φ ( n ) is a Thomason subset of Spec R for all n ∈ Z .A Thomason filtration Φ will be called: • bounded below if there exists k ∈ Z such that Φ ( n ) = Φ ( k ) for all n ≤ k ; • bounded above if there exists k ∈ Z such that Φ ( k + 1) = ∅ .In these cases, we say that Φ is bounded below k or bounded above k , respectively. • A Thomason filtration bounded both below and above will be called aThomason filtration of finite length . Let ℓ ∈ N ∪ { } ; a Thomason filtrationbounded below − ℓ and bounded above 0, such that Φ ( − ℓ + 1) = Φ ( − ℓ ), issaid to be of length ℓ .In [Hrb18] the author classifies all compactly generated t-structures in the derivedcategory of a commutative ring R , generalising the results in [AJS10] concerningthe case of a noetherian commutative ring. More precisely, [Hrb18, Theorem 5.1]exhibits a bijective correspondence between compactly generated t-structures in D ( R ) and Thomason filtrations of Spec R , given explicitly by the assignments Φ ( U Φ , U ⊥ Φ [1]) and ( U , V ) Φ U , where U Φ = { M ∈ D ( R ) | Supp H n ( M ) ⊆ Φ ( n ) , ∀ n ∈ Z } = { M ∈ D ( R ) | H n ( M ) ∈ T Φ ( n ) , ∀ n ∈ Z } , and Φ U ( n ) = [ M ∈ R -Mod M [ − n ] ∈U Supp M, for all n ∈ Z . Remark 3.5. (1) We want to study the local coherence of the hearts associated with Thoma-son filtrations of finite length. Our instance makes sense thanks to [SˇS20](see Remark 1.4), and in particular we shall use Theorem 2.1 once we detectsome TTF triples within such hearts. Nonetheless, as we shall see, crucialinformations and results in this sense will be achieved for more generalThomason filtrations, even arbitrary.(2) Henceforth, when a filtration Φ is fixed we will denote the t-structure andthe heart it gives rise respectively by ( U , V ) and H , i.e. omitting any sub-script referring to Φ , for it will not create confusion. Moreover, the torsionpair associated with each Thomason subset Φ ( n ) will be denoted just by ( T n , F n ); in turn, the relevant adjunctions to the inclusions in R -Mod willbe denoted by T n ֒ −→←−− x n R -Mod y n −−→←− ֓ F n . Corollary 3.6 (Lemma [PS17, Lemma 4.2(3)]) . Let Φ be any Thomason filtrationand let M be a complex in the associated heart H . If r is the least integer suchthat H r ( M ) = 0 , then H r ( M ) ∈ T r ∩ F r +1 ∩ Ker Ext R ( T r +2 , − ) . Proof.
We only need to check that H r ( M ) belongs to the last two classes of thedisplayed intersection. By hypothesis, H r ( M )[ − r ] ∼ = τ ≤ r ( M ), hence for every X ∈ T r +1 we obtainHom R ( X, H r ( M )) ∼ = Hom D ( R ) ( X [ − r ] , H r ( M )[ − r ]) ∼ = Hom D ( R ) ( X [ − r ] , τ ≤ r ( M )) . The latter group is zero since X [ − r ] ∈ U [1], hence its covariant hom functor appliedon exact triangle τ >r ( M )[ − → τ ≤ r ( M ) → M + → (given by the standard approxi-mation of M ) yields a zero exact sequence by the axioms of t-structure. Therefore,the least nonzero cohomology of M is an object of F r +1 .On the other hand, by Verdier’s thesis [Ver], for every X ∈ T r +2 we haveExt R ( X, H r ( M )) ∼ = Hom D ( R ) ( X [ − r ] , H r ( M )[ − r + 1]) , and the right-hand group is zero by the previous argument, i.e. by applying thehom functor of X [ − r ] ∈ U [2] on the rotation of the above triangle. (cid:3) Slightly diverting from [PS17], we fix the following notation: given a Thomasonfiltration Φ , for any k ∈ Z we set T F k := T k ∩ F k +1 T FT k := T k ∩ F k +1 ∩ Ker Ext R ( T k +2 , − ) . It is readily seen that
T F k is closed under subobjects and that T FT k is closedunder kernels; moreover, we will show in Remark 6.2(1) that the latter categoryhas direct limits, so it will make sense to consider the subcategory of its finitelypresented objects, which we will play a crucial role in the subsequent sections. Proposition 3.7.
Let Φ be a Thomason filtration of Spec R . Then the class HT F n := { M ∈ R -Mod | M [ − n ] ∈ H} is a subcategory of R -Mod closed under direct limits, for every n ∈ Z .Proof. Let ( M i ) i ∈ I ∈ HT F n be a direct system, so that ( M i [ − n ]) i ∈ I is a directsystem of H . The stalk complex R [0] of the ring is a homotopically finitely presentedobject of D ( R ) in the sense of [SˇSV17, Definition 5.1]; furthermore, being H theheart of a compactly generated t-structure in D ( R ), by [ibid., Corollary 5.8] itsdirect homotopy colimits are canonically isomorphic to the underlying direct limits,so we obtain the following chain of isomorphisms: H n (lim −→ i ∈ I H M i [ − n ]) ∼ = Hom D ( R ) ( R [0] , lim −→ i ∈ I H M i [ − n ]) ∼ = Hom D ( R ) ( R [0] , holim −−−→ i ∈ I M i [ − n ]) ∼ = lim −→ i ∈ I Hom D ( R ) ( R [0] , M i [ − n ]) ∼ = lim −→ i ∈ I H n ( M i [ − n ]) = lim −→ i ∈ I M i , OCAL COHERENCE OF THOMASON HEARTS 15 while in any degree different from n the direct limit has no cohomology. Therefore,lim −→ i ∈ I H M i [ − n ] ∼ = (cid:16) lim −→ i ∈ I M i (cid:17) [ − n ]i.e. direct limits of HT F n are computed precisely as in R -Mod. (cid:3) Bounded Above Thomason Filtrations
We study the bounded above Thomason filtrations, since among these there arethe finite length ones, of which we want to characterise the local coherence of theirhearts (see Remark 3.5(1)).
Lemma 4.1.
Let Φ be a Thomason filtration bounded above k . Then HT F k − = T F k − . Proof.
Notice that, by definition of the aisle, the boundedness of Φ ensures that U ⊆ D ≤ k ( R ).This said, let M ∈ HT F k − . Then M = H k − ( M [ − k + 1]), hence by Lemma 3.6we obtain M ∈ T k − ∩ F k .Conversely, let us prove that the stalk concentrated in degree − k + 1 of a module M ∈ T k − ∩ F k belongs to the heart associated with Φ . M [ − k + 1] surely landsin the aisle. On the other hand, M [ − k ] falls in the coaisle U ⊥ , i.e. M [ − k +1] ∈ V , since for every U ∈ U , the standard approximation τ ≤ k − ( U ) → U → H k ( U )[ − k ] + → (provided by the boundedness of Φ ) yields, by [Ver], the desiredvanishing Hom D ( R ) ( U, M [ − k ]) = 0. (cid:3) Remark 4.2. (1) As we shall deduce by Proposition 4.6 (which does not depend on the forth-coming results), the torsion class corresponding to any nonempty Thomasonsubset is a locally finitely presented Grothendieck category. In particular,for a Thomason filtration bounded above k , by Lemma 4.1 and [PSV19,Corollary 4.3] we havefp( T F k − ) = add y k (fp( T k − )) = add y k ( T k − ∩ R -mod) . (2) For any finitely generated ideal J , we will denote by K ( J ) the associatedKoszul complex (see [Nor68, Chap. 8] and [Hrb18]). Lemma 4.3.
Let Φ be a Thomason filtration bounded above k . Then: (i) For every J ∈ B k , it is H H ( K ( J )[ − k ]) ∼ = R/J [ − k ] ; (ii) For every J ∈ B k − , it is H H ( K ( J )[ − k + 1]) ∼ = y k ( R/J )[ − k + 1] , where y k is the torsionfree radical associated with the torsion pair ( T k , F k ) .Proof. Let us recall some basic facts concerning Koszul complexes and their coho-mology (see e.g. [Nor68, Chap. 8]). For any finitely generated ideal J one has:(1) K ( J ) ∈ D [ − n, ( R ), where n = rank J ;(2) H ( K ( J )) ∼ = R/J ;(3) JH − j ( K ( J )) = 0 or, equivalently, Supp H − j ( K ( J )) ⊆ V ( J ), for all j =0 , . . . , n .In our setting, (2) and (3) tell us that the Koszul cohomologies are torsion modulesw.r.t. the torsion pair associated with V ( J ).(i) Let J ∈ B k . Conditions (2) and (3) guarantee that K ( J )[ − k ] ∈ U , so the com-plexes K := K ( J )[ − k ] and M := H H ( K ) fit as the vertexes of the approximatingtriangle U [1] −→ K −→ M + −→ provided by the object U := τ ≤U ( K [ − H k ( M ) ∼ = R/J andthat τ ≤ k − ( M ) = 0, whence the conclusion as exactly as in the proof of Lemma 4.5.Fix r ≤ k − H r ( K ) → H r ( M ) → H r +2 ( U ) in R -Mod. By (3), H r ( K ) is an object of T k , hence of T r +1 , so that H r ( M ) ∈ T r +1 since in turn T r +1 ⊇ T r +2 . It follows τ ≤ k − ( M ) ∈ U [1], and from the triangle τ ≤ k − ( M ) −→ M −→ τ >k − ( M ) + −→ we deduce τ >k − ( M ) ∼ = M ⊕ τ ≤ k − ( M )[1] by [Nee01, Corollary 1.2.7] again, whence τ ≤ k − ( M )[1] ∈ D ≤ k − ( R ) ∩ D ≥ k ( R ) = 0. Now, the first displayed triangle yieldsthe following exact sequence in R -Mod: H k +1 ( U ) −→ H k ( K )( ∼ = R/J ) −→ H k ( M ) −→ H k +2 ( U ) , whence we obtain H k ( M ) ∼ = R/J since Φ ( k + 1) = Φ ( k + 2) = ∅ and U ∈ U .(ii) Let J ∈ B k − , K := K ( J )[ − k + 1] and M := H H ( K [1]). The thesis followsas in the previous part, namely by proving that H r ( M ) = 0 for every r = k − H k − ( M ) ∼ = y k ( R/J ). To this aim, look at the long exact cohomologysequence arising from U [1] → K → M + → , in which U := τ ≤U ( K [ − (cid:3) Corollary 4.4.
Let Φ be a Thomason filtration bounded above k . For every module X ∈ R -Mod : (i) X ∈ fp( T k ) if and only if X [ − k ] ∈ fp( H ) . In particular, H k (fp( H ))[ − k ] ⊆ fp( H ) ; (ii) X ∈ fp( HT F k − ) if and only if X [ − k + 1] ∈ fp( H ) .Proof. (i) Let X be a finitely presented object of T k i.e. an object of fp( T k ) = R -mod ∩ T k .By (the proof of) Proposition 3.2 there exists in T k ∩ R -mod an exact row ( R/J ′ ) n α → ( R/J ) m → X →
0, which can be embedded in the following diagram in D ( R ) bytaking the stalk complexes:(Ker α )[ − k ] / / ( R/J ′ ) n [ − k ] / / (Im α )[ − k ] + / / (cid:15) (cid:15) ( R/J ) m [ − k ] (cid:15) (cid:15) X [ − k ] + (cid:15) (cid:15) By Lemma 4.3(i) and [SˇSV17, Lemma 6.3], for every I ∈ B k the stalk R/I [ − k ] is afinitely presented object of H . Moreover, since the triangles of the diagram are in H ,then they actually are short exact sequences of H , hence X [ − k ] ∼ = Coker H ( α [ − k ])and it is finitely presented being the cokernel of a map between finitely presentedcomplexes.Conversely, let X be a module whose stalk X [ − k ] is a finitely presented complexof the heart. Then clearly X ∈ T k ; moreover, for all direct systems of modules( X i ) i ∈ I in T k , by [Ver] we deduce the natural isomorphismlim −→ i ∈ I Hom R ( X, X i ) ∼ = Hom R ( X, lim −→ i ∈ I X i ) , OCAL COHERENCE OF THOMASON HEARTS 17 whence X ∈ R -mod since ( T k , F k ) is a torsion pair of finite type (see [PSV19,Lemma 1.11]).The second part of the statement readily follows by the previous one, since outof the exact triangle τ ≤ k − ( B ) → B → H k ( B )[ − k ] + → approximating a finitely pre-sented complex B of the heart, by [Ver] we infer that H k ( B ) is a finitely presentedobject of T k .(ii) If X is a module whose stalk X [ − k + 1] is a finitely presented complex of theheart, then by definition of HT F k − and by Proposition 3.7, Lemma 4.1 and [Ver],for every direct system of modules ( M i ) i ∈ I in HT F k − we obtain the followingcommutative diagramlim −→ i ∈ I Hom H ( X [ − k + 1] , M i [ − k + 1]) ∼ = / / ∼ = (cid:15) (cid:15) Hom H ( X [ − k + 1] , lim −→ i ∈ I H M i [ − k + 1]) ∼ = (cid:15) (cid:15) lim −→ i ∈ I Hom R ( X, M i ) / / Hom R ( X, lim −→ i ∈ I M i )showing that X is a finitely presented object of HT F k − .Conversely, let X be a module in fp( T F k − ) = add y k (fp( T k − )) (see Remark 4.2),so that there exists a finitely presented object B of T k − such that X ≤ ⊕ y k ( B ) n for some n ∈ N , hence we shall prove the statement on y k ( B ) n , in particular byshowing that y k ( B )[ − k + 1] ∈ fp( H ). By Proposition 3.2 there is an exact sequence( R/J ′ ) n α → ( R/J ) m → B → R -Mod for some positive integers m, n and ideals J ′ , J in B k − . By Lemma 4.3(ii), we have the exact row H H ( K ( J ′ )[ − k +1]) n y k ( α )[ − k +1] −−−−−→ H H ( K ( J )[ − k +1]) m → Coker H ( y k ( α )[ − k +1]) → H ( y k ( α )[ − k + 1]) actually is a stalk complex as well. Toprove this, consider the canonical short exact sequences of H → Ker H ( y k ( α )[ − k + 1]) → H H ( K ( J ′ )[ − k + 1]) n → Im H ( y k ( α )[ − k + 1]) → → Im H ( y k ( α )[ − k + 1]) → H H ( K ( J )[ − k + 1]) m → Coker H ( y k ( α )[ − k + 1]) → , say them 0 → K → M ′ → L → → L → M → N → k −
1, they yield H k ( L ) = 0 and H k ( N ) = 0, respectively. On the other hand, from the secondexact row, we have H r − ( N ) ∼ = H r ( L ) ∈ T r for all r ≤ k −
2, and H k − ( N ) is asubmodule of H k − ( L ) ∈ T k − . Hence τ ≤ k − ( N ) ∈ U [1], so that N ∼ = τ >k − ( N ) = τ ≥ k − ( N ) = H k − ( N )[ − k + 1]. Therefore, the very first displayed exact row M ′ → M → N → N = Coker H ( y k ( α )[ − k + 1]) ∼ = D [ − k + 1] , for some D ∈ T F k − ; notice that D [ − k + 1] ∈ fp( H ). Once we prove that y k ( B ) ∼ = D , then we get the thesis. By the long exact sequence in cohomology of the previoustwo short exact sequences, we obtain the commutative diagram with exact rows: y k ( R/J ′ ) n / / $ $ $ $ ■■■■■ δ (cid:15) (cid:15) y k ( R/J ) m / / Coker y k ( α ) / / q (cid:15) (cid:15) (cid:15) (cid:15) δ : : : : ✉✉✉✉✉ z z z z ✉✉✉✉✉✉ / / H k − ( L ) / / y k ( R/J ) m p / / D / / where Coker δ = H k ( K ) ∈ T k and p is an epimorphism since H k ( L ) = 0. Wededuce that D ∼ = y k (Coker y k ( α )) =: y k ( C ). On the other hand, we have thefollowing commutative diagram with exact rows:0 / / x k ( R/J ′ ) nx k ( α ) (cid:15) (cid:15) / / ( R/J ′ ) nα (cid:15) (cid:15) / / y k ( R/J ′ ) ny k ( α ) (cid:15) (cid:15) / / / / x k ( R/J ) m (cid:15) (cid:15) (cid:15) (cid:15) / / ( R/J ) m (cid:15) (cid:15) (cid:15) (cid:15) / / y k ( R/J ) m (cid:15) (cid:15) (cid:15) (cid:15) / / x k ( α ) / / B g / / C / / → A → B g → C → x k ( α ) → B through its image A yields that this latter is anobject of T k . Consequently, we deduce D ∼ = y k ( B ) by the Snake Lemma applied onthe following commutative diagram0 / / x k ( B ) / / (cid:15) (cid:15) B g (cid:15) (cid:15) (cid:15) (cid:15) / / y k ( B ) / / (cid:15) (cid:15) / / x k ( C ) / / C / / y k ( C ) / / (cid:3) We conclude this section by studying two crucial cases of Thomason filtrationsof finite length. In particular, we will completely characterise the local coherenceof the heart in case of length 0, and obtain a very interesting example in the caseof length 1. First, let us check this general fact.
Lemma 4.5.
Let Φ be a bounded below k Thomason filtration. Then the associatedheart H is contained in D ≥ k ( R ) . In particular, when Φ has length ℓ we have H ⊆D [ − ℓ, ( R ) .Proof. Let us prove that for every M ∈ H we have τ ≤ k − ( M ) = 0. Notice that, bydefinition, T n = T k for every n ≤ k . Thus, τ ≤ k − ( M ) ∈ U [1] since H j ( τ ≤ k − ( M )[ − ( H j − ( M ) ∈ T j − = T k = T j if j ≤ k j > k .Therefore, in the exact triangle τ ≤ k − ( M ) → M → τ >k − ( M ) + → the first edgeis the zero morphism. By [Nee01, Corollary 1.2.7] we obtain the decomposition τ >k − ( M ) ∼ = M ⊕ τ ≤ k − ( M )[1], thus our claim follows at once by additivity of thestandard cohomology. (cid:3) Thomason filtrations of length 0.
By definition, any Thomason filtration Φ of length 0 has the form Φ : · · · = X = X = · · · = X ⊃ ∅ where X is a fixed Thomason subset. Proposition 4.6.
Let Φ : · · · = X = X ⊃ ∅ be a Thomason filtration of length .Then its heart is equivalent to T X .Proof. We have H = T X [0] by Lemma 4.5 (together with the last paragraph beforethe present subsection). (cid:3) OCAL COHERENCE OF THOMASON HEARTS 19
Consequently, we get that for any Thomason subset X = ∅ , its torsion class T X is a locally finitely presented Grothendieck category by [SˇS20, Theorem 8.20],i.e. for being (equivalent to) the heart of a compactly generated t-structure in D ( R ).Now, the following result completely characterise the local coherence of hearts ofThomason filtrations of length 0. Theorem 4.7.
Let X be a nonempty Thomason subset. The following statementsare equivalent: (a) The torsion class T X is a locally coherent Grothendieck category; that is, T X ∩ R -mod is an exact abelian subcategory of T X . (b) ( J : γ ) is a finitely generated ideal for every J ∈ B X and for all γ ∈ R ; (c) R/J is a coherent commutative ring for every J ∈ B X .Proof. Let us recall that B X is the family of finitely generated ideals in the Gabrielfilter associated with the Thomason subset X .“(a) ⇒ (b)” For every J ∈ B X and for all γ ∈ R , the ideal J + Rγ is in B X hence R/ ( J + Rγ ) is a finitely presented (torsion) module (see Proposition 3.2).In turn, ( J + Rγ ) /J ∼ = Rγ/ ( J ∩ Rγ ) is so, being the kernel of the epimorphism R/J → R/ ( J + Rγ ) in T X ∩ R -mod. The conclusion follows from the short exactsequence 0 → ( J : γ ) → R → Rγ/ ( J ∩ Rγ ) → ⇒ (a)” Let f : M → M ′ be a R -linear map in T X ∩ R -mod. By the well-knownclosure properties of this latter class of modules, we only need to verify that Ker f isa finitely presented module, and clearly it suffices to consider f as an epimorphism.Furthermore, from the following commutative diagram with exact rows0 / / K (cid:15) (cid:15) (cid:15) (cid:15) / / ( R/J ) n / / α (cid:15) (cid:15) (cid:15) (cid:15) M ′ / / / / Ker f / / M f / / M ′ / / α is provided by (the proof of) Proposition 3.2(ii), weargue that a “backward” argument on the extension-closure of the finitely pre-sented modules shows that the claim is equivalent to requiring that Ker α is finitelypresented. Indeed, we have the following exact diagram:0 / / Ker( α ◦ µ ) (cid:15) (cid:15) (cid:15) (cid:15) / / ( R/J ) n − / / (cid:15) (cid:15) µ (cid:15) (cid:15) Im( α ◦ µ ) (cid:15) (cid:15) (cid:15) (cid:15) / / / / Ker α (cid:15) (cid:15) (cid:15) (cid:15) / / ( R/J ) n α / / (cid:15) (cid:15) (cid:15) (cid:15) M (cid:15) (cid:15) (cid:15) (cid:15) / / / / C / / R/J / / C ′ / / µ is the canonical split monomorphism and the third exact row is givenby the Snake Lemma, so that Ker α is finitely presented if C and Ker( α ◦ µ ) areso. Now, once we prove that C is finitely presented, we can repeat the previousargument for each n ≥ k ≥
2, achieving the validity at the base k = 2. In otherwords, Ker α is finitely presented iff C is finitely presented. Let us prove that C isa finitely presented module. It is finitely generated for Ker α being so. Consider now the pullback diagram0 / / J / / J ′ (cid:15) (cid:15) (cid:15) (cid:15) / / P . B . C (cid:15) (cid:15) (cid:15) (cid:15) / / / / J / / R (cid:15) (cid:15) (cid:15) (cid:15) / / R/J (cid:15) (cid:15) (cid:15) (cid:15) / / R/J ′ R/J ′ in which J ′ is a finitely generated ideal by extension closure. Let us prove theclaim by induction on the rank of J ′ . If J ′ = Rγ , then it is J ′ = J + Rγ , sothat C ∼ = J ′ /J ∼ = Rγ / ( J ∩ Rγ ). We conclude by hypothesis (b) applied on theshort exact sequence 0 → ( J : γ ) → R → C →
0. If J ′ = Rγ + Rγ , then again J ′ = J + Rγ + Rγ , and from the exact commutative diagram0 / / J / / J + Rγ (cid:15) (cid:15) (cid:15) (cid:15) / / ( J + Rγ ) /J (cid:15) (cid:15) (cid:15) (cid:15) / / / / J / / J ′ (cid:15) (cid:15) (cid:15) (cid:15) / / C (cid:15) (cid:15) (cid:15) (cid:15) / / J ′ / ( J + Rγ ) J ′ / ( J + Rγ )thus, by the inductive base we see that ( J + Rγ ) /J and J ′ / ( J + Rγ ) are finitelypresented, hence C is so by extension-closure. This argument clearly applies atevery finite rank of J ′ , so C is finitely presented.“(b) ⇒ (c)” Let J ′ /J be a finitely generated ideal of R/J (so that J ′ /J is a finitelygenerated module over R ) and let us prove that it is finitely presented. J ′ is in G X , and by the short exact sequence 0 → J ′ /J → R/J → R/J ′ → R -Mod wededuce that R/J ′ is a finitely presented R -module. By the hypothesis “(b) ⇔ (a)”we get that J ′ /J is finitely presented over R , hence over R/J .“(c) ⇒ (b)” Assume that R/J is a coherent ring for each J ∈ B X , and let γ ∈ R .By the short exact sequence 0 → ( J : γ ) → R → Rγ/ ( J ∩ Rγ ) → Rγ/ ( J ∩ Rγ ) ∼ = ( J + Rγ ) /J is a finitely presented R -module. ( J + Rγ ) /J is afinitely generated hence a finitely presented ideal of R/J , so there is a presentation0 → K → ( R/J ) n → ( J + Rγ ) /J → n ∈ N and K a finitely generated R/J -module. Since the scalar restriction functor
R/J -Mod → R -Mod is exact, andsince K is also a finitely generated R -module, such presentation lifts to R -Mod sothat ( J + Rγ ) /J is finitely presented, as desired. (cid:3) Corollary 4.8.
Let R be a coherent commutative ring and X be a Thomasonsubset. Then T X is a locally coherent Grothendieck category.Proof. It follows by the previous Theorem, since any factor ring
R/J is coherentfor every finitely generated ideal J (see [Lam99, (c) p. 143]). (cid:3) Remark 4.9.
The previous three results extend the class of commutative ringsover which the heart associated with any Thomason subset X is always a locallycoherent Grothendieck category. Indeed, it is known e.g. from [PS17, Sao17] thatthis occurs in the case of a noetherian commutative ring. On the other hand, not allnon-coherent commutative rings satisfy the equivalent conditions of Theorem 4.7,as we will show in the next example. OCAL COHERENCE OF THOMASON HEARTS 21
Example 4.10.
In [BP19, Appendix A] authors consider the ring R = Z ⊕ ( Z / Z ) ( N ) , whose sum is componentwise and its multiplication is defined by( m, a ) · ( n, b ) = ( mn, mb + na + ab ) , where ma = ( ma , ma , . . . ) and ab = ( a b , a b , . . . ). In [ibid., Lemma A.1] it isproved that R is a commutative non-coherent ring, namely for the ideal generatedby any (2 m, a ) is finitely generated and not finitely presented. This fact entails atonce a (somehow trivial) example of a Thomason subset whose torsion class is nota locally coherent Grothendieck category, namely Spec R itself, since the resultingtorsion class is R -Mod.Nonetheless, let us show that, over R as above, there are proper Thomasonsubsets of Spec R and finitely generated ideals which do not satisfy Theorem 4.7(b).For instance, consider J = R (0 , e ) , X = V ( J ) , and γ = (2 , e ) ∈ R, where e n is the standard basis vector of ( Z / Z ) ( N ) , so that J ∈ B X . We compute: J = { ( m, a ) · (0 , e ) | ( m, a ) ∈ R } = { (0 , ( m + a ) e ) | ( m, a ) ∈ R } and ( J : γ ) = { ( m, a ) ∈ R | ( m, a )(2 , e ) ∈ J } = { ( m, a ) ∈ R | (2 m, me + ae ) ∈ J } = { ( m, a ) ∈ R | (2 m, ( m + a ) e ) ∈ J } = Ann R ( γ ) . Now, out of the presentation 0 → Ann R ( γ ) → R → Rγ →
0, since Rγ is not finitelypresented ([BP19, Lemma A.1]), then ( J : γ ) is not finitely generated, as claimed.4.2. An example of Thomason filtration of length 1.
We exhibit an exampleof Thomason filtration of length 1 that allows to realise any HRS heart of a hered-itary torsion pair of finite type of R -Mod as its heart. As a consequence of [SˇS20],this heart is automatically a locally finitely presented Grothendieck category. Thisexample will be resumed in the last part of the paper. Example 4.11.
Let us prove that for the Thomason filtration Φ : Spec R ⊃ X ⊃ ∅ of length 1, the associated heart is precisely the Happel-Reiten-Smalø heart H X arising from the torsion pair ( T X , F X ) (see Example 1.3).Let us prove that H ⊆ H X . For every M ∈ H we have H ( M ) ∈ T X , so it remainsto verify that H − ( M ) ∈ F X . This follows by Corollary 3.6 and Lemma 4.5.Conversely, let us prove the inclusion H X ⊆ H by showing that both the torsionand torsionfree classes F X [1] and T X [0] approximating H X are contained in H ,whence the conclusion by the extension-closure of the heart. The fact that T X [0] ⊆H is clear by definition of the t-structure ( U , V ).On the other hand, let F ∈ F X . Since Supp H − ( F [1]) = Supp F is containedin the spectrum i.e. in Φ ( − H k ( F [1]) = ∅ for all k = −
1, we have F X [1] ⊆ U . Let now M ∈ U ⊆ D ≤ ( R ); out of the exact triangle τ ≤− ( M ) → M → H ( M )[0] + → provided by the standard t-structure of D ( R ), applying thecohomological functor Hom D ( R ) ( − , F [0]) we obtain the exact sequenceHom D ( R ) ( H ( M )[0] , F [0]) → Hom D ( R ) ( M, F [0]) → Hom D ( R ) ( τ ≤− ( M ) , F [0]) whence the remaining inclusion F X [1] ⊆ U ⊥ [1], for the left hand term is zero by[Ver] and since ( T X , F X ) is a torsion pair in R -Mod, and for the right hand termbeing clearly zero as well.5. Arbitrary Thomason Filtrations
In the previous section we dealt with bounded above Thomason filtrations. Wenow present an effective way of using certain bounded below filtrations naturallyassociated with an arbitrary Thomason filtration. In fact, these “sub-filtrations”are the TTF classes we are looking for in order to specialise Theorem 2.1 to thehearts of Thomason filtrations of finite length.Let Φ be an arbitrary Thomason filtration of Spec R ; define for any k ∈ Z Φ k ( n ) = ( Φ ( k ) for all n < kΦ ( n ) for all n ≥ k .Thus, Φ k is a bounded below k Thomason filtration, naturally associated with Φ .We will denote by ( U k , V k ) and H k , respectively, the t-structure and the heart of Φ k ,. It is clear that at all the degrees in which Φ k and Φ have the same Thomasonsubsets, namely for all n ≥ k , their corresponding torsion pairs coincide as well; inthis case we will denote these latter just as ( T n , F n ), i.e. as those associated with Φ ( n ). Lemma 5.1.
Let Φ be a Thomason filtration. Then H k ⊆ H .Proof. Given M ∈ H k , then clearly M ∈ U so that it remains to prove that M [ − ∈U ⊥ . This follows immediately by applying the functor Hom D ( R ) ( − , M [ − τ ≤ k − ( U ) → U → τ >k − ( U ) + → of an arbitrary object U ∈ U withinthe shifted standard t-structure of D ( R ), bearing in mind that H k ⊆ D ≥ k ( R )(Lemma 4.5) and that τ >k − ( U ) ∈ U k . (cid:3) Corollary 5.2.
Let Φ be a Thomason filtration. For every M ∈ H the followingassertion hold: (i) there exists in H a short exact sequence → A → M → B → with A ∈ ⊥ H k ( the orthogonal being computed w.r.t. H ) and B ∈ H k ; (ii) there exists in H a short exact sequence → A → M → B → with A ∈ H k and B ∈ H ⊥ k ( the orthogonal being computed w.r.t. H ) .Proof. (i) Let M ∈ H , and consider the octahedron: τ ≤ k − ( M ) / / A (cid:15) (cid:15) / / U [1] (cid:15) (cid:15) + / / τ ≤ k − ( M ) / / M (cid:15) (cid:15) / / τ >k − ( M ) (cid:15) (cid:15) + / / H H k ( τ >k − ( M )) + (cid:15) (cid:15) H H k ( τ >k − ( M )) + (cid:15) (cid:15) provided by U := τ ≤U k ( τ >k − ( M )[ − A (notice that τ >k − ( M ) ∈ U k ).Since B := H H k ( τ >k − ( M )) actually is in H k , hence in H by Lemma 5.1, we onlyhave to check that A belongs to H and that it is left orthogonal to H k in H . Fromthe first vertical triangle we see that A ∈ V , whereas by the first horizontal one OCAL COHERENCE OF THOMASON HEARTS 23 we deduce that A ∈ U . Moreover, using once again the first horizontal triangle,we infer that A ∈ ⊥ H k since H k ⊆ D ≥ k ( R ), as desired. Thus, the first verticaltriangle yields the stated short exact sequence of H .(ii) Consider the approximation A → M → B + → of M within the t-structure( U k , V k ), thus surely B is right orthogonal to H k in H . It remains to check that A ∈ V k and that B ∈ H . The first claim holds true by extension-closure of thecoaisle applied on the rotated triangle B [ − → A [ − → M [ − + → , and since U k ⊆ U . On the other hand, B belongs to the aisle U in view of the rotated triangle M → B → A [1] + → , while for every U ∈ U , by the approximation τ ≤ k − ( U ) −→ U −→ τ >k − ( U ) + −→ , we have τ >k − ( U ) ∈ U k , whence Hom D ( R ) ( τ >k − ( U ) , B [ − D ( R ) ( τ ≤ k − ( U ) , B [ − τ ≤ k − ( U ) on the triangle M [ − → B [ − → A + → , bearing in mind that A ∈ H k ⊆ D ≥ k ( R ). (cid:3) Corollary 5.3.
Let Φ be a Thomason filtration. Then the heart H k is closed in H under taking products and coproducts.Proof. Let ( M i ) i ∈ I be a family of objects of H k with product ( Q i ∈ I M i , ( π i ) i ∈ I ) in H k . We have to prove that such pair satisfies the universal property of the productin H . So, let M ∈ H and ( f i ) i ∈ I be a family of morphisms f i : M → M i in H . ByCorollary 5.2(i) we obtain the following commutative diagram, A (cid:15) (cid:15) α / / M f i (cid:15) (cid:15) β / / B g i (cid:15) (cid:15) + / / / / M i M i + / / hence a family of morphisms g i : B → M i in H k inducing a unique morphism g : B → Q i ∈ I M i such that π i ◦ g = g i for all i ∈ I . The composition g ◦ β yields theexistence of a morphism M → Q i ∈ I M i in H such that π i ◦ ( g ◦ β ) = f i for all i ∈ I .Uniqueness of g ◦ β w.r.t. the latter property is a byproduct of the construction of thetriangle made in Corollary 5.2, namely for both A and B are uniquely determinedup to isomorphism, together with the fact that β is an epimorphism in H .The proof concerning the coproduct is dual. (cid:3) Proposition 5.4.
Let Φ be a Thomason filtration. Then the heart H k is a TTFclass of finite type in H .Proof. In order to prove that H k is a TTF class in H , by the previous Corollary weonly have to show that the former heart is closed under subobjects, quotient objectsand extensions. The closure under extensions is obvious since both the aisle andthe coaisle fulfil it. So, let 0 → L → M → N → H with M ∈ H k . Clearly, L and N belong to V k . By Lemma 4.5, for all j < k − R -module isomorphisms H j ( N ) ∼ = H j +1 ( L ) ∈ T j +1 ( ∗ ); moreover, H k − ( N )is a submodule of H k ( L ), i.e. it belongs to T k . It follows τ ≤ k − ( N ) ∈ U [1] andconsequently, by the usual argument of the proof of Lemma 4.5, that τ ≤ k − ( N ) = 0.By ( ∗ ), we infer τ ≤ k − ( L ) = 0 as well. Thus, N, L ∈ H ∩ D ≥ k ( R ) ⊆ U ∩ D ≥ k ( R ) ⊆U k .Let us now prove that the torsion pair ( H k , H ⊥ k ) is of finite type, i.e. thatthe torsionfree class is closed under direct limits. Since ( U k , V k ) is a compactlygenerated t-structure of D ( R ), which is the base of a strong and stable derivator,by [SˇSV17, Proposition 5.6] it is homotopically smashing; that is, its coaisle is closed under homotopy filtered colimits. On the other hand, H ⊥ k is contained (asa subcategory of H ) in V k [ − M ∈ H ⊥ k and apply the functorHom D ( R ) ( − , M ) on the exact triangle τ ≤U k ( U [ − → U → H H k ( U ) + → associatedwith an arbitrary object U ∈ U k to get Hom D ( R ) ( U, M ) = 0. Now, for any directsystem ( M i ) i ∈ I in H ⊥ k , hence in V k [ − −−−→ i ∈ I M i ∼ = lim −→ i ∈ I H M i . Therefore, the right hand object belongs to V k [ − ∩ H and, in particular, to H ⊥ k . (cid:3) Corollary 5.5.
Let Φ be a Thomason filtration bounded above k . Then T k [ − k ] isa TTF class of finite type in H .Proof. Thanks to the boundedness of Φ , we have H k = T k [ − k ], so the conclusionfollows by the previous Proposition. Notice that in this case the left constituent ofthe TTF triple is ( τ ≤ k − ( H ) , T k [ − k ]), for there are no nonzero morphisms betweenthe members of the pair and, by Corollary 5.2(i), for every M ∈ H its standardapproximation τ ≤ k − ( M ) → M → H k ( M )[ − k ] + → yields a functorial short exactsequence in H . (cid:3) Remark 5.6.
As we have seen in the proof of Theorem 2.1, the existence in H ofa TTF triple of finite type carries useful information, both on the members of thetriple and on the local coherence of H itself. More precisely:(1) By Corollary 5.2(i), the torsion class ⊥ H k consists of those complexes M of H which fit in an exact triangle τ ≤ k − ( M ) → M → U [1] + → for someobject U ∈ U k .(2) The torsion class H k is a locally finitely presented category by [SˇS20], more-over we have fp( ⊥ H k ) , fp( H k ) ⊆ fp( H ) by [PSV19, Lemma 1.11]. Further-more, by Theorem 2.1, they both ⊥ H k and H k are locally coherent in case H is so.(3) In order to distinguish the torsion radicals and coradicals of each torsionpair ( ⊥ H k , H k ) of H to those of each torsion pair ( T k , F k ) of R -Mod wedealt with so far, we will use the following notation ⊥ H k ֒ −→←−− x k H y k −−→←− ֓ H k ;furthermore, we will drop the index in case the value of the integer is clearfrom the context.6. Thomason Filtrations of Finite Length
The present section is devoted to deepen the approximation theory of the heart H associated with a Thomason filtration Φ of finite length, in order to characteriseits local coherence. In this vein, the main tool is given by the TTF classes H k detected in Proposition 5.4, for they allow to specialise Theorem 2.1. Bearing inmind Remark 5.6, it is then natural to seek for a recursive characterisation, namelya result which takes in account the local coherence of each heart H k . Therefore,we set ℓ + 1 to be the length of Φ . Lemma 6.1.
Let Φ be a Thomason filtration of length ℓ + 1 . Then: (i) For every X ∈ T − ℓ − , we have H H ( X [ ℓ + 1]) ∈ ⊥ H − ℓ ; (ii) For every X ∈ T FT − ℓ − there exist U ∈ U − ℓ +2 and a triangle U [1] → X [ ℓ + 1] → H H ( X [ ℓ + 1]) + → . In particular, H − ℓ − ( H H ( X [ ℓ + 1])) = X . OCAL COHERENCE OF THOMASON HEARTS 25 (iii) for all M ∈ ⊥ H − ℓ , there exists in H a functorial short exact sequence → L → W → M → , in which L ∈ H − ℓ +1 and W ∼ = H H ( X [ ℓ + 1]) ,where X = H − ℓ − ( M ) ; (iv) ⊥ H − ℓ = Gen( H H ( K ( J )[ ℓ + 1]) | J ∈ B − ℓ − ) .Proof. We will often exploit the characterisation of the torsion class ⊥ H − ℓ deducedfrom Corollary 5.2 (see Remark 5.6(1)).(i) Given X ∈ T − ℓ − , let M = H H ( X [ ℓ + 1]) and consider the exact triangle U [1] → X [ ℓ + 1] → M + → given by some object U ∈ U . Let us show that M satisfies the aforementioned characterisation of the torsion class ⊥ H − ℓ . Applyingthe standard cohomology on the above triangle we obtain H j ( M ) ∼ = H j +2 ( U ) forall j ≥ − ℓ , and these latter are modules in the torsion class T j +2 . We claim that τ ≥− ℓ ( M )[ − ∈ U − ℓ , whence the conclusion thanks to the triangle H − ℓ − ( M )[ ℓ + 1] −→ M −→ τ ≥− ℓ ( M ) + −→ . Indeed, we have H j ( τ ≥− ℓ ( M )[ − H j − ( τ ≥− ℓ ( M )) = ( j − < − ℓ,H j − ( M ) if j − ≥ − ℓ ,hence, when j − ≥ − ℓ , we have H j − ( M ) ∼ = H j +1 ( U ) ∈ T j +1 ⊆ T j , as desired.(ii) Let X ∈ T FT − ℓ − and U [1] → X [ ℓ + 1] → M + → as in part (i). The long exactsequence in standard cohomology yields0 −→ H − ℓ ( U ) −→ X −→ H − ℓ − ( M ) −→ H − ℓ +1 ( U ) −→ H − ℓ ( U ) = 0 for it belongs simultaneously to T − ℓ and F − ℓ byassumption on X . Moreover, the resulting extension of H − ℓ − ( M ) is split byassumption on X again, meaning that H − ℓ +1 ( U ) = 0 as well. Consequently, U ∈D ≥− ℓ +2 ( R ) ∩ U , as desired.(iii) Let M ∈ ⊥ H − ℓ , so that by Lemma 4.5 and Remark 5.6(1) there exists U ∈ U − ℓ and an exact triangle H − ℓ − ( M )[ ℓ + 1] → M → U [1] + → , in which we set X = H − ℓ − ( M ). The long exact sequence in standard cohomology yields in particular U ∈ U − ℓ ∩ D [ − ℓ +1 , ( R ). The approximation of U within ( U − ℓ +1 , V − ℓ +1 ) gives thefollowing octahedron U ′ [1] (cid:15) (cid:15) U ′ [1] (cid:15) (cid:15) U / / (cid:15) (cid:15) X [ ℓ + 1] / / (cid:15) (cid:15) M + / / L / / + (cid:15) (cid:15) W + (cid:15) (cid:15) / / M + / / for some U ′ ∈ U − ℓ +1 , so that L ∼ = H H − ℓ +1 ( U ), and a cone W , which actually belongsto H by extension-closure applied on the second horizontal triangle. Applyingthe t-cohomological functor H H on the second vertical triangle we obtain W ∼ = H H ( X [ ℓ + 1]), hence the former triangle is a functorial short exact sequence of H ;indeed, it is the image under H H of the first horizontal triangle, which is in turnfunctorial.(iv) Let M and X be as in part (iii). By Proposition 3.2 we know that there exista family ( J i ) i ∈ I of finitely generated ideals in the Gabriel filter associated with the torsion class T − ℓ − , and an epimorphism ϕ : L i ∈ I ( R/J i ) ( α i ) → X . Applying H H on the associated triangle of the stalk complexes concentrated in degrees − ℓ − D ( R ), we obtain the exactsequence of H H H (Ker( ϕ )[ ℓ + 1]) −→ M i ∈ I H H ( R/J i [ ℓ + 1]) ( α i ) −→ ∼ = W z }| { H H ( X [ ℓ + 1]) −→ . Thus, our claim follows once we prove that H H ( K ( J )[ ℓ + 1]) ∼ = H H ( R/J [ ℓ + 1]) forall J ∈ B − ℓ − , since M is an epimorphic image of W in H . Shifting by ℓ + 1 thestandard approximation τ ≤− ( K ( J )) → K ( J ) → R/J [0] + → of the Koszul complex K ( J ), we see that τ ≤− ( K ( J ))[ ℓ + 1] = ( τ ≤− ( K ( J )[ ℓ ]))[1] is an object of the aisle U [1]. Therefore, applying the functor H H on the resultin triangle, we conclude. (cid:3) Remark 6.2. (1) For all Thomason filtration (not necessarily of finite length) and k ∈ Z , theclass T FT k is closed under direct limits (of R -Mod), so it is an additivecategory with direct limits. Indeed, let ( X i ) i ∈ I be a direct system in T FT k ,and for all i ∈ I consider H H k ( X i [ − k ]) ∼ = H H ( X i [ − k ]); since Φ k is boundedbelow, by using the proof of Lemma 6.1(ii) get that H k ( H H ( X i [ − k ])) ∼ = X i . On the other hand, lim −→ H H H ( X i [ − k ]) belongs to H k , which in turn iscontained in D ≥ k ( R ), and consequently H k (lim −→ H H H ( X i [ − k ])) ∈ T FT k byLemma 3.6. But this latter module is isomorphic to lim −→ H k ( H H ( X i [ − k ])) ∼ =lim −→ X i , as desired (see also the proof of Proposition 3.7).(2) For every X ∈ T FT − ℓ − and M ∈ H − ℓ +1 we have Ext H ( H H ( X [ ℓ +1]) , M ) = 0. Indeed, by Lemma 6.1(ii) there are U ∈ U − ℓ +2 and a triangle U [1] → X [ ℓ + 1] → H H ( X [ ℓ + 1]) + → , hence applying Hom D ( R ) ( − , M [1])on the triangle we obtain, by [Ver], the desired vanishing of the ext-groupsince in the exact sequenceHom D ( R ) ( U [2] , M [1]) −→ Ext H ( H H ( X [ ℓ + 1]) , M ) −→ Hom D ( R ) ( X [ ℓ + 1] , M [1])the first term is zero by axioms of t-structure, as well as the third since M [1] ∈ D ≥− ℓ ( R ). Lemma 6.3.
Let Φ be a Thomason filtration of length ℓ + 1 , and X ∈ T − ℓ − . Then H H ( X [ ℓ + 1]) ∈ fp( H ) if and only if the functor Hom R ( X, − ) commutes with directlimits of direct systems in T FT − ℓ − .In particular, for all B ∈ fp( T FT − ℓ − ) we have H H ( B [ ℓ + 1]) ∈ fp( H ) .Proof. “ ⇒ ” Let X ∈ T − ℓ − and suppose that H H ( X [ ℓ + 1]) is a finitely presented objectof H . Let ( X i ) i ∈ I be a direct system in T FT − ℓ − . By approximating each complex X i [ ℓ +1] ∈ U within ( U , V ), we have a triangle U i [1] → X i [ ℓ +1] → H H ( X i [ ℓ +1]) + → ,say M i its last vertex, in which X i ∼ = H − ℓ − ( M i ) for all i ∈ I by Lemma 6.1(ii).On the other hand, by the approximating triangle U [1] → X [ ℓ + 1] → M + → of X [ ℓ + 1] within ( U , V ), we obtain the commutative diagram with exact rows0 / / lim −→ i ∈ I Hom D ( R ) ( M, M i ) / / ∼ = (cid:15) (cid:15) lim −→ i ∈ I Hom D ( R ) ( X [ ℓ + 1] , M i ) / / (cid:15) (cid:15) / / Hom D ( R ) ( M, lim −→ i ∈ I H M i ) / / Hom D ( R ) ( X [ ℓ + 1] , lim −→ i ∈ I H M i ) / / OCAL COHERENCE OF THOMASON HEARTS 27 in which the left hand vertical homomorphism is bijective by hypothesis, thus theright hand one is so. Eventually, by Lemma 4.5 we have a triangle H − ℓ − (lim −→ i ∈ I H M i )[ ℓ + 1] −→ lim −→ i ∈ I H M i −→ τ > − ℓ − (lim −→ i ∈ I H M i ) + → whose first vertex is (lim −→ i ∈ I H − ℓ − ( M i ))[ ℓ + 1] since the standard cohomologiescommute with direct limits, and applying Hom D ( R ) ( X [ ℓ + 1] , − ) on such trianglewe see, by [Ver], that the right hand isomorphism of the previous diagram actuallyis lim −→ i ∈ I Hom R ( X, H − ℓ − ( M i )) −→ Hom R ( X, lim −→ i ∈ I H − ℓ − ( M i ))i.e. the desired one showing that Hom R ( X, − ) commutes with direct limits of directsystems in T FT − ℓ − .“ ⇐ ” Let X ∈ T − ℓ − be a module whose functor Hom R ( X, − ) commutes with directlimits of direct systems in T FT − ℓ − . Let ( M i ) i ∈ I be a direct system in H , andconsider the direct system of approximating triangles ( H − ℓ − ( M i )[ ℓ + 1] → M i → τ > − ℓ − ( M i ) + → ) i ∈ I in D ( R ). Applying Hom D ( R ) ( X [ ℓ + 1] , − ) we obtain, as in theprevious part of the proof, the commutative diagram with exact rows0 / / lim −→ i ∈ I Hom R ( X, H − ℓ − ( M i )) / / ∼ = (cid:15) (cid:15) lim −→ i ∈ I Hom D ( R ) ( X [ ℓ + 1] , M i ) / / (cid:15) (cid:15) / / Hom R ( X, lim −→ i ∈ I H − ℓ − ( M i )) / / Hom D ( R ) ( X [ ℓ + 1] , lim −→ i ∈ I H M i ) / / D ( R ) ( − , M i )’s on the approximating triangle U [1] → X [ ℓ + 1] → M + → of X [ ℓ + 1] within ( U , V ), we obtain again that the right handisomorphism of the previous diagram is the desired one. (cid:3) Remark 6.4.
Let Φ be any Thomason filtration, and k ∈ Z . Then the composition H − k ◦ H H ◦ [ k ] defines a functor Σ − k : T F − k −→ T FT − k X H − k ( H H ( X [ k ]))equipped with a functorial monomorphism σ : id → Σ − k such that Coker σ X ∈T − k +2 . Indeed, for every X ∈ T F − k , i.e. X ∈ T − k ∩ F − k +1 , its stalk X [ k ] isan object of U ∩ U − k , hence H H ( X [ k ]) ∼ = H H − k ( X [ k ]) so that the least nonzerocohomology of the latter complex is at degree − k by Lemma 4.5. Therefore, byLemma 3.6, Σ − k is well-defined on objects. Let now f : X → X ′ be a morphism in T F − k . Then we have a diagram U [1] / / (cid:15) (cid:15) X [ k ] f [ k ] (cid:15) (cid:15) / / H H ( X [ k ]) + / / h (cid:15) (cid:15) U ′ [1] / / X ′ [ k ] / / H H ( X ′ [ k ]) + / / for some U, U ′ ∈ U , which can be completed to a morphism of triangles since U [1] → X [ k ] f [ k ] → X ′ [ k ] → H H ( X ′ [ k ]) is the zero map. We have h = H H ( f [ k ]),hence Σ − k actually is a functor. This said, apply the standard cohomology H − k on the first triangle of the previous commutative diagram, to obtain the exactsequence 0 −→ H − k ( U [1]) −→ X σ X −→ Σ − k ( X ) −→ H − k +1 ( U [1]) −→ in which H − k ( U [1]) ∈ T − k +1 ∩ F − k +1 = 0 by assumption on X . Therefore, the σ X ’s are monomorphisms, moreover they form a natural transformation in view ofthe construction of the functor Σ − k . Finally, Coker σ X ∈ T − k +2 being isomorphicto H − k +1 ( U [1]). Lemma 6.5.
Let Φ be a Thomason filtration, k ∈ Z and X ∈ T − k . The followingassertion are equivalent: (a) H H ( X [ k ]) ∈ fp( H ) ; (b) H H ( y − k +1 ( X )[ k ]) ∈ fp( H ) ; (c) H H ( Σ − k ( y − k +1 ( X ))[ k ]) ∈ fp( H ) ; (d) Σ − k ( y − k +1 ( X )) ∈ fp( T FT − k ) .The subclass of T − k of modules satisfying the previous equivalent conditions will bedenoted by Σ T − k .Proof. “(a) ⇔ (b)” Consider the approximation 0 → x − k +1 ( X ) → X → y − k +1 ( X ) → X within the torsion pair ( T − k +1 , F − k +1 ) of R -Mod. Then x − k +1 ( X )[ k ] ∈ U [1]hence applying the functor H H on the triangle involving the stalk complexes of thesequence, we obtain H H ( X [ k ]) ∼ = H H ( y − k +1 ( X )[ k ]), and we are done.“(b) ⇔ (c)” Since y − k +1 ( X ) ∈ T F − k , in view of Remark 6.4 we have a short exactsequence 0 −→ y − k +1 ( X ) −→ Σ − k ( y − k +1 ( X )) −→ Coker σ y − k +1 ( X ) −→ → Y → Z → C → C ∈ T − k +2 . By applying thefunctor H H on the triangle involving the stalk complexes of such sequence, weobtain H H ( Y [ k ]) ∼ = H H ( Z [ k ]), whence the thesis.“(c) ⇔ (d)” Recall that we have fp( H − k ) ⊆ fp( H ) and notice, using the samenotation of the previous part, that H H ( Z [ k ]) ∼ = H H − k ( Z [ k ]). Now the claim followsby the proof of Lemma 6.3. (cid:3) Corollary 6.6.
Let Φ be a Thomason filtration of length ℓ + 1 . For every B ∈ fp( T FT − ℓ − ) , there exist n ∈ N , ideals J , . . . , J n ∈ B − ℓ − , and (i) an epimorphism in H n M k =1 H H ( K ( J k )[ ℓ + 1]) − ։ H H ( B [ ℓ + 1]);(ii) integers k , . . . , k n , and a homomorphism in R -Mod f : n M i =1 Σ − ℓ − ( y − ℓ ( R/J i ) k i ) −→ B with Coker f ∈ T − ℓ .Proof. (i) By Lemma 6.3 we know that H H ( B [ ℓ + 1]) is a finitely presented object of theheart. On the other hand, by Lemma 6.1(iv) there are families ( J i ) i ∈ I of ideals in B − ℓ − , a set Λ and an epimorphism p : (cid:16)M i ∈ I H H ( K ( J i )[ ℓ + 1]) (cid:17) ( Λ ) − ։ H H ( B [ ℓ + 1]) . For every finite subset ¯ I ⊂ I , every i ∈ ¯ I , and every finite subset A ⊂ Λ , considerthe composition H H ( K ( J i )[ ℓ + 1]) ( A ) ε Ai −→ (cid:16)M i ∈ I H H ( K ( J i )[ ℓ + 1]) (cid:17) ( Λ ) p − ։ H H ( B [ ℓ + 1]) OCAL COHERENCE OF THOMASON HEARTS 29 where ε Ai is the split monomorphism. Then H H ( B [ ℓ + 1]) = Im p = X ¯ I ⊂ IA ⊂ Λ Im( p ◦ ε Ai ) , hence being the former a finitely presented complex, there exist finite subsets ¯ I ⊂ I and A ⊂ Λ such that H H ( B [ ℓ + 1]) = P i ∈ ¯ I Im( p ◦ ε Ai ), as desired.(ii) Let p be as in part (i) and define f := H − ℓ − ( p ). In view of the proof ofLemma 6.1(iv), in the heart H we have exact rows H H (Ker( f )[ ℓ + 1]) −→ n M i =1 H H ( R/J i [ ℓ + 1]) β −→ H H (Im( f )[ ℓ + 1]) −→ H H (Im( f )[ ℓ + 1]) α −→ H H ( B [ ℓ + 1]) −→ H H (Coker( f )[ ℓ + 1]) −→ α ◦ β = p , whence α is an epimorphism, so that H H (Coker( f )[ ℓ + 1]) = 0.Consequently, the usual triangle of D ( R ) ending in this latter complex of H showsthat Coker( f )[ ℓ + 1] is isomorphic to the object U [1] for some U ∈ U , meaning thatCoker f ∼ = H − ℓ − ( U [1]) ∈ T − ℓ . (cid:3) We now pass to consider some necessary conditions to the local coherence of theheart of a Thomason filtration of finite length.
Proposition 6.7.
Let Φ be a Thomason filtration of length ℓ + 1 . If H is a locallycoherent Grothendieck category and P ∈ ⊥ H − ℓ , then P ∈ fp( ⊥ H − ℓ ) if and onlyif the following conditions hold true: (i) H − ℓ − ( P ) ∈ fp( T FT − ℓ − ) ; (ii) Hom D ( R ) ( τ ≥− ℓ ( P )[ − , − ) commutes with direct limits of direct systems in H − ℓ +1 .Proof. “ ⇒ ” Let P ∈ fp( ⊥ H − ℓ ). By Lemma 6.1(iii) there exists L ∈ H − ℓ +1 and a shortexact sequence 0 → L → H H ( X [ ℓ + 1]) → P → H , in which X = H − ℓ − ( P ).Set W = H H ( X [ ℓ + 1]), and consider the exact sequence of covariant functors0 → Hom H ( P, − ) → Hom H ( W, − ) → Hom H ( L, − ) → Ext H ( P, − ) → Ext H ( W, − ) . When we restrict these functors to H − ℓ +2 , we obtain Hom H ( W, − ) ↾ = 0 by Lem-ma 6.1(i), hence Hom H ( P, − ) ↾ = 0, moreover Ext H ( W, − ) ↾ = 0 by Remark 6.2(2).Therefore, there is a natural isomorphism Hom H ( L, − ) ↾ ∼ = Ext H ( P, − ) ↾ , and bylocal coherence of H together with [Sao17, Proposition 3.5(2)] we get that L ∈ fp( H − ℓ +2 ) ⊆ fp( H ). By extension-closure of fp( H ) (see [PSV19, Corollary 1.4]),we have that W is a finitely presented object of H , whence X ∈ fp( T FT − ℓ − ) byLemma 6.3. This proves part (i), so let us show part (ii). By Lemma 4.5 we havean exact triangle H − ℓ − ( P )[ ℓ + 1] → P → τ ≥− ℓ ( P ) + → , say X [ ℓ + 1] the first vertex,as in part (i). By Remark 5.6(1), we know that τ ≥− ℓ ( P ) ∈ U [1]. Thus, by applyingthe functor H H on such triangle we obtain the exact row0 −→ H H ( τ ≥− ℓ ( P )[ − −→ H H ( X [ ℓ + 1]) −→ P −→ , which actually coincides with the short exact sequence 0 → L → W → P → H together withLemma 6.3, H H ( τ ≥− ℓ ( P )[ − U [1] → τ ≥− ℓ ( P )[ − → H H ( τ ≥− ℓ ( P )[ − + → provided by U := τ ≤U ( τ ≥− ℓ ( P )[ − M i ) i ∈ I of complexes in H − ℓ +1 and applying the functors F = lim −→ i ∈ I Hom D ( R ) ( − , M i ) and G = Hom D ( R ) ( − , lim −→ i ∈ I H M i )on the previous triangle, we obtain the commutative diagram with exact rows0 / / F ( H H ( τ ≥− ℓ ( P )[ − ∼ = (cid:15) (cid:15) / / F ( τ ≥− ℓ ( P )[ − (cid:15) (cid:15) / / / / G ( H H ( τ ≥− ℓ ( P )[ − / / G ( τ ≥− ℓ ( P )[ − / / L ∈ fp( H ).“ ⇐ ” Let P ∈ ⊥ H − ℓ and consider the short exact sequence 0 → L → W → P → H provided by Lemma 6.1(iii). Then L ∈ fp( H ) by what we said at the end ofthe proof of the previous part (ii), whereas W ∈ fp( H ) by Lemma 6.3. Therefore, P is finitely presented as well, being a cokernel of a morphism in fp( H ). (cid:3) Corollary 6.8.
Let Φ be a Thomason filtration such that its heart H is a locallycoherent Grothendieck category. If B ∈ fp( H ) and r is the least nonzero cohomologydegree of B , then we have H r ( B ) ∈ fp( T FT r ) .Proof. By definition of r and by Lemma 4.5, we have B ∈ H r . Moreover, since H is locally coherent, so is H r being a TTF class of finite type. In particular,the torsion pair ( ⊥ H r +1 , H r +1 ) of H r restricts to fp( H r ) (see Theorem 2.1), hencethe approximation 0 → x ( B ) → B → y ( B ) → B within the torsion pair(see Remark 5.6(3)) actually is in fp( H r ). By the proof of Proposition 6.7, we get H r ( x ( B )) ∈ fp( T FT r ), and being y ( B ) ∈ H r +1 ⊆ D ≥ r +1 ( R ), it follows H r ( B ) ∼ = H r ( x ( B )), and we are done. (cid:3) Proposition 6.9.
Let Φ be a Thomason filtration of length ℓ + 1 . If the heart H is locally coherent, then (i) fp( T FT − ℓ − ) is closed under kernels ( in R -Mod) ; (ii) For all B ∈ fp( T FT − ℓ − ) , there exists a R -linear map f : n M i =1 Σ − ℓ − ( y − ℓ ( R/J i ) k i ) −→ B with Coker f ∈ Σ T − ℓ ; (iii) For all morphisms f in fp( T FT − ℓ − ) with Coker f ∈ T − ℓ , then Coker f ∈ Σ T − ℓ .Proof. (i) Given f : B → B ′ a homomorphism in fp( T FT − ℓ − ), we have to show thatKer f ∈ fp( T FT − ℓ − ). Consider the following diagram in D ( R ) obtained by ap-proximating the stalk complexes of the modules within the t-structure ( U , V ): U [1] a / / (cid:15) (cid:15) B [ ℓ + 1] / / f [ ℓ +1] (cid:15) (cid:15) H H ( B [ ℓ + 1]) + / / q (cid:15) (cid:15) U ′ [1] / / B ′ [ ℓ + 1] b ′ / / H H ( B ′ [ ℓ + 1]) + / / Since b ′ ◦ f [ ℓ + 1] ◦ a = 0, the dotted vertical maps actually exist and they completethe diagram to a morphism of triangles (see e.g. [Mil, Proposition 1.4.5]). Now,by Lemma 6.1(i) we have that H H ( B [ ℓ + 1]) =: M and H H ( B ′ [ ℓ + 1]) =: M ′ are complexes of ⊥ H − ℓ , whereas by Lemma 6.3 we have that q is a morphism in OCAL COHERENCE OF THOMASON HEARTS 31 fp( ⊥ H − ℓ ). This said, by the hypothesis of local coherence of ⊥ H − ℓ we infer that x (Ker H ( q )) is a finitely presented object of ⊥ H − ℓ , so that of H . Moreover, noticethat H − ℓ − ( q ) = f , and that the standard cohomology sequences associated withthe following two sequences of H / / x (Ker H ( q )) / / Ker H ( q ) / / (cid:15) (cid:15) (cid:15) (cid:15) y (Ker H ( q )) / / M q (cid:15) (cid:15) M ′ yield H − ℓ − ( x (Ker H ( q )) = H − ℓ − (Ker H ( q ))= Ker H − ℓ − ( q ) = Ker f, where the second equality follows by applying the functor H − ℓ − to commutativediagram of H obtained by the factorisation of q through its kernel and image.Now, since Ker f = H − ℓ − ( x (Ker H ( q ))), by Proposition 6.7 we infer that Ker f ∈ fp( T FT − ℓ − ), as desired.(ii) We already know the existence of a homomorphism f : L ni =1 Σ − ℓ − ( y − ℓ ( R/J i ) k i ) → B having cokernel in T − ℓ (see Corollary 6.6(ii)). Let us rename the correspondingcanonical exact sequence by 0 → K → N f → B → C →
0, and let L = Im f . Since N, B ∈ fp( T FT − ℓ − ), by part (i) we know that K ∈ fp( T FT − ℓ − ) as well. In turn, L ∈ T F − ℓ − and H H ( L [ ℓ + 1]) is finitely presented being a cokernel in fp( H ), byLemma 6.3. On the other hand, since C ∈ T − ℓ , we have C [ ℓ + 1] ∈ U [1] whence H H ( C [ ℓ + 1]) = 0. This said, in the heart we have the commutative diagram withexact row H H ( B [ ℓ ]) / / (cid:31) (cid:31) (cid:31) (cid:31) ❃❃❃❃❃❃❃ H H ( C [ ℓ ]) / / (cid:30) (cid:30) (cid:30) (cid:30) ❃❃❃❃❃❃❃ H H ( L [ ℓ + 1]) / / H H ( B [ ℓ + 1]) / / H @ @ @ @ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) H ′ = = = = ③③③③③③③③ in which H H ( B [ ℓ ]) ∈ H − ℓ +2 (as we will show at the end of the proof), so that also H belongs to such TTF class of H ; then H H ( C [ ℓ ]) ∈ ⊥ H − ℓ +1 by an adaptationof Lemma 6.1(i), so that also H ′ belongs to such torsion class of H ; eventually,the remaining terms of the diagram belong to ⊥ H − ℓ ∩ fp( H ) by Lemmata 6.1(i)and 6.3. Since H is locally coherent by assumption, we infer that H ′ ∈ fp( H ).Now we take the standard cohomologies of the extension of H H ( C [ ℓ ]) to see that H − ℓ ( H H ( C [ ℓ ])) ∼ = H − ℓ ( H ′ ) ∈ fp( T FT − ℓ ) by Corollary 6.8. On the other hand, thestandard cohomology sequence of the triangle U [1] → C [ ℓ ] → H H ( C [ ℓ ]) + → providedby some object U ∈ U yields H − ℓ +1 ( U ) −→ C −→ H − ℓ ( H H ( C [ ℓ ])) −→ H − ℓ +2 ( U ) −→ C within the torsion pair ( T − ℓ +1 , F − ℓ +1 ). Therefore, H − ℓ ( H H ( C [ ℓ ])) = Σ − ℓ ( y − ℓ +1 ( C )) by means of the monomorphism induced by the natural transfor-mation σ (see Remark 6.4), and we are done.As announded above, let us now show that H H ( B [ ℓ ]) ∈ H − ℓ +2 . By Lemma 6.1(ii)there exist U ∈ U − ℓ +2 and a triangle U [1] → B [ ℓ + 1] → H H ( B [ ℓ + 1]) + → , whence H H ( H H ( B [ ℓ + 1])[ j ]) = 0 for j = − , −
2, meaning that H H ( B [ ℓ ]) = H H ( U ) = H H − ℓ +2 ( U ), as claimed. (iii) This is a consequence of part (ii). (cid:3) In order to guarantee that the heart H of a Thomason filtration of finite lengthis locally coherent, so that of study the subcategory fp( H ), one crucial issue isto determine those complexes that are finitely presented w.r.t. each torsion class ⊥ H − k of the left constituent of the TTF triples we detected in H . The followingLemma establishes the necessary conditions on the standard cohomologies of thefinitely presented objects in ⊥ H − k (cf. Theorem 6.11). Lemma 6.10.
Let Φ be a Thomason filtration of length ℓ +1 . Let L = S ℓi =0 fp( ⊥ H − i ) ,and for every object B ∈ L let r be the least nonzero cohomology degree of B . As-sume the following three hypotheses: (1) The heart H − ℓ is locally coherent; (2) For every P ∈ fp( ⊥ H − ℓ ) , in the functorial short exact sequence → L → W → P → of Lemma 6.1, we have L ∈ fp( H ) ; (3) The torsion pair ( ⊥ H − ℓ , H − ℓ ) restricts to fp( H ) .Then: (i) H r ( B ) ∈ fp( T FT r ) , and (ii) for every j = r + 1 , . . . , , it is H j ( B ) ∈ T j +2 or y j +2 ( H j ( B )) ∈ Σ T j +1 (see Lemma 6.5).Proof. Firstly, notice that since ⊥ H − i ⊆ H − i − and Hom( ⊥ H − i , H − i ) = 0 for all i = 0 , . . . , ℓ (we have H − ℓ − = H ), then for any B ∈ L there exists a unique i suchthat B ∈ fp( H ) ∩ ⊥ H − i . Secondly, we observe that it suffices to prove the claimfor r = − ℓ − j = − ℓ, . . . , i = 0 , . . . , ℓ the heart H − i is a TTF class of finite type of H − i − , thus hypothesis (1) is transferred to eachsuch heart by H − ℓ (see Remark 5.6(2)) so that we can repeat the first argumentfor every value of r and j .(i) This follows by Lemma 6.3 using hypothesis (2).(ii) We shall prove that by negating one of the claims we deduce the other one.The proof will follow several steps. First step . Assume that j = r + 1 = − ℓ . Then we have B ∈ ⊥ H − ℓ and byLemma 6.1(iii) there exist L ∈ H − ℓ +1 and a short exact sequence 0 → L → W → B → H − ℓ with L ∈ fp( H − ℓ +1 ) by hypothesis (2). As aforementioned, wesuppose that H − ℓ ( B ) / ∈ T − ℓ +2 to prove that y − ℓ +2 ( H − ℓ ( B )) ∈ Σ T − ℓ +1 . Applyingthe standard cohomology functors on the previous short exact sequence, we obtainby Lemma 4.5 an isomorphism H − ℓ − ( W ) ∼ = H − ℓ − ( B ) and an exact row0 −→ H − ℓ ( W ) −→ H − ℓ ( B ) d −→ H − ℓ +1 ( L ) f −→ H − ℓ +1 ( W ) . The R -linear map d cannot be zero, otherwise H − ℓ ( W ) ∼ = H − ℓ ( B ), but being W = H H ( H − ℓ − ( B )[ ℓ + 1]), its − ℓ th cohomology belongs to T − ℓ +2 (see the proofof Lemma 6.1(i)), contradiction by our assumption. In turn, H − ℓ +1 ( L ) is nonzero,hence − ℓ + 1 is the least nonzero cohomology degree of L , so by hypothesis (1)and Corollary 6.8 we infer that H − ℓ +1 ( L ) ∈ fp( T FT − ℓ +1 ). On the other hand,since H − ℓ +1 ( L ) ∈ F − ℓ +2 and H − ℓ ( W ) ∈ T − ℓ +2 , we have Im d = y − ℓ +2 ( H − ℓ ( B )).Moreover, Im f ∈ T − ℓ +3 , so by the functorial construction made in Remark 6.4 weobtain H − ℓ +1 ( L ) = Σ − ℓ +1 ( y − ℓ +2 ( H − ℓ ( B )))and this proves our claim. Second step . Assume that j = r + 2 = − ℓ + 1. Let us suppose that H − ℓ +1 ( B ) / ∈T − ℓ +3 to prove that y − ℓ +3 ( H − ℓ +1 ( B )) ∈ Σ T − ℓ +2 . Using the same notation of thefirst step for B, W, L , we have the exact row H − ℓ +1 ( W ) a −→ H − ℓ +1 ( B ) b −→ H − ℓ +2 ( L ) c −→ H − ℓ +2 ( W ) OCAL COHERENCE OF THOMASON HEARTS 33 where b , whence H − ℓ +2 ( L ), cannot be zero otherwise H − ℓ +1 ( B ) ∈ T − ℓ +3 , contra-diction. Consider the approximation 0 → x ( L ) → L → y ( L ) → L within theleft constituent of the TTF triple of H − ℓ +1 given by the TTF class H − ℓ +2 . Itslong exact cohomology sequence yields the isomorphism H − ℓ +1 ( x ( L )) ∼ = H − ℓ +1 ( L )together with the exact row0 −→ H − ℓ +2 ( x ( L )) −→ H − ℓ +2 ( L ) f −→ H − ℓ +2 ( y ( L )) d −→ H − ℓ +3 ( x ( L ))in which we have f = 0 as well, since otherwise H − ℓ +2 ( L ) ∼ = H − ℓ +2 ( x ( L )) ∈ T − ℓ +3 and consequently, by the previous display, also H − ℓ +1 ( B ) would be in such torsionclass, contradiction. In particular, we have H − ℓ +2 ( y ( L )) = 0. Now we have Im d ∈T − ℓ +4 and Im f ∈ T F − ℓ +2 , whence H − ℓ +2 ( y ( L )) = Σ − ℓ +2 ( y − ℓ +3 ( H − ℓ +2 ( L ))) . On the other hand, we have a commutative diagram with exact rows0 / / x − ℓ +3 (Im b ) (cid:15) (cid:15) (cid:15) (cid:15) / / Im b (cid:15) (cid:15) (cid:15) (cid:15) / / y − ℓ +3 (Im b ) λ (cid:15) (cid:15) / / / / x − ℓ +3 ( H − ℓ +2 ( L )) (cid:15) (cid:15) (cid:15) (cid:15) / / H − ℓ +2 ( L ) (cid:15) (cid:15) (cid:15) (cid:15) / / y − ℓ +3 ( H − ℓ +2 ( L )) (cid:15) (cid:15) (cid:15) (cid:15) / / / / M ′′ / / Im c / / N ′′ / / λ actually is a monomorphism, sinceits kernel belongs to F − ℓ +3 ∩ T − ℓ +3 . Let us rename the last exact column of thediagram by 0 → N ′ λ → N → N ′′ →
0. Since N ′′ [ ℓ − ∈ U [2] (for Im c ∈ T − ℓ +4 ),in the heart H we have an isomorphism H H ( N ′ [ ℓ − ∼ = H H ( N [ ℓ − H by Lemma 6.5 and theprevious display. Eventually, let us consider the short exact sequence 0 → Im a → H − ℓ +1 ( B ) → Im b →
0. We have the following commutative diagram with exactrows0 / / H ′ (cid:15) (cid:15) (cid:15) (cid:15) / / Im a (cid:15) (cid:15) (cid:15) (cid:15) / / K ′ (cid:15) (cid:15) / / x − ℓ +3 ( H − ℓ +1 ( B )) (cid:15) (cid:15) / / H − ℓ +1 ( B ) (cid:15) (cid:15) (cid:15) (cid:15) / / y − ℓ +3 ( H − ℓ +1 ( B )) γ (cid:15) (cid:15) (cid:15) (cid:15) / / / / x − ℓ +3 (Im b ) / / Im b / / N ′ / / γ is a monomorphism, hence an isomorphism, since K ′ ∈ F − ℓ +3 ∩ T − ℓ +3 .Therefore, we obtain H H ( y − ℓ +3 ( H − ℓ +1 ( B ))[ ℓ − ∼ = H H ( N ′ [ ℓ − Third step . The argument of the previous step can be repeated for all the remainingvalues of j (using again that hypothesis (1) on H − ℓ is inherited by all the hearts H − i , i = 0 , . . . , ℓ ), namely starting by approximating the complex y ( L ) withinthe left constituent of the TTF triple of H − ℓ +2 having H − ℓ +3 as TTF class, anditerating the argument step by step. (cid:3) The main results.
We are now ready to state and prove the characterisationof the local coherence of the heart associated with a Thomason filtration of finitelength of the prime spectrum of a commutative ring.
Theorem 6.11.
Let Φ be a Thomason filtration of length ℓ +1 . Then H is a locallycoherent Grothendieck category if and only if the following conditions hold true: (1) H − ℓ is a locally coherent Grothendieck category; (2) For every B ∈ fp( T FT − ℓ − ) , the functor Ext H ( H H ( B [ ℓ + 1]) , − ) commuteswith direct limits of direct systems in H − ℓ ; (3) For every B ∈ fp( T FT − ℓ − ) , the functor Ext H ( H H ( B [ ℓ + 1]) , − ) commuteswith direct limits of direct systems in ⊥ H − ℓ ; (4) For all P ∈ fp( ⊥ H − ℓ ) , in the functorial short exact sequence → L → W → P → of Lemma 6.1, we have L ∈ fp( H ) ; (5) The torsion pair ( ⊥ H − ℓ , H − ℓ ) restricts to fp( H ) .Proof. Let us assume that the heart H associated with Φ is a locally coherentGrothendieck category, and let us show that the five stated conditions hold true.(1) H − ℓ is a locally coherent Grothendieck category since it is a TTF class offinite type in H . (2) and (3) follow by Lemma 6.3 and [Sao17, Proposition 3.5(2)].(4) follows by the proof of Proposition 6.7. (5) holds true by hypothesis on H andsince the torsion pair ( H − ℓ , H ⊥ − ℓ ) is of finite type.Conversely, let us show that the five stated conditions imply the local coherenceof the heart H . More in details, we want to exploit Theorem 2.1 which charac-terises the local coherence of an arbitrary Grothendieck category equipped with aTTF triple of finite type. Notice that hypothesis (iii)’ of Theorem 2.1 coincideswith our hypothesis (5)Concerning condition (i) of Theorem 2.1, thanks to our hypothesis (1) we need tocheck that the torsion class ⊥ H − ℓ is locally coherent. We know that H is a locallyfinitely presented Grothendieck category by [SˇS20], and by imitating the proof of“(a) ⇒ (b)” in Theorem 2.1 we deduce that ⊥ H − ℓ is locally finitely presented aswell, thus it remains to prove that fp( ⊥ H − ℓ ) is closed under taking kernels; inparticular, it suffices to check that for every epimorphism p : P → P ′ in fp( ⊥ H − ℓ ),we have x (Ker H ( f )) ∈ fp( ⊥ H ) ⊆ fp( H ). The following diagram provided byLemma 6.1(iii) 0 / / L α (cid:15) (cid:15) / / W β (cid:15) (cid:15) / / P / / p (cid:15) (cid:15) (cid:15) (cid:15) / / L ′ / / W ′ / / P ′ / / D ( R ) the composition W → P p → P ′ → L [1] yield an element of Ext H ( W, L ), which is zero by Remark 6.2(2);consequently α is defined by the universal property of the kernel. By condition (4),the objects L, L ′ are finitely presented complexes of H , while W, W ′ are so byextension-closure. Moreover, since W ′ ∈ ⊥ H − ℓ by Lemma 6.1(i), then β is anepimorphism since its cokernel in H is a quotient in both the torsion classes ⊥ H − ℓ and H − ℓ . The Snake Lemma applied on the previous commutative diagram givesus the exact row0 −→ Ker H ( α ) −→ Ker H ( β ) −→ Ker H ( p ) −→ Coker H ( α ) −→ H := Ker H ( β ) is a finitely presented object. Let X = H − ℓ − ( P ) so that W = H H ( X [ ℓ + 1]) (similarly for W ′ ), and consider f := H − ℓ − ( β ), with K :=Ker f , N := Im f and C := Coker f . By applying the functor H H to the triangles K [ ℓ + 1] → X [ ℓ + 1] → N ′ [ ℓ + 1] + → and C [ ℓ ] → N [ ℓ + 1] → X ′ [ ℓ + 1] + → obtainedout of the canonical short exact sequences in R -Mod associated to f , we get the OCAL COHERENCE OF THOMASON HEARTS 35 commutative diagram of H with exact rows H H ( K [ ℓ + 1]) (cid:15) (cid:15) x x x x ♣♣♣♣♣ M ′ ' ' ' ' ❖❖❖❖❖❖❖ M ′ ' ' ' ' ❖❖❖❖❖❖❖ / / H / / (cid:15) (cid:15) % % % % ❑❑❑❑❑ W β / / δ (cid:15) (cid:15) (cid:15) (cid:15) W ′ / / M ' ' ' ' ◆◆◆◆◆◆ H H ( C [ ℓ ]) / / : : : : ttttt H H ( N [ ℓ + 1]) α / / W ′ / / β = α ◦ δ (this also implies H H ( C [ ℓ + 1]) = 0 i.e. that C ∈ T − ℓ , whence H H ( C [ ℓ ]) ∈ H − ℓ ) and the epimorphism H → M is provided by the universal prop-erty of the kernel. Moreover, by the Snake Lemma, the image M ′ of the morphism H H ( K [ ℓ + 1]) → W induces the short exact sequence 0 → M ′ → H → M → H within the torsion pair ( ⊥ H − ℓ , H − ℓ ) (seeLemma 6.1(i)). Thus, we reduced our claim to check that M ′ , M ∈ fp( H ). We have M ′ ∈ fp( ⊥ H − ℓ ) ⊆ fp( H ) by hypothesis (3) applied on the short exact sequence0 → M ′ → W → H H ( N [ ℓ + 1]) →
0, whereas M ∈ fp( H − ℓ ) ⊆ fp( H ) thanks tohypothesis (2) applied on the short exact sequence 0 → M → H H ( N [ ℓ + 1]) → W ′ → P ∈ fp( ⊥ H − ℓ )we have a short exact sequence 0 → L → W → P → L ∈ fp( H ) by hypothesis (4). The sequence yields an exact row0 → Hom H ( P, − ) → Hom H ( W, − ) → Hom H ( L, − ) − · · ·· · · → Ext H ( P, − ) → Ext H ( W, − ) → Ext H ( L, − )of covariant functors H →
Ab. When restricted to H − ℓ , the first three functors com-mute with direct limits, whereas the last two do so respectively by hypotheses (2)and (1), and by [Sao17, Proposition 3.5(2)], so that Ext H ( P, − ) ↾ H − ℓ commutes withthe desired direct limits. (cid:3) Remark 6.12.
The previous Theorem provides a recursive argument for the con-struction of a Thomason filtration of finite length whose heart is a locally coherentGrothendieck category. However, one practical issue is to check conditions (2)and (3) when the length of the filtration, i.e. ℓ , is greater than 2. Nonetheless,for 0 ≤ ℓ ≤ R -Mod and certain HRS hearts, as we have already seen),the conditions of the Theorem simplify so that most of them can be rephrased inmodule-theoretic ones, as we will show in the following results.The length zero case has been treated in subsection 4.1, and it consists in acharacterisation of the local coherence of the torsion class T X associated with theunique proper Thomason subset X of the filtration. Corollary 6.13.
Let Φ be a Thomason filtration of length . Then H is a locallycoherent Grothendieck category if and only if the following conditions are satisfied: (1) T is locally coherent; (2) For all P ∈ fp( T FT − ) , the functor Hom R ( P, − ) commutes direct limits ofdirect systems in T ; (3) For all P ∈ fp( T FT − ) , the functor Ext R ( P, − ) commutes direct limits ofdirect systems in T FT − ; (4) For all Q ∈ fp( T ) , the functor Ext R ( Q, − ) commutes with direct limits ofdirect systems in T FT − . Proof.
First, notice that H H ( P [1]) = P [1] for all P ∈ T FT − , that H = T [0] andthat ⊥ H = T FT − [1]. Thus, the stated conditions (1), (2) and (3) are exactlythe corresponding ones of Theorem 6.11, since ℓ = 0. In turn, condition (4) ofthe Theorem is clearly satisfied since L ∈ H = 0 (see Lemma 6.1). Let us checkcondition (5) of the Theorem. We claim that it is implied by our condition (4). Let B ∈ fp( H ) and consider its approximation 0 → H − ( B )[1] → B → H ( B )[0] → ⊥ H , H ) = ( T FT − [1] , T [0]); we have to prove that theouter terms are finitely presented objects of H . We recall that H ( B )[0] ∈ fp( H )by Corollary 4.4(i); in particular, we have H ( B ) ∈ fp( T ). Let ( X i ) i ∈ I be a directsystem of modules in T FT − . Applying the functors F k = lim −→ i ∈ I Ext H ( − , X i [1]) and F k = Ext H ( − , lim −→ i ∈ I X i [1]) ( k ∈ N ∪ { } )on the previous approximation, say it 0 → Y [1] → B → X [0] → / / F ( X [0]) f (cid:15) (cid:15) / / F ( B ) f (cid:15) (cid:15) / / F ( Y [1]) f (cid:15) (cid:15) / / F ( X [0]) f (cid:15) (cid:15) / / F ( B ) f (cid:15) (cid:15) / / G ( X [0]) / / G ( B ) / / G ( Y [1]) / / G ( X [0]) / / G ( B )in which, using [Ver, BBD82], f is an isomorphism by Corollary 4.4(i), f is isoand f is monic, and f is an isomorphism by hypothesis (4), so we are done by theFive Lemma.In order to conclude, it remains to prove that if H is locally coherent, then ourhypothesis (4) is satisfied. Let Q ∈ fp( T ). By Corollary 4.4(i) again, we have Q [0] ∈ fp( H ), hence Ext H ( Q [0] , − ) preserves direct limits by [Sao17, Proposition 3.5(2)];in particular, it commutes with direct limits of T FT − [1], which is our thesis by[Ver, BBD82]. (cid:3) Corollary 6.14.
Let Φ be a Thomason filtration of length . Then H is a locallycoherent Grothendieck category if and only if the following conditions are satisfied: (1) H − is locally coherent (cf. Corollary 6.13); (2) For all P ∈ fp( T FT − ) , the functor Hom R ( P, − ) preserves direct limits ofdirect systems in T FT − ; (3) The following conditions hold true: (3.i)
For all J ∈ B − , the functor Ext H ( Σ − ( y − ( R/J ))[2] , − ) preservesdirect limits of direct systems in ⊥ H − ; (3.ii) fp( T FT − ) is closed under kernels in R -Mod . (3.iii) For all morphisms f in fp( T FT − ) , we have Σ − (Im f ) / Im f ∈ R -mod ; (4) For all exact sequences of R -Mod of the form → Y → M f → N → X → such that Y ∈ fp( T FT − ) , X ∈ fg( T ) and Cone( f [1]) ∈ H , we have X ∈ fp( T ) . (5) For all P ∈ fp( H ) , the following conditions hold true: (5.i) H − ( P ) ∈ fp( T FT − ) ; (5.ii) x ( H − ( P )) ∈ fp( T ) .Proof. It is clear that our hypothesis (1) corresponds exactly to condition (1) ofTheorem 6.11.Let us prove that our hypothesis (2) is equivalent to Theorem 6.11(2). No-tice again that for all P ∈ T FT − we have H H ( P [2]) = P [2]. This said, anydirect system ( M i ) i ∈ I of H − is approximated by (0 → H − ( M i )[1] → M i → H ( M i )[0] → i ∈ I within the left constituent of the TTF triple given by the OCAL COHERENCE OF THOMASON HEARTS 37
TTF class H (see the proof of Corollary 6.13). Thus, by applying the cohomo-logical functor Hom D ( R ) ( P [2] , − ) on the direct limit of the previous approximationand using [Ver, BBD82], we obtain the commutative diagram with exact rows0 / / lim −→ i ∈ I Hom R ( P, H − ( M i )) (cid:15) (cid:15) / / lim −→ i ∈ I Ext H ( P [2] , M i ) (cid:15) (cid:15) / / / / Hom R ( P, lim −→ i ∈ I H − ( M i )) / / Ext H ( P [2] , lim −→ i ∈ I M i ) / / M ∈ H − and Y ∈ T FT − , wehave H − ( M ) ∈ T FT − and Y [1] ∈ H − .Let us show that Theorem 6.11 implies our condition (3).(3.i) Let J ∈ B − . By the approximating triangle τ ≤− ( K ( J )[2]) → K ( J )[2] → R/J [2] + → of the Koszul complex K ( J )[2] within the standard t-structure of D ( R ),since the first vertex belongs to U [3] by the proof of Lemma 6.1(iv), we obtain H H ( K ( J )[2]) ∼ = H H ( R/J [2]), and these are finitely presented objects of H by[SˇSV17, Lemma 6.3]. Let us call M such complex; it fits in an exact triangle U [1] → K ( J )[2] → M + → provided by some U ∈ U , whose standard cohomologyexact sequence yields0 −→ H − ( U ) −→ R/J d −→ H − ( M ) −→ H ( U ) −→ . On the one hand we infer that M is a stalk, i.e. M ∼ = H − ( M )[2], whence in turn H − ( M ) ∈ fp( T FT − ) by Lemma 6.3; on the other hand, we have Im d ∈ T F − and H ( U ) ∈ T , thus H − ( M ) ∼ = Σ − (Im d ) ∼ = Σ − ( y − ( R/J ))and we conclude by Lemma 6.5 and [Sao17, Proposition 3.5(2)].(3.ii) It follows by Proposition 6.9(i).(3.iii) Let f : B → B ′ be a morphism in fp( T FT − ). In view of Remark 6.4, wehave to prove that Coker σ Im f is a finitely presented R -module. We have Ker f ∈ fp( T FT − ) by part (3.ii), so by the exact sequence0 −→ H H (Im( f )[1]) −→ Ker( f )[2] −→ B [2] −→ H H (Im( f )[2]) −→ H we obtain that the outer terms are finitely presented, in particularwe infer Σ − (Im f ) ∈ fp( T FT − ) by Lemma 6.5. On the other hand, from theshort exact sequence 0 → Im f → Σ − (Im f ) → Coker σ Im f → Σ − (Im f )[0] −→ Coker( σ Im f )[0] −→ Im( f )[1] −→ Σ − (Im f )[1]whence H H (Im( f )[1]) ∼ = H H (Coker( σ Im f )[0]) = Coker( σ Im f )[0]and the latter term belongs to fp( H ). Then, by Corollary 4.4(i) we obtain thatCoker σ Im f ∈ R -mod, as desired.Conversely, let us prove that our hypotheses (2) and (3) implies Theorem 6.11(3).Let B ∈ fp( T FT − ). By Corollary 6.6(ii) there exists an R -linear map f : n M i =1 Σ − ( y − ( R/J i ) k i ) −→ B, which we rename f : N → B , with cokernel C ∈ T − and kernel K ∈ fp( T FT − )by hypothesis (3.ii). Let f = µ ◦ β be the canonical factorisation of f through its image L . Consider the following commutative diagram with exact rows in H H H ( C [1]) (cid:15) (cid:15) γ (cid:15) (cid:15) / / H H ( L [1]) λ / / K [2] (cid:15) (cid:15) v [2] / / N [2] H H ( β [2]) / / H H ( L [2]) H H ( µ [2]) (cid:15) (cid:15) (cid:15) (cid:15) / / / / H / / N [2] f [2] / / B [2] / / f [2] is an epimorphism since its cone in D ( R ) belongs to U [1], whereas λ and γ are monomorphisms since H H ( N [1]) = 0 and H H ( B [1]) = 0, respectively;moreover, notice that H H ( C [1]) ∼ = y − ( C )[1], in particular it belongs to H − . TheSnake Lemma yields a short exact sequence 0 → Im H ( v [2]) → H → y − ( C )[1] → H ( v [2]) is finitely presented for being a cokernel in fp( H ); indeed, H H ( L [1]) ∼ = Coker( σ L )[0] is finitely presented by hypothesis (3.iii) and Corol-lary 4.4(i). On the other hand, we have H H ( L [2]) = H H ( Σ − ( L )[2]) = Σ − ( L )[2]and Σ − ( L ) ∈ fp( T FT − ) by Lemma 6.5; moreover, by our condition (2) (i.e. The-orem 6.11(2)) we infer that y − ( C )[1] ∈ fp( H − ) ⊆ fp( H ). By extension-closure,we have H ∈ fp( H ) as well. Thus, the second exact row of the previous diagraminduces the exact sequence of covariant functors0 → Hom H ( B [2] , − ) → Hom H ( N [2] , − ) → Hom H ( H, − ) − · · ·· · · → Ext H ( B [2] , − ) → Ext H ( N [2] , − ) → Ext H ( H, − )in which, since Ext H ( N [2] , − ) restricted to ⊥ H − preserves direct limits by (3.i),then also Ext H ( B [2] , − ) ↾ does so, as desired.Let us prove that Theorem 6.11 implies our condition (4). First notice that if X ∈ fg( T ), then there exists B ∈ fp( T ) and an epimorphism p : B → X , whencea short exact sequence 0 → Ker( p )[0] → B [0] → X [0] → H , which shows that X [0] ∈ fg( H ). Let now 0 → Y → M f → N → X → D ( R ) Y [2] (cid:15) (cid:15) M [1] f [1] / / N [1] / / Cone( f [1]) + / / (cid:15) (cid:15) X [1] + (cid:15) (cid:15) and the rotation of the vertical triangle is a short exact sequence of H by hypoth-esis on the cone. In particular, by 0 → X [0] → Y [2] → Cone( f [1]) →
0, being X [0] ∈ fg( H ) and Y [2] ∈ fp( H ) (see Lemma 6.5(d)), we infer that Cone( f [1])is a finitely presented object of H . By [Sao17, Proposition 3.5(2)], the functorExt H (Cone( f [1]) , − ) commutes with direct limits, in particular those of T [0], butthe relevant restriction of the functor is naturally isomorphic to Hom R ( X, − ) ↾ T ,and we are done. OCAL COHERENCE OF THOMASON HEARTS 39
Let us prove that our conditions (4) and (5.i) implies Theorem 6.11(4). Let P ∈ fp( ⊥ H − ), and consider the associated short exact sequence 0 → L ε → W → B → L ∈ H and W = H − ( B )[2]. By hypothesis (5.i)and Lemma 6.5 we know that W ∈ fp( H ), thus L ∈ fg( H ). Therefore, there existsan epimorphism Q → L originating in a finitely presented complex Q of H , whencewe have the epimorphism H ( Q ) → H ( L ) originating in H ( Q ) ∈ fp( T ), whence H ( L ) ∈ fg( T ). Now, since ε is a morphism inHom D ( R ) ( L, W ) ∼ = Hom D ( R ) ( H ( L )[0] , H − ( W )[2]) ∼ = Ext R ( H ( L ) , H − ( W ))it is represented by an exact sequence0 −→ H − ( W ) −→ X f −→ X −→ H ( L ) −→ R -Mod, in which Cone( f [1]) ∼ = B . By (4.ii), we deduce that H ( L ) ∈ fp( T ), i.e. L ∼ = H ( L )[0] ∈ fp( H ) by Corollary 4.4(i).It remains to treat condition (5). Part (5.i) has been proved in Corollary 6.8.On the other hand, for any P ∈ fp( H ) consider the approximation 0 → x ( P ) → P → y ( P ) → ⊥ H − , H − ). Its cohomology long exactsequence breaks up in the following exact rows of R -Mod:0 −→ H − ( x ( P )) −→ H − ( P ) −→ −→ H − ( x ( P )) −→ H − ( P ) −→ H − ( y ( P )) −→ −→ H ( P ) −→ H ( y ( P )) −→ H ( x ( P )) = 0, but this follows since theepimorphism x ( P ) → H ( x ( P ))[0] is zero by axiom of torsion pair. This said,we have H − ( x ( P )) ∈ T since x ( P ) ∈ ⊥ H − , and H − ( y ( P )) ∈ T FT − ⊆ F .Therefore, by the second displayed exact row we deduce H − ( x ( P )) ∼ = x ( H − ( P )).Moreover, by rotating the approximation of x ( P ) within the standard t-structureof D ( R ) we obtain the short exact sequence0 −→ x ( H − ( P ))[0] −→ H − ( P )[2] −→ x ( P ) −→ H . Now, bearing in mind part (5.i), if H is locally coherent, then our condi-tion (5.ii) holds true by Corollary 4.4(i); conversely, if x ( H − ( P )) ∈ fp( T ), then x ( P ) ∈ fp( H ) for being a cokernel of a morphism in fp( H ). (cid:3) Applications
We apply Corollary 6.13 in the case of the HRS heart H associated with a torsionpair ( T , F ) in R -Mod; indeed, in Example 4.11 we saw that H can be realised asthe heart associated with the Thomason filtration Φ of length 1 defined by Φ ( n ) = Spec R if n ≤ − ,Φ (0) ∅ if n ≥ Φ (0) is the Thomason subset that corresponds to the torsion class T .The following crucial necessary condition for the local coherence ensures that( T , F ) must be hereditary of finite type. Notice that this follows by [HˇS17, Propo-sition 2.6] since the locally finite presentability of the heart is equivalent to T =lim −→ fp( T ) by [PSV19]; however, we now achieve such result with a different argu-ment. Proposition 7.1.
Let ( T , F ) be a torsion pair in R -Mod . If the associated HRS heart H is a locally finitely presented Grothendieck category, then ( T , F ) is hereditary (offinite type). Proof.
By [PS15] the torsion pair is necessarily of finite type; moreover, since H is locally finitely presented, by [PSV19, Theorem 5.1, Proposition 1.14] we have inparticular T = lim −→ ( T ∩ R -mod). Therefore, T ∩ R -mod is a set (up to isomorphism),whose right orthogonal in R -Mod coincides with F , hence by [BP18, Theorem 3.3]( T , F ) is a tCG torsion pair ; that is, its HRS t-structure ( U , V ) in D ( R ) is com-pactly generated. Consequently, by [Hrb18, Theorem 5.1] there exists a Thomasonfiltration Φ such that ( U , V ) = ( U Φ , V Φ ). We claim that T = T , whence T turnsout to be a hereditary torsion class. This readily follows thanks to the equality U = U Φ , namely by taking the 0th cohomology of the stalk X [0] for a module X either in T or in T . (cid:3) Remark 7.2.
We recall that the converse of the previous result is known in theliterature (see [GP08, Theorem 2.2] and [Hrb18, PSV19, SˇS20]).
Corollary 7.3.
Let ( T , F ) be a torsion pair in R -Mod , say with adjunctions T ֒ −→←−− x R -Mod y −−→←− ֓ F . The associated HRS heart H is a locally coherent Grothendieck category if and onlyif ( T , F ) is hereditary of finite type and the following four conditions hold: (i) The torsion class T is locally coherent; (ii) For every B ∈ R -mod , the functor Hom R ( y ( B ) , − ) commutes with directlimits of direct systems in T ; (iii) For all B ∈ R -mod , the functor Ext R ( y ( B ) , − ) commutes with direct limitsof direct systems of F ; (iv) For every finitely generated ideal J in the Gabriel filter associated with T ,the functor Ext R ( R/J, − ) commutes with direct limits of F .Proof. The necessity of the torsion pair being hereditary and of finite type hasbeen proved in Proposition 7.1; this said, we shall prove the present Corollary byshowing that the listed four conditions are equivalent to the corresponding ones ofCorollary 6.13.It is clear that our hypothesis (i) is precisely Corollary 6.13(1). On the otherhand, we have
T FT − = T F − = F , thus fp( T FT − ) = add y ( R -mod) (see Re-mark 4.2). The previous equality together with the additivity of the bifunctorsHom R ( − , − ) and Ext R ( − , − ) shows that also our hypotheses (ii) and (iii) areequivalent to the corresponding conditions of Corollary 6.13. Moreover, it is clearthat Corollary 6.13(4) implies our condition (iv). Let us prove that our hypothe-ses (i) and (iv) implies Corollary 6.13(4). Let Q ∈ fp( T ) and let ( Y i ) i ∈ I be adirect systems of modules in F . By Proposition 3.2 there exist a finitely generatedideal J in the Gabriel filter of the torsion pair ( T , F ) and a short exact sequence0 → X → ( R/J ) n → Q → n ∈ N and X a torsion module (we have X ∈ fp( T ) by (i)). By applying the functors L k = lim −→ i ∈ I Ext kR ( − , Y i ) and Γ k = Ext kR ( − , lim −→ i ∈ I Y i ) ( k ≥ / / L ( Q ) g (cid:15) (cid:15) / / L (( R/J ) n ) g (cid:15) (cid:15) / / L ( X ) g (cid:15) (cid:15) / / L ( Q ) g (cid:15) (cid:15) / / L (( R/J ) n ) g (cid:15) (cid:15) / / L ( X ) g (cid:15) (cid:15) / / Γ ( Q ) / / Γ (( R/J ) n ) / / Γ ( X ) / / Γ ( Q ) / / Γ (( R/J ) n ) / / Γ ( X ) OCAL COHERENCE OF THOMASON HEARTS 41 in which g , g , g are isomorphisms since Q [0] , ( R/J ) n [0] , X [0] ∈ fp( H ) by Corol-lary 4.4(i), g is iso by condition (iv), while g is a monomorphism by [PSV19,Lemma 1.3], so that g is iso as well. This concludes the proof. (cid:3) Remark 7.4.
A more general characterisation of the local coherence of the HRS heartshas been achieved in [PSV19, Sec. 6] in the context of locally finitely presentedGrothendieck categories.7.1.
When the ring is coherent.
When the ring is coherent, our previous char-acterisations furtherly lighten, as we shall prove in Corollary 7.6. Let us start withan interesting example.
Example 7.5.
Let R be a commutative coherent ring, and let ( X , Y ) be a torsionpair in the abelian category R -mod; by [CB94, p. 1666] (lim −→ X , lim −→ Y ) =: ( T , F ) isa torsion pair (of finite type) in R -Mod. We claim that the associated HRS heartin D ( R ) is a locally coherent Grothendieck category, namely by showing that thetorsion pair is hereditary and satisfies the four conditions of the previous Corollary.The torsion pair ( T , F ) is hereditary by the same argument of the proof ofProposition 7.1, namely for it is a t CG torsion pair.(i) Since R -Mod is a locally coherent Grothendieck category, then T is so (seeRemark 5.6).On the other hand, the torsion pair ( T , F ) restricts to R -mod, so y ( B ) is afinitely presented module for all B ∈ R -mod, whence it is clear that conditions (ii),(iii) and (iv) of the Corollary hold true, since R is coherent. Corollary 7.6.
Let R be a commutative coherent ring and ( T , F ) be a torsion pairin R -Mod . Then the HRS heart of the torsion pair is a locally coherent Grothendieckcategory if and only if (i) The torsion pair is hereditary of finite type; (ii)
For all B ∈ R -mod , the functor Hom R ( y ( B ) , − ) commutes with direct lim-its of direct systems in T ; (iii) For all B ∈ R -mod , the functor Ext R ( y ( B ) , − ) commutes with direct limitsof direct systems of F . Question 7.7.
Let R be a commutative coherent ring and ( T , F ) be a torsionpair whose HRS heart is a locally coherent Grothendieck category. Then does thetorsion pair necessarily restrict to R -mod?7.2. When the torsion pair is stable.
We equip the torsion pairs of R -Mod witha homological condition, i.e. we consider the case of stable torsion pairs, so thateven the torsion classes are closed under taking injective envelopes. As we shallsee, such a homological condition translates into a finiteness one and, in particu-lar, the necessary and sufficient conditions for the local coherence of the involvedHRS hearts simplifies furtherly. In fact, our assumption is consistent and indepen-dent from the previous subsection, thanks to the following example which exhibitsa non-trivial stable torsion pair over a non-coherent commutative ring. Example 7.8.
Consider the non-coherent commutative ring R = Z ⊕ ( Z / Z ) ( N ) introduced in Example 4.10. For any nonzero tuple a ∈ ( Z / Z ) ( N ) , the non unitaryelement e = (1 , a ) is idempotent, and the ideal J = Re is idempotent as well.Therefore, J gives rise to a TTF triple ( E , T , F ) in R -Mod which is split ; thatis (see [Ste75, Proposition VI.8.5]), in which E = F and both the torsion pairs( T , F ) and ( F , T ) are hereditary. In particular, ( T , F ) is of finite type for being F = Ker Hom R ( R/J, − ), and stable for ( F , T ) being hereditary.We need some auxiliary preliminary results, which in fact specialise the condi-tions of Corollary 7.3 within the stability assumption. Lemma 7.9. If ( T , F ) is a stable hereditary torsion pair of R -Mod , then for every X, Y ∈ T we have
Ext kR ( X, Y ) ∼ = Ext k T ( X, Y ) , for all k ∈ N ∪ { } .Proof. By the adjuction : T ֒ −→←−− R -Mod : x and by [NS14, Proposition 2.28], wehave the adjoint pair L : D ( T ) −−→←−− D ( R ) : R x of derived functors. In particular, for all X, Y ∈ T , regarding the stalk of X as anobject of D ( T ) and the stalk of Y as a complex of D ( R ), being x an exact functorby hereditariness, we have the natural isomorphismHom D ( R ) ( L ( X [0]) , Y [ n ]) ∼ = Hom D ( T ) ( X [0] , R x ( Y [ n ]))= Hom D ( T ) ( X [0] , x ( i Y [ n ])) ∼ = Hom D ( T ) ( X [0] , i Y [ n ]) ∼ = Hom D ( T ) ( X [0] , Y [ n ]) , where i is the homotopically injective coresolution functor, computed equivalentlyeither on D ( R ) or in D ( T ), for T being a stable torsion class and an exact sub-category of R -Mod. By [Ver], the latter group of the display is isomorphic toExt n T ( X, Y ), so we claim that the first displayed group is isomorphic to Ext nR ( X, Y ).Indeed, we haveHom D ( R ) ( L ( X [0]) , Y [ n ]) = Hom D ( R ) ( ( p X [0]) , Y [ n ]) ∼ = Hom D ( R ) ( p X [0] , Y [ n ]) ∼ = Hom D ( R ) ( X [0] , Y [ n ]) , where p : D ( R ) → K ( R ) is the homotopically projective resolution functor. (cid:3) Lemma 7.10.
Let ( T , F ) be a stable torsion pair of R -Mod . Assume that con-ditions (i) and (iv) of Corollary 7.3 hold true. Then, for every finitely generatedideal J in the Gabriel filter associated with T , it is R/J ∈ FP ( R ) , i.e. the functors Ext kR ( R/J, − ) commute with direct limits for k = 0 , , .Proof. Let ( M i ) i ∈ I be a direct system in R -Mod and consider the direct system(0 → X i → M i → Y i → i ∈ I formed by the approximations of its members within( T , F ). Since R/J is a finitely presented torsion module (so that
R/J [0] is a finitelypresented object of H ), by applying the functors L k = lim −→ i ∈ I Ext kR ( R/J, − ) and Γ k = Ext kR ( R/J, lim −→ i ∈ I ( − )) ( k ∈ N ∪ { } )on the latter direct system, we find at once that L ( M i ) ∼ = Γ ( M i ); moreover, inthe following commutative diagram with exact rows, L ( X i ) f (cid:15) (cid:15) / / / / L ( M i ) f (cid:15) (cid:15) / / L ( Y i ) f (cid:15) (cid:15) / / L ( X i ) f (cid:15) (cid:15) / / L ( M i ) f (cid:15) (cid:15) / / L ( Y i ) f (cid:15) (cid:15) / / L ( X i ) f (cid:15) (cid:15) Γ ( X i ) / / / / Γ ( M i ) / / Γ ( Y i ) / / Γ ( X i ) / / Γ ( M i ) / / Γ ( Y i ) / / Γ ( X i )the canonical maps f , f , f are isomorphisms by hypothesis (i) and Lemma 7.9, f is an isomorphism by hypothesis (iv), while f is iso as well by [Ver, BBD82]and since R/J [0] ∈ fp( H ). Therefore, by the Five Lemma we deduce that f and f are isomorphisms, as desired. (cid:3) Remark 7.11.
By [GT12, Lemma 2.14] every indexing set I is the union of a well-ordered chain of directed subposets ( I α | α < λ ), where each I α has cardinality less OCAL COHERENCE OF THOMASON HEARTS 43 than I . Moreover, for every direct system ( M i ) i ∈ I of R -modules, (lim −→ i ∈ I α M i | α < λ )is a well-ordered direct system satisfyinglim −→ i ∈ I M i = lim −→ α<λ lim −→ i ∈ I α M i . Lemma 7.12.
Let ( T , F ) be a stable torsion pair in R -Mod . Assume that condi-tion (ii) of Corollary 7.3 holds true. Then, for every B ∈ R -mod and every directsystem ( M i ) i ∈ I in T , the canonical homomorphism lim −→ i ∈ I Ext R ( y ( B ) , M i ) −→ Ext R ( y ( B ) , lim −→ i ∈ I M i ) is injective.Proof. We formerly prove the statement in case I is a well ordered directed poset.If I is a finite set, there exist indeces ¯ ı, ¯ ∈ I such that lim −→ i ∈ I M i = M ¯ ı andlim −→ i ∈ I Ext R ( y ( B ) , M i ) = Ext R ( y ( B ) , M ¯ ); moreover, there exists k ≥ ¯ ı, ¯ makingthe displayed canonical map an isomorphism indeed.If I is infinite, by [Hrb18, Lemma 3.5] there exists a direct system (0 → M i → E i → C i → i ∈ I in which E i is the injective envelope of M i , so that the directsystem is in T by the stability hypothesis. Therefore, the canonical homomorphismdisplayed in the statement factors through the kernel of the mapExt R ( y ( B ) , lim −→ i ∈ I M i ) −→ Ext R ( y ( B ) , lim −→ i ∈ I E i )by means of an isomorphism, thanks to the Snake Lemma and the assumption on y ( B ). In other words, our statement is true for well ordered directed posets.This said, the general case follows as soon as we write I = S α<λ I α as in Re-mark 7.11; indeed, by the argument of the previous part (applied twice) and byAB-5 condition of abelian groups, we obtain the following composition of monomor-phismslim −→ α<λ lim −→ i ∈ I α Ext R ( y ( B ) , M i ) ֒ −→ lim −→ α<λ Ext R ( y ( B ) , lim −→ i ∈ I α M i ) ֒ −→ Ext R ( y ( B ) , lim −→ α<λ lim −→ i ∈ I α M i ) , which coincides with the natural map of the statement. (cid:3) Corollary 7.13.
Let ( T , F ) be a stable torsion pair in R -Mod . Then its HRS heart H is a locally coherent Grothendieck category if and only if the torsion pair is offinite type and the following three conditions hold: (i) fp( T ) ⊆ FP ( R ) ; (ii) fp( F ) ⊆ R -mod ; (iii) For all B ∈ R -mod , the functor Ext R ( y ( B ) , − ) commutes with direct limitsof direct systems of F .Proof. We shall prove that the stated conditions are equivalent to the ones of Corol-lary 7.3. Let us start by proving that our three hypotheses imply the conditions ofthe Corollary.(i) By Proposition 4.6, Proposition 7.1, and [SˇS20], T is a locally finitely pre-sented Grothendieck category. It remains to show that fp( T ) is an exact abeliansubcategory of T , and this follows by condition (i), namely for the kernel of anyepimorphism in fp( T ) is finitely presented as well.(ii) It follows immediately by our hypothesis (ii).(iv) It follows immediately by our hypothesis (i).Let us prove that the four conditions of Corollary 7.3 imply our hypotheses (i)and (ii).(i) Let B ∈ fp( T ); by Corollary 4.4 there exist finitely generated ideals J ′ , J in theGabriel filter associated with T and an exact row ( R/J ′ ) n α → ( R/J ) m → B → for some n, m ∈ N . By Lemma 7.10, R/J and
R/J ′ are objects of FP ( R ), thus,being Ker α a finitely presented torsion module by Corollary 7.3(i), in view e.g. of[BPz16] we infer that Im α ∈ FP ( R ) and consequently that B ∈ FP ( R ).(ii) Since fp( F ) = add y ( R -mod), we shall prove our assertion (ii) on torsionfreemodules of the form y ( B ), where B ∈ R -mod. Let ( M i ) i ∈ I be a direct systemin R -Mod and consider its approximation (0 → X i → M i → Y i → i ∈ I within( T , F ). By applying the functors L k = lim −→ i ∈ I Ext R ( y ( B ) , − ) and Γ k = Ext kR ( y ( B ) , lim −→ i ∈ I ( − )) ( k ∈ N ∪ { } )on the latter direct system, we obtain0 / / L ( X i ) g (cid:15) (cid:15) / / L ( M i ) g (cid:15) (cid:15) / / L ( Y i ) g (cid:15) (cid:15) / / L ( X i ) g (cid:15) (cid:15) / / Γ ( X i ) / / Γ ( M i ) / / Γ ( Y i ) / / Γ ( X i )in which g is an isomorphism by Corollary 7.3(ii), g is isomorphism since y ( B ) ∈ fp( F ), and g is monic by Lemma 7.12. By the Five Lemma, we conclude that y ( B ) is a finitely presented module. (cid:3) Example 7.14.
Let us show that the torsion pair ( T , F ) of Example 7.8, eventhough restricts and splits, does not have a locally coherent HRS heart. Assume,by contradiction, that such H is locally coherent. Since the ring R is non-coherent,there exists an R -linear epimorphism f : M → N in R -mod such that Ker f is notfinitely presented. In the exact row0 −→ Ker x ( f ) −→ Ker f a −→ Ker y ( f ) −→ Coker x ( f ) −→ y ( f ) ∈ fp( F ) by conditions (ii) and (iii)of the previous Corollary. On the other hand, Ker x ( f ) , Coker x ( f ) ∈ fp( T ) sincethe torsion pair is split, so that x ( M ) and x ( N ) are finitely presented objects of T ,which is locally coherent (as proved in the previous Corollary). By hypotheses (i)and (ii) we infer that Im a ∈ R -mod, thus we get the contradiction Ker f ∈ R -modby the extension-closure of the finitely presented modules. References [AJS10] L. Alonso Tarr´ıo, A. L´opez Jeremias, M. Saor´ın,
Compactly generated t-structures onthe derived category of a noetherian ring , Journal of Algebra, (3):313–346, 2010[AHH16] L. Angeleri H¨ugel, M. Hrbek,
Silting Modules over Commutative Rings , InternationalMathematics Research Notices, Vol. 2017, No. 13, pp. 4131–4151[AHMV17] L. Angeleri H¨ugel, F. Marks, J. Vit´oria,
Torsion pairs in silting theory , Pacific Journalof Mathematics, Vol. (2017), No. 2, 257–278[BBD82] A. Beilinson, J. Bernstein, P. Deligne,
Faisceaux Pervers , in “
Analysis and topologyon singular spaces ”, I, Luminy 1981, Ast`erisque , Soc. Math. France, Paris (1982),5–171[BR07] A. Beligiannis, I. Reiten,
Homological and Homotopical Aspects of Torsion Theories ,Mem. Amer. Math. Soc. (2007), no. 883, viii+207[Bon19] M. Bondarko,
On perfectly generated weight structures and adjacent t -structures ,Preprint available at arXiv:1909.12819[BP18] D. Bravo, C. E. Parra, tCG torsion pairs , J. Algebra Appl. Vol. , no. 17 (2019)[BP19] D. Bravo, C. E. Parra, Torsion pairs over n -hereditary rings , Communications inAlgebra, 47:5, 1892–1907 (2019)[BPz16] D. Bravo, M. P´erez, Finiteness conditions and cotorsion pairs , Journal of Pure andApplied Algebra, (2017), 1249–1267[CGM07] R. Colpi, E. Gregorio, F. Mantese,
On the Heart of a faithful torsion theory , Journalof Algebra, (2007), 841–863[CMT11] R. Colpi, F. Mantese, A. Tonolo,
When the heart of a faithful torsion pair is a modulecategory , Journal of Pure and Applied Algebra, (2011), 2923–2936
OCAL COHERENCE OF THOMASON HEARTS 45 [CB94] W. Crawley-Boevey,
Locally finitely presented additive categories , Comm. Algebra, (5):1641–1674, 1994[GP08] G. Garkusha, M. Prest, Torsion classes of finite type and spectra , K-theory andNoncommutative Geometry, pages 393–412, 2008[GT12] R. G¨obel, J. Trlifaj,
Approximations and Endomorphism Algebras of Modules , Twovolumes, De Gruyter Expositions in Mathematics 41, 2012[Gro13] M. Groth,
Derivators, pointed derivators and stable derivators , Algebraic & Geomet-ric Topology (2013), 313–374[Gro57] A. Grothendieck, Sur quelques points d’alg`ebre homologique , Tˆohoku Math. J. , vol. 9no. 2, 119–221, 1957[HRS96] D. Happel, I. Reiten, S. O. Smalø, Tilting in abelian categories and quasitilted alge-bras , Mem. Amer. Math. Soc. (1996)[Her97] I. Herzog,
The Ziegler spectrum of a locally coherent Grothendieck category , Proc.Lond. Math. Soc. (3) 1997, 503–558[Hrb18] M. Hrbek, Compactly generated t-structures in the derived category of a commutativering , Mathematische Zeitschrift, month 6, 2019[HˇS17] M. Hrbek, J. ˇSˇtov´ıˇcek,
Tilting classes over commutative rings , Preprint available atArXiv:1701.0534v1[Kra97] H. Krause,
The spectrum of a locally coherent category , J. Pure Appl. Algebra (3),1997, 259–271[Kra00] H. Krause,
Smashing subcategories and the telescope conjecture — an algebraic ap-proach , Invent. math. , 99–133 (2000)[Lak18] R. Laking,
Purity in compactly generated derivators and t-structures withGrothendieck hearts , Mathematische Zeitschrift (2019), https://doi.org/10.1007/-s00209-019-02411-9 [Lam99] T. Y. Lam,
Lectures on Modules and Rings , GTM 189, Springer-Verlag New York,1999[MT12] F. Mantese, A. Tonolo,
On the heart associated with a torsion pair , Topology and itsApplications, (2012), 2483–2489[Mil] D. Miliˇci´c,
Lectures on Derived Categories , eprint available at [Nee01] A. Neeman,
Triangulated Categories , Ann. Math. Stud., vol. 148, Princeton UniversityPress, 2001[NS14] P. Nicol´as, M. Saor´ın,
Classical derived functors as fully faithful embeddings ,Proc. 46th Japan Symposium on Ring Theory and Representation Theory (Tokyo2013), editor Kikumasa, I. Yamaguchi University (2014), 137–187[Nor68] D. G. Northcott,
Lessons on Rings, Modules and Multiplicities , Cambridge UniversityPress, 1968[PS15] C. E. Parra, M. Saor´ın,
Direct limits in the heart of a t-structure: the case of a torsionpair , J. Pure Appl. Algebra , 4117–4143 (2015)[PS16a] C. E. Parra, M. Saor´ın,
Addendum to “Direct limits in the heart of a t-structure: thecase of a torsion pair” , J. Pure Appl. Algebra , 2567–2469 (2016)[PS16] C. E. Parra, M. Saor´ın,
On hearts which are module categories , J. Math. Soc. Japan,Volume , Number 4 (2016), 1421-1460.[PS17] C. E. Parra, M. Saor´ın, Hearts of t-structures in the derived category of a commutativenoetherian ring , Trans. Amer. Math. Soc. (2017), 7789–7827[PS20] C. E. Parra, M, Saor´ın,
The HRS tilting process and Grothendieck hearts of t-structures , Contemporary Mathematics, to appear (2021). Preprint available atarXiv:2001.08638[PSV19] C. E. Parra, M. Saor´ın, S. Virili,
Locally finitely presented and coherent hearts ,preprint available at arXiv:1908.00649v1[Pop73] N. Popescu,
Abelian Categories with Applications to Rings and Modules , AcademicPress, London, 1973[Sao17] M. Saor´ın,
On locally coherent hearts , Pacific Journal of Mathematics, Vol. (2017), No. 1, 199–221[SˇS20] M. Saor´ın, J. ˇSˇtov´ıˇcek, t-Structures with Grothendieck hearts via functor categories ,preprint available at arXiv:2003.01401v1[SˇSV17] M. Saor´ın, J. ˇSˇtov´ıˇcek, S. Virili, t-Structures on stable derivators and Grothendieckhearts , preprint available at arXiv:1708.07540v2[Ste75] B. B. Stenstr¨om,
Rings of Quotients , Grundleheren der Math. 217, Springer-Verlag,1975[ˇSto14] J. ˇSˇtov´ıˇcek,
Derived equivalences induced by big cotilting modules , Adv. Math. (2014):45–87 [Ver] J. L. Verdier,
Des Cat´egories D´eriv´es des Cat´egories Ab´eliennes , Ast`erisque , Soc.Math. France, Marseilles (1996)(Lorenzo Martini)
Dipartimento di Matematica, Universit`a di Trento, Via Sommarive14, 38123 Povo (Trento), Italy
Email address : [email protected] Dipartimento di Informatica – Settore di Matematica, Universit`a di Verona, Stradale Grazie 15 – Ca’ Vignal, I-37134 Verona, Italy
Email address : [email protected] (Carlos E. Parra) Universidad Austral de Chile, Instituto de Ciencias F´ısicas y Mate-m´aticas, Edificio Emilio Pugin, Campus Isla Teja, 5090000 Valdivia, Chile
Email address ::