Gluing compactly generated t-structures over stalks of affine schemes
aa r X i v : . [ m a t h . A C ] J a n GLUING COMPACTLY GENERATED T-STRUCTURES OVER STALKS OFAFFINE SCHEMES
MICHAL HRBEK, JIANGSHENG HU, AND RONGMIN ZHU
Abstract.
We show that compactly generated t-structures in the derived category of a com-mutative ring R are in a bijection with certain families of compactly generated t-structures overthe local rings R m where m runs through the maximal ideals in the Zariski spectrum Spec( R ) .The families are precisely those satisfying a gluing condition for the associated sequence ofThomason subsets of Spec( R ) . As one application, we show that the compact generation of ahomotopically smashing t-structure can be checked locally over localizations at maximal ideals.In combination with a result due to Balmer and Favi, we conclude that the ⊗ -Telescope Con-jecture for a quasi-coherent and quasi-separated scheme is a stalk-local property. Furthermore,we generalize the results of Trlifaj and Şahinkaya and establish an explicit bijection betweencosilting objects of cofinite type over R and compatible families of cosilting objects of cofinitetype over all localizations R m at maximal primes. Introduction
The notion of a t-structure was introduced by Be˘ılinson, Bernstein, and Deligne [13] in theirstudy of perverse sheaves on an algebraic or analytic variety as a tool for constructing cohomo-logical functors. Later, t-structures turned out to be a natural framework for tilting theory oftriangulated categories, see [4], [6] and [30]. Such t-structures usually satisfy some kind of finite-ness condition, see e.g. [28]. The compactly generated t-structures have been studied in depthand in some cases are known to allow for a full classification. For derived categories of commuta-tive noetherian rings, a bijective correspondence between compactly generated t-structures andfiltrations of the Zariski spectrum by supports was established by Alonso Tarrío, Jeremías Lópezand Saorín [1]. This was further generalized by the first author to arbitrary commutative rings[20] using filtrations by Thomason sets.Silting theory can be viewed as an adaptation of tilting theory to triangulated categories, andthe modern versions of silting theory rely heavily on the notion of a t-structure. Indeed, anycosilting object C is up to equivalence determined by the cosilting t-structure ( ⊥ ≤ C, ⊥ > C ) .A strong relation between cosilting t-structures and compactly generated t-structures followsfrom a result by Laking [26, Theorem 4.6]. In particular, any compactly generated t-structureis induced by a cosilting object if and only if it is non-degenerate. As a consequence, we calla cosilting object C of cofinite type if the t-structure induced by it is compactly generated,see [3]. Such cosilting objects are abundant, for example, any bounded cosilting complex over Mathematics Subject Classification.
Primary: 13D09, 18G80; Secondary: 13D30, 13B30, 13C05.
Key words and phrases.
Derived category, Thomason set, telescope conjecture, silting complex, cosiltingcomplex.Michal Hrbek was supported by the GAČR project 20-13778S and RVO: 67985840. Jiangsheng Hu was sup-ported by NSFC grant 11771212. Rongmin Zhu was supported by NSFC grant 11771202. commutative noetherian ring is of cofinite type [21, Corollary 2.14]. Recently, Trlifaj andŞahinkaya [38] constructed a bijective correspondence between (equivalence classes of) n-cotiltingmodules over a commutative noetherian ring R and (equivalence classes of) compatible familiesof their colocalizations in all maximal ideals of R . This result is the starting point of our inquiryto gluing properties of compactly generated t-structures. The notion of colocalization in Mod - R is due to Melkerson and Schenzel (cf. [40, p.118]) with similar constructions already used in [15],and a derived version of colocalization will play an essential role in our approach as well.A Bousfield localization of a triangulated category is called smashing if it commutes withall coproducts. The Telescope Conjecture (TC) originates from the work of Ravenel in alge-braic topology [32] and asks whether any such smashing localization is generated by compactobjects. It is a landmark result of Neeman [29] that (TC) holds in the derived category D ( R ) of a commutative noetherian ring R . On the other hand, Keller [23] established examples ofnon-noetherian commutative rings for which (TC) fails. For non-stable t-structures, a general-ization of the smashing property was introduced by Saorín, Šťovíček and Virili [35]. This classof homotopically smashing t-structures encompasses both the smashing Bousfield localizationsand the t-structures induced by pure-injective cosilting objects, this follows from the work ofKrause [25] and Laking [27]. Recently, the first author and Nakamura showed in [21] that anyhomotopically smashing t-structure in the derived category of a commutative noetherian ring iscompactly generated, which generalizes the validity of (TC) for commutative noetherian rings.As a consequence, we say that the derived category D ( R ) of a (not necessarily noetherian) ring R satisfies the Semistable Telescope Conjecture (STC) if any homotopically smashing t-structureis compactly generated.The aim of this paper is to glue compactly generated t-structures and cosilting objects ofcofinite type over all (co)localizations at maximal ideals, and to study the stalk-local propertiesof the (Semistable) Telescope Conjecture. For this purpose, we use the description of these t-structures of [1] and [20] by geometric invariants and first introduce the “compatibility” conditionfor the family { X ( m ) | m ∈ mSpec( R ) } of Thomason filtrations, which we then demonstrate tocorrespond precisely to the case in which this collection naturally glues over the cover of Spec( R ) by the subsets homeomorphic to Spec( R m ) (see Definition 3.9 and Proposition 3.11). We remarkthat our gluing condition Definition 3.9 is nothing but a suitable generalization of the conditionused in [38] to the setting of rings which are not necessarily noetherian, see Remark 3.10.Let us briefly list the highlights of the present paper, which are all obtained using the gluingtechnique described above.(1) In Theorem 3.15 we glue (non-degenerate) compactly generated t-structures over all lo-calizations at maximal ideals which is based on the gluing of the corresponding Thomasonfiltrations via Proposition 3.11. More specifically, it is proved that there is a bijective cor-respondence between (non-degenerate) compactly generated t-structures ( U , V ) in D ( R ) and compatible families { ( U ( m ) , V ( m )) | m ∈ mSpec( R ) } of (non-degenerate) compactlygenerated t-structures.(2) In Theorem 4.5 we obtain a stalk-local criterion for (Semistable) Telescope Conjectureby applying the local-global property of compact generation established in Proposition .13. More precisely, for any commutative ring R , it is proved that (Semistable) Tele-scope Conjecture holds in D ( R ) if and only if (Semistable) Telescope conjecture holdsin D ( R m ) for any maximal ideal m of R . One corollary is that both (TC) and (STC)hold for any commutative ring R all of which stalks R m are noetherian. Examples ofsuch rings include non-noetherian rings like von Neumann regular rings, recovering aresult of Bazzoni-Šťovíček, see [11, §7]. Our result also has consequences for non-affineschemes. Indeed, in combination with the result of Balmer and Favi we obtain that the ⊗ -Telescope Conjecture ( ⊗ TC) for a quasi-coherent and quasi-separated scheme is notjust affine-local, but even a stalk-local property, see §4.1.(3) We establish a bijective correspondence between cosilting objects in D ( R ) of cofinite typeup to equivalence and compatible families { C ( m ) | m ∈ mSpec( R ) } of cosilting objects ofcofinite type up to equivalence (see Theorem 5.15). We give applications to pure-injectivecosilting objects, n -term cosilting objects, cotilting modules, and cosilting modules overcommutative noetherian rings (see Corollaries 5.18, 5.19 and 5.23). It should be notedthat our correspondence here restricts to one of [38, Corollary 3.6], and the notions ofequivalence and compatible condition on families here restrict perfectly well to the notionsused in [38]. In Section 6, we obtain a similar result for silting objects under slightlystronger assumptions. Unlike in the cosilting setting however, the gluing of local siltingobject to form a global one is not explicit, see Remark 6.5. Notation.
All subcategories are always considered to be full and closed under isomorphisms.Complexes are written using the cohomological notation, meaning that the degree increases inthe direction of differential maps. The n -th power of the suspension functor of any triangulatedcategory will be denoted as [ n ] for n ∈ Z .2. Preliminaries
In this section, T will always denote a compactly generated triangulated category underly-ing a Grothendieck derivator. We refer to reader to [21, Appendix] for a source of referencesand basic terminology about such categories well-suited for our objectives. In particular, thisassumption implies that T has all set-indexed products and coproducts. The enrichment of theGrothendieck derivator allows for computing of homotopy limits and colimits. In fact, we will bemostly interested in the case when T = D ( R ) is the unbounded derived category of the modulecategory Mod - R of a (not necessarily noetherian) commutative ring R . Also, the only homotopyconstruction we will be interested in computing is that of a directed homotopy colimits . In thecase of D ( R ) , the directed homotopy colimits are precisely the direct limits (= directed colimits)constructed in C ( R ) , the Grothendieck category of cochain complexes of R -modules. In partic-ular, a subcategory C of D ( R ) is closed under directed homotopy colimits if it is closed underdirect limits computed in C ( R ) .Recall that an object S ∈ T is compact if the covariant functor Hom T ( S, − ) : D ( R ) → Ab preserves coproducts. The symbol T c will denote the subcategory of T consisting of all compactobjects of T . Recall that an object S ∈ D ( R ) belongs to D ( R ) c if and only if S is isomorphic in D ( R ) to a bounded complex of finitely generated projective R -modules. .1. t-structures. A t-structure is a pair ( U , V ) of full subcategories of T which satisfy thefollowing axioms:(t1) Hom T ( U , V ) = 0 ,(t2) U [1] ⊆ U (and V [ − ⊆ V ),(t3) for any X ∈ T there is a triangle U → X → V → U [1] with U ∈ U and V ∈ V .The class U is called the aisle and V is called the coaisle of the t-structure. We recall thewell-known fact that the triangle in axiom (t3) is uniquely determined and functorial — indeed,it follows from the axioms that U is a coreflective subcategory of T and the map U → X isprecisely the U -coreflection of X , see [24]. The dual statement is valid for the coaisle V and themap X → V as well — in particular, V is a reflective subcategory of T .A t-structure ( U , V ) is called: • stable if U [ − ⊆ U (or equivalently, V [1] ⊆ V ); • non-degenerate if T n ∈ Z U [ n ] = 0 and T n ∈ Z V [ n ] = 0 .The aisles of stable t-structures are precisely the kernels of Bousfield localization functors.The non-degeneracy condition holds precisely when the cohomological functor induced by thet-structure detects zero object. Clearly, the two conditions are mutually exclusive whenever T contains non-zero objects.2.2. Purity and definable subcategories in triangulated setting.
We briefly recall thetheory of purity in T , first introduced by Beligiannis [12] and Krause [25]. We call a map f in T a pure monomorphism (resp. pure epimorphism ) provided that Hom T ( S, f ) is a monomorphism(resp. an epimorphism) for any S ∈ T c . Note that, in the triangulated world, a pure monomor-phism does not have to be a categorical monomorphism. An object E ∈ T is called pure-injective provided that any pure monomorphism starting in E is a split monomorphism in T . Let C bea full subcategory of T . We say that C is closed under pure monomorphisms if for any puremonomorphism f : X → Y with Y ∈ C we also have X ∈ C and we define the analogous notionof subcategory closed under pure epimorphisms similarly.A subcategory C is definable provided that there is a set Φ of maps between objects of T c suchthat C = { X ∈ T | Hom R ( f, X ) is surjective for any f ∈ Φ } . Similarly to their more classicalcounterparts in module categories, definable subcategories of T can be characterized by theirclosure properties. Indeed, we have the following result due to Laking and Laking-Vitória. Theorem 2.1. ([27, Theorem 3.11], [26, Theorem 4.7])
The following are equivalent for a sub-category C of T :(i) C is definable,(ii) C is closed under products, pure monomorphisms, and pure epimorphisms.(iii) C is closed under products, pure monomorphisms, and directed homotopy colimits. .3. Homotopically smashing t-structures.
A t-structure ( U , V ) is called smashing if thecoaisle V is closed under coproducts. The aisles of stable smashing t-structures are preciselythe kernels of smashing localization functors of T , see [21, A.5]. For t-structures which are non-stable, a stronger condition is often needed, here we follow [35]. A t-structure ( U , V ) is called homotopically smashing if V is closed under directed homotopy colimits. For stable t-structures,this is equivalent to the smashing property [25]. A priori, this is a weaker condition than requiringthe coaisle V to be a definable subcategory. However, a recent result due to Saorín and Šťovíček[34] shows that at least in the algebraic setting (in particular, in the case T = D ( R ) ), these twoconditions coincide. Furthermore, Angeleri-Hügel, Marks, and Vitória [5] showed that coaislesof such t-structures are fully determined by their closure properties.Recall that a subcategory C of T is suspended (respectively, cosuspended ) if C is closed underextensions and C [1] ⊆ C (respectively, C [ − ⊆ C ). Then we can summarize the two abovementioned results about homotopically smashing t-structures in derived categories. Theorem 2.2. [34, 5]
The following conditions are equivalent for a subcategory V of D ( R ) :(i) V is the coaisle of a homotopically smashing t-structure ( U , V ) ,(ii) V is definable and cosuspended.Proof. Any coaisle is clearly a cosuspended subcategory. If ( U , V ) is homotopically smashingt-structure then V is definable by [34, Remark 8.9]. On the other hand, if V is a definableand cosuspended subcategory then it is a coaisle of a t-structure by [5, Lemma 4.8], and sucht-structure is clearly homotopically smashing. (cid:3) Orthogonal subcategories.
Let C be a subcategory of T . We write Hom T ( C , X ) = 0 asa shorthand for the statement Hom T ( C, X ) = 0 for all C ∈ C . Given a subset I of Z , we alsouse the following notation for subcategories degreewise orthogonal to C : C ⊥ I = { X ∈ T | Hom T ( C , X [ i ]) = 0 ∀ i ∈ I } . The role of I will be played by the symbols , ≤ , < , ≥ , > , or Z with their obviousinterpretations as subsets of Z . The symbols Hom T ( X, C ) = 0 and ⊥ I C are defined analogously,in particular: ⊥ I C = { X ∈ T | Hom T ( X, C [ i ]) = 0 ∀ i ∈ I } . If C = { C } is a singleton for some object C ∈ T , we will omit the brackets and write just C ⊥ et cetera.2.5. Compactly generated t-structures.
We say that a t-structure ( U , V ) in T is compactlygenerated if there is a set S ⊆ T c of compact objects such that V = S ⊥ . Any compactlygenerated t-structure is homotopically smashing, see [35, Proposition 5.4].The compactly generated t-structures in D ( R ) admit a geometrical classification in termsof invariants coming from the dual topology on Spec( R ) , which we recall now. A subset X of Spec( R ) is called Thomason provided that there is a set I of finitely generated ideals of R such that X = S I ∈I V ( I ) . We remark that a subset of the spectrum is Thomason preciselyif it is an open subset with respect to the Hochster dual topology on Spec( R ) , as explained inthe discussion [22, §2]. Also note that if R is noetherian, Thomason subsets are precisely the pecialization closed subsets of Spec( R ) , that is, the upper subsets of the poset (Spec( R ) , ⊆ ) . A Thomason filtration is a sequence X = ( X n | n ∈ Z ) of Thomason subsets of Spec( R ) which isdecreasing in the sense that X n ⊇ X n +1 for each n ∈ Z .It turns out that any compactly generated t-structure in D ( R ) is generated by distinguishedcompact objects of D ( R ) , the Koszul complexes. Recall that if x ∈ R is an element then the Koszul complex of x is the complex of the form K ( x ) = ( R · x −→ R ) concentrated in degrees -1and 0. If ¯ x = ( x , x , . . . , x n ) is a finite sequence of elements then we define the Koszul complex K (¯ x ) = N ni =1 K ( x i ) . Let I be a finitely generated ideal of R . Then we will abuse the notation andwrite K ( I ) for the Koszul complex on any fixed finite sequence ¯ x of generators of I . Recall that K ( I ) is always a compact object in D ( R ) and also that its cohomology modules are supportedon V ( I ) . Although the change of choice of generators may alter K ( I ) even when considered asan object of D ( R ) up to isomorphism, these complexes generate the same t-structure regardlessof the choice of generators, and therefore the abuse in notation is harmless in our application.The following results generalizes the classification in the noetherian case considered in [1]. Theorem 2.3. ([20, Theorem 5.1])
Let R be a commutative ring. There is a bijective correspon-dence between the following collections:(i) compactly generated t-structures ( U , V ) in D ( R ) , and(ii) Thomason filtrations X = ( X n | n ∈ Z ) on Spec( R ) .This correspondence assigns to a Thomason filtration X the coaisle of the form V = { K ( I )[ − n ] | V ( I ) ⊆ X n , n ∈ Z } ⊥ . Hereditary torsion pairs in
Mod - R . Recall that a torsion pair in Mod - R is a pair of fullsubcategories ( T , F ) of Mod - R which are maximal with respect to the property Hom R ( T , F ) = 0 .A torsion pair ( T , F ) is called hereditary if the torsion class T is closed under submodules, orequivalently, if the torsion-free class F is closed under taking injective envelopes. A hereditarytorsion pair ( T , F ) is of finite type if F is closed under direct limits. Hereditary torsion pairsof finite type in Mod - R were proved by Garkusha and Prest to be in bijection with Thomasonsubsets of Spec( R ) . Theorem 2.4. ([16], see also [22, Proposition 2.11])
There is a bijection ( Hereditary torsion pairs ( T , F ) in Mod - R ) − ←−→ ( Thomason subsets X of Spec( R ) ) provided by the mutually inverse assignments T 7→ X = [ { V ( I ) | R/I ∈ T } and X
7→ T = { M ∈ Mod - R | Supp( M ) ⊆ X } . (Co)localization and t-structures Definition 3.1.
For every object X ∈ D ( R ) and every prime ideal p of R , we denote by X p theobject X ⊗ L R R p = X ⊗ R R p , the localization of X at p and by X p the object R Hom R ( R p , X ) ;we call it the colocalization of X at p . Similarly, for a subcategory C of D ( R ) we consider thefollowing subcategories of D ( R p ) : C p = { X p | X ∈ C} and C p = { X p | X ∈ C} . emark 3.2. Recall that for any prime ideal p , the derived category D ( R p ) is naturally a (full)subcategory of D ( R ) . By the same token, we can naturally consider any subcategory C p of D ( R p ) as a subcategory of D ( R ) . Moreover, there is a (stable) TTF-triple ( L , D ( R p ) , K ) in D ( R ) , whichamounts to saying that there are two adjacent t-structures ( L , D ( R p )) and ( D ( R p ) , K ) (both ofwhich are necessarily stable). It follows that the inclusion of D ( R p ) into D ( R ) admits both theleft and the right adjoint, and these are realized by the functors − ⊗ L R R p and R Hom R ( R p , − ) ,respectively. For details, see e.g. [3, Theorem 4.3] and references therein. Lemma 3.3.
Let p be a prime ideal of R . We have the following natural isomorphisms: ( i ) For every X ∈ D ( R p ) and Y ∈ D ( R ) , we have Hom D ( R p ) ( X, Y p ) ∼ = Hom D ( R ) ( X, Y ) . ( ii ) For every Y ∈ D ( R p ) and X ∈ D ( R ) , we have Hom D ( R p ) ( X p , Y ) ∼ = Hom D ( R ) ( X, Y ) . ( iii ) For
X, Y ∈ D ( R ) , we have Hom D ( R ) ( X p , Y ) ∼ = Hom D ( R ) ( X, Y p ) . Proof.
For any object X ∈ D ( R p ) and maximal ideal p , we have X p = X ⊗ R R p ∼ = X ⊗ L R R p ∼ = X ⊗ L R p R p ∼ = X and X p = R Hom R ( R p , X ) ∼ = R Hom R p ( R p , X ) ∼ = X . The isomor-phism in ( i ) follows by Hom D ( R p ) ( X, Y p ) ∼ = Hom D ( R p ) ( X, R Hom R ( R p , Y )) ∼ = Hom D ( R ) ( X ⊗ L R R p , Y ) ∼ = Hom D ( R ) ( X, Y ) . Similarly, the isomorphism in ( ii ) follows by Hom D ( R p ) ( X p , Y ) ∼ =Hom D ( R ) ( X, R Hom R ( R p , Y )) ∼ = Hom D ( R ) ( X, Y ) . Applying the isomorphisms from ( i ) and ( ii ) ,we obtain Hom D ( R ) ( X p , Y ) ∼ = Hom D ( R p ) ( X p , Y p ) ∼ = Hom D ( R ) ( X, Y p ) . (cid:3) Lemma 3.4.
Let ( U , V ) be a t-structure in D ( R ) and p ∈ Spec( R ) . Then:(i) U p = U ∩ D ( R p ) and V p = V ∩ D ( R p ) .(ii) ( U p , V p ) is a t-structure in D ( R p ) .(iii) If ( U , V ) is in addition homotopically smashing then V p = V p .Proof. ( i ) : By [20, Proposition 2.2], we have that U p ∈ U for any U ∈ U and V p ∈ V for any V ∈ V , which easily yields the desired equalities. ( ii ) : The only non-trivial step is to check the existence of canonical triangles with respect to ( U p , V p ) . Let X be an object of D ( R p ) ⊆ D ( R ) and consider the canonical triangle U → X r −→ V → U [1] with respect to the t-structure ( U , V ) in D ( R ) . Consider the natural colocalization morphism c : V p → V . Recall from Remark 3.2 that c is the D ( R p ) -coreflection of V . Together with X ∈ D ( R p ) this yields that the morphism r factors through c by a map f : X → V p . On theother hand, as V p belongs to V by ( i ) , the map f factors through the V -reflection map r by amap g : V → V p . The situation is captured in the following commutative diagram: X VV p rf gc y the construction, we have cgr = cf = r . Since r is the V -reflection morphism of X , themap cg has to be the identity on V , and so g is a split monomorphism. It follows that V ∈ D ( R p ) ,which implies also U ∈ D ( R p ) . We conclude that U → X r −→ V → U [1] is already the desiredapproximation triangle in D ( R p ) . ( iii ) : If ( U , V ) is homotopically smashing then V is closed under the localization functor − ⊗ R R p , and therefore V p = V ∩ D ( R p ) . By ( i ) , V p = V p . (cid:3) Lemma 3.5.
Let V be a definable subcategory of D ( R ) . Then the following conditions areequivalent for any object X ∈ D ( R ) :(i) X ∈ V ,(ii) X m ∈ V m for any m ∈ mSpec( R ) ,(iii) X m ∈ V for any m ∈ mSpec( R ) .Proof. ( i ) = ⇒ ( ii ) : This is just the definition of the subcategory V m . ( ii ) = ⇒ ( iii ) : Since V is definable, it is closed under directed homotopy colimits. But since X m = X ⊗ R R m , X m can be represented as a directed homotopy colimit of a coherent diagramconsisting of finite coproducts of copies of X . Therefore, V m is a subcategory of V . ( iii ) = ⇒ ( i ) : Consider the natural map f : X → Q m ∈ mSpec( R ) X m . If S ∈ D ( R ) c is a com-pact object, we have that Hom D ( R ) ( S, f ) : Hom D ( R ) ( S, X ) → Hom D ( R ) ( S, Q m ∈ mSpec( R ) X m ) ∼ = Q m ∈ mSpec( R ) Hom D ( R ) ( S, X ) m is a monomorphism in Mod - R , and therefore f is a pure monomor-phism in D ( R ) . Since V contains X m for any m ∈ mSpec( R ) and is closed under products andpure monomorphisms, we conclude that X ∈ V . (cid:3) Aisles of compactly generated t-structures. If R is noetherian, a more explicit de-scription of the compactly generated t-structure ( U , V ) in terms of the associated Thomasonfiltration is available in [1, Theorem 3.11]. In this case, the aisle is described cohomologicallyas U = { X ∈ D ( R ) | Supp H n ( X ) ⊆ X n ∀ n ∈ Z } . For general commutative ring and gen-eral Thomason filtration, such description seems currently unavailable apart from special cases[33, Proposition 6.6], [22, Lemma 3.10]. However, we can extract the following slightly weakerstructural information even in the general situation which we record here for later use.Let X = ( X n | n ∈ Z ) be a Thomason filtration of Spec( R ) inducing a compactly generatedt-structure ( U , V ) . Let D + ( R ) denote the subcategory of D ( R ) consisting of all objects X with H i ( X ) = 0 for i ≪ . Lemma 3.6.
Let ( U , V ) be a compactly generated t-structure in D ( R ) corresponding to a Thoma-son filtration X = ( X n | n ∈ Z ) . Put U = { X ∈ D ( R ) | Supp H n ( X ) ⊆ X n ∀ n ∈ Z } . Then:(i) There is a set E of shifts of stalks of injective R -modules such that U = ⊥ E .(ii) The category U is an aisle of a t-structure ( U , V ) .(iii) U ⊆ U and V ⊆ V ,(iv) U ∩ D + ( R ) = U ∩ D + ( R ) and V ∩ D + ( R ) = V ∩ D + ( R ) .Proof. ( i ) : Recall from §2.6 that T n = { M ∈ Mod - R | Supp( M ) ⊆ X n } is a torsion class ofa hereditary torsion pair ( T n , F n ) , which amounts to saying that there is a set E n of injective R -modules such that T n = { M ∈ Mod - R | Hom R ( M, E ) = 0 ∀ E ∈ E n } . Then U = ⊥ E , where E = S n ∈ Z E n [ − n ] (see [20, Lemma 3.2]). ii ) : This follows from ( i ) , [26, Corollary 5.4], and the easy observation that U is a suspendedsubcategory of D ( R ) . ( iii ) : Since V = U ⊥ and V = U ⊥ , it is clearly enough to show the first claim. Recallfrom [1, §1.2] that U is the smallest suspended subcategory of U containing all objects K ( I )[ − n ] where V ( I ) ⊆ X n , n ∈ Z closed under coproducts. Since Supp H n ( K ( I )) ⊆ V ( I ) for any finitelygenerated ideal I and any n ∈ Z , we see that all the objects K ( I )[ − n ] belong to U . Finally,since U is an aisle, we conclude that U ⊆ U ′ . ( iv ) : By ( iii ) we already have inclusions U ∩ D + ( R ) ⊆ U ∩ D + ( R ) and V ∩ D + ( R ) ⊇V ∩ D + ( R ) . Let X ∈ U ∩ D + ( R ) . Since the subcategory U is defined by cohomology, weclearly have that any soft (cohomological) truncation τ Let ( U , V ) be a compactly generated t-structure in D ( R ) corresponding to a Thoma-son filtration X = ( X n | n ∈ Z ) . Then the following statements hold true:(i) For any p ∈ Spec( R ) , κ ( p )[ − n ] ∈ U if and only if p ∈ X n .(ii) For any X ∈ V ∩ D + ( R ) we have E ( H inf( X ) ( X ))[ − inf( X )] ∈ V , where inf( X ) = min { k ∈ Z | H k ( X ) = 0 } ,(iii) Let E n be the subcategory of Mod - R consisting of all injective R -modules E such that E [ − n ] ∈ V . Then for any p ∈ Spec( R ) we have Hom R ( κ ( p ) , E n ) = 0 if and only if p ∈ X n .Proof. ( i ) : By Lemma 3.6(iv), κ ( p )[ − n ] ∈ U if and only if κ ( p )[ − n ] ∈ U , or equivalently, Supp( κ ( p )) = { p } ⊆ X n . ( ii ) : This follows from [20, Lemma 3.3]. ( iii ) : If E [ − n ] belongs to V then Hom R ( κ ( p ) , E ) = Hom D ( R ) ( κ ( p )[ − n ] , E [ − n ]) = 0 for any p ∈ X n because κ ( p )[ − n ] ∈ U by ( i ) . For the converse, assume that p X n and let X ∈ U .Since E ( κ ( p )) is injective, we have Hom D ( R ) ( X, E ( κ ( p ))[ − n ]) = Hom R ( H n ( X ) , E ( κ ( p ))) by [20,Lemma 3.2]. By Lemma 3.6(iii), Supp H n ( X ) ⊆ X n , and so Hom R ( H n ( X ) , E ( κ ( p ))) = 0 . Thisshows that E ( κ ( p ))[ − n ] ∈ U ⊥ = V , and therefore E ( κ ( p )) ∈ E n . Then clearly Hom R ( κ ( p ) , E n ) =0 . (cid:3) Compatible families of Thomason sets. Our next step is to show how Thomason setscan be glued together from local data over all localizations at maximal ideals. Let X ( m ) bea Thomason subset of Spec( R m ) for each m ∈ mSpec( R ) . If Y is any subset of Spec( R m ) , wewill denote by Y ∗ its image under the natural inclusion Spec( R m ) ֒ −→ Spec( R ) induced by the ocalization map (the choice of maximal ideal m will always be clear from the context). Notethat Y = { p m | p ∈ Y ∗ } ⊆ Spec( R m ) . We consider the following condition for the family { X ( m ) | m ∈ mSpec( R ) } :( † ) ∀ m , m ′ ∈ mSpec( R ) : { p ∈ X ( m ) ∗ | p ⊆ m ′ } = { p ∈ X ( m ′ ) ∗ | p ⊆ m } . Lemma 3.8. In the setting as above, put X = S m ∈ mSpec( R ) X ( m ) ∗ . Then the following conditionsare equivalent:(i) The set { X ( m ) | m ∈ mSpec( R ) } satisfies condition ( † ) and X is a Thomason subset of Spec( R ) ,(ii) X = S I ∈I V ( I ) where I is the set of all finitely generated ideals of R such that V ( I m ) ⊆ X ( m ) for all m ∈ Spec(R) .Proof. ( i ) = ⇒ ( ii ) : Set X ′ = S I ∈I V ( I ) and let us show X = X ′ . Fix I ∈ I , p ∈ V ( I ) ,and m ∈ mSpec( R ) . Since V ( I m ) ⊆ X ( m ) , we clearly have p m ∈ X ( m ) , showing that X ′ ⊆ X .Conversely, assume that p ∈ X ( m ) ∗ ⊆ X and let m ′ be another maximal ideal of R . Thenthere are two possibilities. Either p ⊆ m ′ , and then p ∈ X ( m ′ ) ∗ by the condition ( † ). Theother possibility is that p m ′ , and then p m ′ = R m ′ . Together we proved that X = { p ∈ Spec( R ) | V ( p m ) ⊆ X ( m ) ∀ m ∈ mSpec( R ) } . Since X is Thomason, for any p ∈ X there isa finitely generated ideal I of R such that p ∈ V ( I ) ⊆ X . For any m ∈ mSpec( R ) we have V ( I m ) = { p m | p ∈ V ( I ) } , and so V ( I m ) ⊆ X ( m ) , as desired. ( ii ) = ⇒ ( i ) : First, the condition ( ii ) clearly implies that X is Thomason. We are left withproving that the set { X ( m ) | m ∈ mSpec( R ) } satisfies condition ( † ). Let p ∈ X ( m ) ∗ satisfy p ⊆ m ′ and let us show that p ∈ X ( m ′ ) ∗ . By ( ii ) there is a finitely generated ideal of R suchthat p ∈ V ( I ) ⊆ X and such that V ( I m ′ ) ⊆ X ( m ′ ) . This already testifies that p ∈ X ( m ′ ) ∗ . (cid:3) Definition 3.9. Let X ( m ) be a Thomason subset of Spec( R m ) for each m ∈ mSpec( R ) . Thefamily { X ( m ) | m ∈ mSpec( R ) } is called compatible if it satisfies the equivalent conditions ofLemma 3.8.Let X ( m ) = ( X ( m ) n | n ∈ Z ) be a Thomason filtration of Spec( R m ) for each m ∈ mSpec( R ) .The family { X ( m ) | m ∈ mSpec( R ) } is called compatible if { X ( m ) n | m ∈ mSpec( R ) } is acompatible family of Thomason subsets for each n ∈ Z . Remark 3.10. If R is noetherian then the assumption of X being a Thomason set in condition(i) of Lemma 3.8 is superfluous. Indeed, the proof of the lemma shows that X = { p ∈ Spec( R ) | V ( p m ) ⊆ X ( m ) ∀ m ∈ mSpec( R ) } holds even without such assumption, and this is enough tosee that X is a specialization closed subset of Spec( R ) , which for noetherian rings amounts to X being Thomason. It follows that our compatibility condition generalizes the one used in [38,Definition 2.3]. Indeed, as we just observed, for noetherian rings our compatibility condition fora family of Thomason sets boils down to ( † ), which is precisely the condition used by Trlifaj andŞahinkaya.On the other hand, the condition of X being Thomason is not superfluous for rings which arenot noetherian, in general. Indeed, let R = k ω be the countably infinite product of a field k andchoose a non-principal maximal ideal m of R . Set X ( m ) = { m m } and X ( m ′ ) = ∅ for any maximal deal m ′ = m . Clearly, this family satisfies ( † ). However X = S n ∈ mSpec( R ) X ( n ) ∗ = { m } , whichis well-known not to be a Thomason set, as there is no idempotent e of R with V ( e ) = { m } . Proposition 3.11. There is a bijective correspondence ( Thomason subsets X of Spec( R ) ) − ←−→ ( Compatible families { X ( m ) | m ∈ mSpec( R ) } of Thomason subsets ) induced by the mutually inverse assignments X X ( m ) = X m = { p m | p ∈ X } ∀ m ∈ mSpec( R ) and { X ( m ) | m ∈ mSpec( R ) } 7→ X = [ m ∈ mSpec( R ) X ( m ) ∗ . This bijection naturally extends to a bijection ( Thomason filtrations X of Spec( R ) ) − ←−→ ( Compatible families { X ( m ) | m ∈ mSpec( R ) } of Thomason filtrations ) . Proof. Let X be a Thomason subset of Spec( R ) and put X m = { p m | p ∈ X } for each max-imal ideal m . Clearly, X ∗ m = X ∩ Spec( R m ) ∗ and so the condition ( † ) is satisfied. Also, X = S m ∈ mSpec( R ) X ∗ m is Thomason, showing that the family { X m | m ∈ mSpec( R ) } is com-patible. It follows from Lemma 3.8 that if a given family { X ( m ) | m ∈ mSpec( R ) } is compatible,then X ( m ) = X m = { p m | p ∈ X } , which establishes that the assignments are mutually inverse.For the claim about Thomason filtrations, it is enough to notice that for two Thomasonsubsets X, Y we have X ⊆ Y if and only if X m ⊆ Y m for each maximal ideal m , as the sets Spec( R m ) ∗ , m ∈ mSpec( R ) form a cover of Spec( R ) . (cid:3) Local-global property of compact generation. The gluing condition on Thomason setsallows us to prove the following useful local characterization of when a homotopically smashing t-structure is compactly generated. Indeed, it allows us to construct a family of compact generatorsout of a family of compact generators for each localization at maximal ideal. Before that, weneed a relatively straightforward observation on the injective R -modules living in torsion-freeclasses of hereditary torsion pairs.Let Inj - R denote the subcategory of all injective R -modules. We will consider subcategories E of Inj - R which reflect the closure properties of torsion-free classes of hereditary torsion pairsof finite type. Two obvious conditions are that E needs to be closed under products and directsummands. Furthermore, the fact that the torsion-free class is closed under direct limit will bereflected by the following condition on the subcategory E ⊆ Inj - R :( †† ) For any direct system ( E i | i ∈ I ) in E , the injective envelope of its colimit belongs to E . Lemma 3.12. There is a bijection ( Hereditary torsion pairs ( T , F ) in Mod - R ) − ←−→ ( Subcategories E of Inj - R closed under products and direct summands ) provided by the mutually inverse assignments F 7→ E = Inj - R ∩ F nd E 7→ T = { M ∈ Mod - R | Hom R ( M, E ) = 0 } . Furthermore, the torsion pair ( T , F ) is of finite type (that is, F is closed under direct limits) ifand only if the corresponding subcategory E satisfies condition ( †† ).Proof. First, it is routine to check that both the assignments are well-defined. If ( T , F ) is ahereditary torsion pair then F is closed under injective envelopes, and therefore T = { M ∈ Mod - R | Hom R ( M, E ) = 0 } . To establish that the assignments are mutually inverse, the onlynon-trivial step is to check that if E is closed under products and direct summands then thecorresponding torsion pair ( T , F ) satisfies F ∩ Inj - R ⊆ E . To see this, it is enough to provethat F = Sub( E ) , the subcategory of all submodules of modules from E . Since F is the smallesttorsion-free class containing E , it is further enough to show that Sub( E ) is a torsion-free classin Mod - R . Clearly, Sub( E ) is closed under subobjects, and since E is closed under products,so is Sub( E ) . Therefore, we only need to see that Sub( E ) is closed under extensions. Let → A → B → A → be an exact sequence with A i ⊆ E i where E i belongs to E for both i = 0 , . The injectivity of E allows to extend the inclusion A ⊆ E to a map B → E , whicheasily yields an embedding of B into E ⊕ E . Since E is closed under products, this shows B ∈ Sub( E ) , as desired.Finally we show that F is closed under direct limits if and only if E satisfies ( †† ). If F isclosed under direct limits then for any direct system E i , i ∈ I in E we have lim −→ i ∈ I E i ∈ F . Since F is closed under injective envelopes, the injective envelope of this direct limit belongs to E .Conversely, let E satisfy ( †† ). To establish that F is closed under direct limits, it suffices to showthat it is closed under direct limits of well-ordered systems [17, Lemma 2.14]. Let ( F α | α < λ )be such a direct system in F and denote its direct limit by F = lim −→ α<λ F α . By [20, Lemma 3.5],there is a direct system E α , α < λ such that E α is the injective envelope of F α for each α < λ ,and such that the envelope embeddings F α ⊆ E α induce a map between the two direct systems.Since E satisfies ( †† ), the injective envelope E of the direct limit M = lim −→ α<λ E α belongs to E .By the properties of the direct systems constructed and by exactness of the direct limit functor,we have a natural limit monomorphism F → M , and therefore F embeds into E . We concludethat F ∈ F as desired. (cid:3) We are ready to prove a key result of this section showing that the property of a homotopicallysmashing t-structure being compactly generated is a local-global property with respect to thecover Spec( R ) = S m ∈ mSpec( R ) Spec( R m ) ∗ . Proposition 3.13. Let ( U , V ) be a homotopically smashing t-structure in D ( R ) . For each max-imal ideal m , let ( U m , V m ) be the localized t-structure in D ( R m ) of Lemma 3.4. Then:(i) The t-structure ( U , V ) is compactly generated in D ( R ) if and only if the t-structure ( U m , V m ) is compactly generated in D ( R m ) for each m ∈ mSpec( R ) .(ii) Assume that ( U , V ) is compactly generated and corresponds to a Thomason filtration X .Let X ( m ) be a Thomason filtration corresponding to the t-structure ( U m , V m ) for each m ∈ mSpec( R ) . Then { X ( m ) | m ∈ mSpec( R ) } is a compatible family of Thomasonfiltrations corresponding via Proposition 3.11 to the Thomason filtration X on Spec( R ) . roof. ( i ) : If ( U , V ) is compactly generated then there is a set S of compact objects such that V = S ⊥ . By Lemma 3.3 it is easy to see that V m = { S m | S ∈ S} ⊥ . Since S m is a compactobject of D ( R m ) for any S ∈ D ( R ) c , we see that ( U m , V m ) is compactly generated.For the converse, let X ( m ) = ( X ( m ) n | n ∈ Z ) be a Thomason filtration corresponding viaTheorem 2.3 to the compactly generated t-structure ( U m , V m ) for each m ∈ mSpec( R ) . Put X n = S m ∈ mSpec( R ) ( X ( m ) n ) ∗ for each n ∈ Z . By Lemma 3.7(i), X ( m ) n = { p ∈ Spec( R m ) | κ ( p )[ − n ] ∈ U m } . Since U m = U ∩ D ( R m ) by Lemma 3.4, we have ( X ( m ) n ) ∗ = { p ∈ Spec( R ) | p ⊆ m and κ ( p )[ − n ] ∈ U } . Then it easily follows that the family { X ( m ) n | m ∈ mSpec( R ) } satisfiescondition ( † ) for each n ∈ Z . Next, let E n = { E ∈ Mod - R | E [ − n ] ∈ V and E is injective } for each n ∈ Z . Clearly, E n is closed under direct summands and products as a subcategory of Inj - R . We claim that E n satisfies condition ( †† ) for each n ∈ Z . Indeed, let ( E i | i ∈ I ) be adirect system of modules from E n . Denote its direct limit by M = lim −→ i ∈ I E i . Since E α ∈ E n , wehave E α [ − n ] ∈ V for each n ∈ V . Therefore, M [ − n ] = hocolim α<λ E α [ − n ] belongs to V , as V isclosed under directed homotopy colimits. Let E be the injective envelope of M , we need to showthat E [ − n ] ∈ V . For any m ∈ mSpec( R ) , M m [ − n ] belongs to V m , and so the injective envelope G m of M m in Mod - R m also satisfies G m [ − n ] ∈ V m , here we use Lemma 3.7(ii). Since G m is alsoinjective as an R -module, we have G m ∈ E n . Finally, observe that E is a direct summand of Q m ∈ mSpec( R ) G m , and so E ∈ E n .Let ( T n , F n ) be the hereditary torsion pair of finite type in Mod - R corresponding to E n viaLemma 3.12, and let X ′ n be the Thomason set in Spec( R ) corresponding to ( T n , F n ) via The-orem 2.4. Note that since T n = { M ∈ Mod - R | Hom R ( M, E n ) = 0 } and Supp( κ ( p )) = { p } ,we have X ′ n = { p ∈ Spec( R ) | Hom R ( κ ( p ) , E n ) = 0 } . We claim that X ′ n = X n . The inclusion X n ⊆ X ′ n is clear as E n [ − n ] ⊆ V . On the other hand, if p ∈ X ′ n then consider any maximal ideal m which contains p . Since Hom R ( κ ( p ) , E n ) = 0 , we have in particular that Hom R ( κ ( p ) , E n ( m )) = 0 where E n ( m ) consists of all injective R m -modules E such that E [ − n ] ∈ V m . But since the t-structure ( U m , V m ) is compactly generated, this implies p m ∈ X m n , and so p ∈ ( X m n ) ∗ ⊆ X n byLemma 3.7(iv).We have proved that { X ( m ) | m ∈ mSpec( R ) } is a compatible family of Thomason filtrationscorresponding to a Thomason filtration X = { X n | n ∈ Z } via Proposition 3.11. Put S = { K ( I )[ − n ] | V ( I ) ⊆ X n , n ∈ Z } . It follows from Theorem 2.3 that S m = { K ( I )[ − n ] ⊗ R R m | V ( I ) ⊆ X n , n ∈ Z } = { K ( I m )[ − n ] | V ( I m ) ⊆ X ( m ) n , n ∈ Z } is a set of compact generators for ( U m , V m ) in D ( R m ) , and thus satisfies ( S m ) ⊥ = V m . By Lemma 3.3, we have ( S ⊥ ) m = ( S m ) ⊥ = V m . But since both S ⊥ and V are definable subcategories of D ( R ) , Lemma 3.5 shows that S ⊥ = V , and so ( U , V ) is compactly generated. ( ii ) : This follows easily either from the proof of ( i ) or directly from Lemma 3.7(i). (cid:3) Gluing of compactly generated t-structures.Definition 3.14. Let ( U ( m ) , V ( m )) be a compactly generated t-structure in D ( R m ) for each m ∈ mSpec( R ) corresponding to a Thomason filtration X ( m ) in Spec( R m ) . The family { ( U ( m ) , V ( m )) | m ∈ mSpec( R ) } is said to be compatible if the family { X ( m ) | m ∈ mSpec( R ) } of Thomasonfiltrations is compatible. e say that a Thomason filtration X = ( X n | n ∈ Z ) is non-degenerate if T n ∈ Z X n = ∅ and S n ∈ Z X n = Spec( R ) . Theorem 3.15. Let R be a commutative ring. There is a bijective correspondence between(i) compactly generated t-structures ( U , V ) in D ( R ) , and(ii) compatible families { ( U ( m ) , V ( m )) | m ∈ mSpec( R ) } of compactly generated t-structures,which restricts to a bijective correspondence between(i’) non-degenerate compactly generated t-structures ( U , V ) in D ( R ) , and(ii’) compatible families { ( U ( m ) , V ( m )) | m ∈ mSpec( R ) } of non-degenerate compactly gener-ated t-structures.Proof. The bijection between ( i ) and ( ii ) follows from Theorem 2.3, Proposition 3.11, and Propo-sition 3.13 ( ii ) . Recall that ( U , V ) is a non-degenerate t-structure if it satisfies T n ∈ Z U [ n ] = 0 = T n ∈ Z V [ n ] . To prove the equivalence of ( i ′ ) and ( ii ′ ) , by the bijection between ( i ) and ( ii ) , itsuffices to show that a compactly generated t-structure ( U , V ) is non-degenerate if and only ifthe corresponding Thomason filtration X corresponding to ( U , V ) is non-degenerate.Let U = { X ∈ D ( R ) | Supp H n ( X ) ⊆ X n , n ∈ Z } be the aisle of Lemma 3.6, recall that U ⊆ U . Now if T n ∈ Z X n = ∅ then T n ∈ Z U [ n ] = { X ∈ D ( R ) | Supp H ∗ ( X ) = ∅} = 0 , whichimples T n ∈ Z U [ n ] = 0 . On the other hand, assume that T n ∈ Z X n contains a prime ideal p . Then U [ n ] contains κ ( p )[0] for each n ∈ Z by Lemma 3.7(i), and so T n ∈ Z U [ n ] = 0 .Now we consider the second non-degeneracy condition. Put S = U ∩ D ( R ) c . Since the t-structure is compactly generated we have S ⊥ = V . Observe that T n ∈ Z V [ n ] = 0 is equivalentto S ⊥ Z = ( S n ∈ Z S [ n ]) ⊥ = 0 . We let ( L , C ) be the t-structure generated by the set S n ∈ Z S [ n ] ,so that C = S ⊥ Z . The t-structure ( L , C ) is compactly generated and so it corresponds to aThomason filtration Y = ( Y n | n ∈ Z ) via Theorem 2.3. Since the t-structure ( L , C ) is stable(recall that this means that L is closed under the cosuspension functor [ − ), the Thomasonfiltration Y is constant in the sense that there is a single Thomason set Y such that Y n = Y forall n ∈ Z , cf. [20, Theorem 5.3]. It also follows from Theorem 2.3 that Y = S n ∈ Z X n , as L isgenerated by all shifts of the Koszul complexes of the form K ( I ) with V ( I ) ⊆ X n for any n ∈ Z .Finally, we have that C = 0 if and only if L = D ( R ) , which happens if and only if the Thomasonset Y is equal to Spec( R ) . (cid:3) Stalk-locality of Telescope Conjecture Definition 4.1. A triangulated category T satisfying the assumptions of §2 is said to • satisfy the Telescope Conjecture (TC) if any stable (homotopically) smashing t-structurein T is compactly generated. • satisfy the Semistable Telescope Conjecture (STC) if any homotopically smashing t-structure in T is compactly generated. Remark 4.2. The formulation (TC) of Definition 4.1 is equivalent to a more customary form ofthe Telescope Conjecture which asks for any kernel of a smashing localization of the triangulatedcategory D ( R ) to be generated by compact objects. More precisely, a stable t-structure ( L , C ) is homotopically smashing if and only if U is a smashing subcategory , that is, if U is a localizing ubcategory of D ( R ) such that U ⊥ = V is closed under coproducts, see e.g. [21, Appendix] fordetails and further references.Clearly, the validity of (STC) in D ( R ) implies (TC).The fact that (TC) holds in the derived category of a commutative noetherian ring was estab-lished by Neeman [29]. The following recent development shows that also (STC) is valid in thesame setting. Theorem 4.3. ([21, Theorem 1.1]) Let R be a commutative noetherian ring. Then D ( R ) satisfies(STC). We are ready to formulate the local-global property of Proposition 3.13 in terms of telescopeproperties of D ( R ) . Lemma 4.4. Let R be a commutative ring and p ∈ Spec( R ) . Then:(i) If (TC) holds in D ( R ) then (TC) holds in D ( R p ) .(ii) If (STC) holds in D ( R ) then (STC) holds in D ( R p ) .Proof. Assume that (STC) holds in D ( R ) and let p ∈ mSpec( R ) . Let ( U , V ) be a homotopicallysmashing t-structure in D ( R p ) . Consider V as a subcategory of D ( R ) . Since V is a cosuspendeddefinable subcategory of D ( R p ) (see [34, Remark 8.9]) and D ( R p ) is a cosuspended definablesubcategory of D ( R ) , we infer that V is a cosuspended definable subcategory of D ( R ) , and sothere is a homotopically smashing t-structure ( U ′ , V ) in D ( R ) , see [5, Proposition 4.5]. Since D ( R ) satisfies (STC), there is a set of compact objects S of D ( R ) such that V = S ⊥ in D ( R ) .By Lemma 3.3, we have the equality V = S ⊥ p in D ( R p ) , and so ( U , V ) is compactly generatedin D ( R p ) . The version of this implication for (TC) follows easily as the closure of V undersuspensions is also checked equivalently in D ( R p ) and D ( R ) . (cid:3) Theorem 4.5. Let R be a commutative ring.(i) (TC) holds in D ( R ) if and only if (TC) holds in D ( R m ) for any maximal ideal m of R .(ii) (STC) holds in D ( R ) if and only if (STC) holds in D ( R m ) for any maximal ideal m of R .Proof. By Lemma 4.4, we only need to prove the backward implication of both statements. Ifa t-structure ( U , V ) is homotopically smashing in D ( R ) then clearly so is ( U m , V m ) in D ( R m ) .Therefore, if (STC) holds for each localization R m at maximal ideals then (STC) holds in D ( R ) by Proposition 3.13(i). Also, if ( U , V ) is in addition stable then so is clearly ( U m , V m ) . As aconsequence, we also get that if (TC) holds for each localization R m then it holds in D ( R ) . (cid:3) Corollary 4.6. Let R be a commutative ring such that R m is noetherian for any m ∈ mSpec( R ) .Then (STC) holds in D ( R ) . In particular, (TC) holds in D ( R ) .Proof. Combine Proposition 3.13 and the main result of [21]. (cid:3) Remark 4.7. There are many examples of non-noetherian commutative rings which are locallynoetherian in the sense of Corollary 4.6, see e.g. [18], [8]. A well-known class of examples of suchrings consists of the (commutative and not semi-simple) von Neumann regular rings — indeed, hese can be characterized as rings R such that R m is a field for any m ∈ Spec( R ) . In this way,our local-global criterion recovers some recent results for von Neumann regular rings [11, §7],[37], [10, Corollary 3.12].4.1. Stalk-locality of ⊗ -Telescope Conjecture for schemes. Here we follow the setting of[9, Examples 1.2(2)], for basic terminology about schemes we adhere to [39]. Let X be a quasi-compact and quasi-separated scheme X and let D ( X ) denote the derived category of complexesof O X -modules with quasi-coherent cohomology. Then D ( X ) together with the usual derivedtensor product ⊗ L X is a compactly generated tensor triangulated category. If X is in additionseparated then D ( X ) is naturally equivalent to D (Qcoh(X)) , the usual derived category of thecategory of quasi-coherent sheaves over X . For non-affine schemes, one can only expect goodlocal behavior from localizations which respect the tensor structure. The following formulationof tensor-friendly Telescope Conjecture is a special case of the general formulation for tensortriangulated categories of Balmer and Favi [9, Definition 4.2], building on previous work ofHovey, Palmieri and Strickland [19]. Definition 4.8. Let X be a quasi-compact and quasi-separated scheme. We say that D ( X ) satisfies the ⊗ -Telescope Conjecture ( ⊗ TC) if any stable (homotopically) smashing t-structure ( L , C ) such that L is a ⊗ -ideal, meaning that L satisfies in addition the condition(4.1) L ⊗ L X M ∈ L for all L ∈ L , M ∈ D ( X ) , is compactly generated. It is well-known that in case X is an affine scheme the tensor condition(4.1) is vacuous, and so for affine schemes, ( ⊗ TC) is equivalent to (TC) by taking the ring ofglobal sections.Balmer and Favi [9, Corollary 6.8] showed that ( ⊗ TC) is a local-global property with respectto any cover X = S i ∈ I U i by open and quasi-compact sets. In particular, one can check ( ⊗ TC)locally on any cover of X by open affine sets. We remark that Balmer and Favi worked ina much broader generality of compactly generated tensor triangulated categories in terms ofBalmer spectra. This was further generalized by Stevenson [36] to the relative setting of asuitable action triangulated categories. In algebrogeometric context, Antieau [7] established thelocal-global criterion for étale covers of (derived) schemes. Note these results do not include thecase of the cover Spec( R ) = S m ∈ mSpec( R ) Spec( R m ) ∗ we consider in Theorem 4.5 ( i ) — indeed,this cover is not even fpqc whenever mSpec( R ) is an infinite set.Given a scheme X and x ∈ X , we denote the stalk of X at x by O X,x , recall that O X,x isalways a local commutative ring. Combining Theorem 4.5 ( i ) with the Balmer and Favi result,we obtain that ( ⊗ TC) for D ( X ) is a stalk-local property in the following sense. Theorem 4.9. Let X be a quasi-compact and quasi-separated scheme. Then the following state-ments are equivalent:(i) ( ⊗ TC) holds in D ( X ) ,(ii) ( ⊗ TC), or equivalently (TC), holds in D ( O X,x ) for all x ∈ X ,(iii) ( ⊗ TC), or equivalently (TC), holds in D ( O X,x ) for all closed points x ∈ X . roof. Let X = S i ∈ I U i be a cover of X by open affine sets, let λ i : U i ∼ = Spec( R i ) be home-omorphisms where R i is a appropriate commutative ring for each i ∈ I . By [9, Corollary 6.8],( ⊗ TC) holds in D ( X ) if and only if ( ⊗ TC) holds in D ( U i ) for each i ∈ I . Then ( ⊗ TC)holds in D ( X ) if and only if (TC) holds in D ( R i ) for each i ∈ I . By Theorem 4.5 ( i ) weknow that (TC) holds in D ( R i ) if and only if it holds in D (( R i ) p ) for any p ∈ Spec( R i ) . But D (( R i ) p ) ∼ = D ( O U i ,λ i ( p ) ) ∼ = D ( O X,λ i ( p ) ) . This establishes the equivalence ( i ) ⇐⇒ ( ii ) .It remains to prove the implication ( iii ) = ⇒ ( ii ) . Let x ∈ X be any point. Since X isquasi-compact, there is a closed point c contained in the closure of x in X . By the assumption,(TC) holds in D ( O X,c ) . Since the ring O X,x is isomorphic to a localization of O X,c at some primeideal, Lemma 4.4 implies that (TC) holds in D ( O X,x ) . (cid:3) As a consequence, ( ⊗ TC) holds if X is stalk-noetherian in the following sense. As mentionedalready in Remark 4.7, the property of a scheme being noetherian is not a stalk-local property,even for affine schemes [18],[8]. Corollary 4.10. Let X be a quasi-compact and quasi-separated scheme such that the stalk O X,x is noetherian for any point x ∈ X (equivalently, for any closed point x ∈ X ). Then D ( X ) satisfies ( ⊗ TC).Proof. This is a direct consequence of Theorem 4.9 together with validity of (TC) in derivedcategories of commutative noetherian rings [29]. (cid:3) Cosilting and cotilting objects The goal of this section is to refine the (co)localization results for t-structures that are inducedby cosilting objects. Definition 5.1. We say that an object T ∈ D ( R ) is silting if the pair ( T ⊥ > , T ⊥ ≤ ) is a t-structure, which we call the silting t-structure induced by T . Two silting objects T, T ′ ∈ D ( R ) are equivalent if they induce the same t-structure.An object C ∈ D ( R ) is cosilting if the pair ( ⊥ ≤ C, ⊥ > C ) forms a t-structure, which we call the cosilting t-structure induced by C . Two cosilting objects C, C ′ are equivalent if they induce thesame t-structure. By [31, Lemma 4.5], C and C ′ are equivalent if and only if Prod( C ) = Prod( C ′ ) ,where Prod( C ) is the subcategory of all direct summands of set-indexed direct products of copiesof C .A silting object is called a bounded silting complex if it is quasi-isomorphic to a boundedcomplex of projective R -modules, and a cosilting object is called a bounded cosilting complex ifit is quasi-isomorphic to a bounded complex of injective R -modules.It is an easy consequence of the definition that both silting and cosilting t-structures are alwaysnon-degenerate. If C is a pure-injective cosilting object then the induced t-structure ( U , V ) ishomotopically smashing. Indeed, the pure-injectivity of C ensures that the coaisle V = ⊥ > C isclosed under both pure monomorphisms and pure epimorphisms. Definition 5.2. We say that a silting object T in D ( R ) is of finite type if there is a set of compactobjects S such that T ⊥ > = S ⊥ . Similarly, we call a cosilting object C in D ( R ) of cofinite type f there is a set of compact objects S such that ⊥ > C = S ⊥ . Note that the a cosilting object iscofinite type precisely when the induced cosilting t-structure ( U , V ) is compactly generated. Theorem 5.3. ([28]) Any bounded silting object is of finite type. Any bounded cosilting object ispure-injective. Theorem 5.4. ([21, A.8, Corollary 2.14]) If D ( R ) satisfies (STC) then any pure-injective cosilt-ing object is of cofinite type. In particular, this is the case if R is a commutative noetherianring. Lemma 5.5. Let C ∈ D ( R ) . Then C is a cosilting object if and only if all the followingconditions hold:(i) C cogenerates D ( R ) , that is, ⊥ Z C = 0 ,(ii) C ∈ ⊥ > C ,(iii) ⊥ > C is closed under products,(iv) ⊥ > C is a coaisle of a t-structure.Furthermore, the condition (iv) follows from the other three conditions provided that C ispure-injective.Proof. Using condition (iv) the proof is dual to that of [31, Proposition 4.13].If C is pure-injective, ⊥ > C is closed under both pure monomorphisms and pure epimorphisms.Since ⊥ > C is clearly cosuspended and by (iii) it is closed under products, we infer by [5, Lemma4.8] that V = ⊥ > C is a coaisle of a t-structure. (cid:3) We are ready to investigate colocalization properties of cosilting objects in D ( R ) . Lemma 5.6. Let C be a cosilting object in D ( R ) and p a prime ideal of R . Then C p is acosilting object in D ( R p ) . Furthermore, if ( U , V ) is the t-structure induced by C then C p inducesthe cosilting t-structure ( U p , V p ) in D ( R p ) .Proof. By Lemma 3.3, for any object X ∈ D ( R p ) we have Hom D ( R p ) ( X, C p ) ∼ = Hom D ( R ) ( X, C ) .From this we easily derive the property (i) and (iii) of Lemma 5.5 applied to the object C p ofthe category D ( R p ) . Also, we can infer from [20, Proposition 2.2] that C p ∈ ⊥ > C , and so theadjunction formula of Lemma 3.3 also yields condition (ii). We are left to show the condition(iv), that is, the subcategory ⊥ > C p is a coaisle in D ( R p ) .We denote V = ⊥ > C and U = ⊥ V so that ( U , V ) is the t-structure induced by the cosiltingobject C in D ( R ) . Recall from Lemma 3.4 that V p = V ∩ D ( R p ) and note that V p is equalas a subcategory of D ( R p ) to ⊥ > C p . But V p is a coaisle of a t-structure by Lemma 3.4, asdesired. (cid:3) Lemma 5.7. Let C be a pure-injective cosilting object in D ( R ) . Then D = Q m ∈ mSpec( R ) C m isa cosilting object in D ( R ) which is equivalent to C .Proof. We will again use Lemma 5.5 to show that D is a cosilting object. Set V = ⊥ > C . Themain observation we make is that for any X ∈ D ( R ) we have X ∈ ⊥ > D if and only if X m ∈ V for each maximal ideal m . This already shows that D is a cogenerator in D ( R ) . Because V is definable, Lemma 3.5 yields that X ∈ V whenever X m ∈ V for all m ∈ mSpec( R ) , and so = ⊥ > D . Because C is pure-injective, so is C m for each maximal ideal m , and therefore D is pure-injective. Finally, C m ∈ V for each maximal ideal m by [20, Proposition 2.2], and thus D ∈ V . Now we can apply Lemma 5.5 to infer that D is a cosilting object, and since V = ⊥ > D ,we have that D is equivalent to C . (cid:3) Definition 5.8. A cosilting object C is cotilting if Prod( C ) ⊆ ⊥ < C . Remark 5.9. The importance of the last definition comes from derived equivalences, here wefollow [31]. Let ( U , V ) be the t-structure induced by C and let H = V ∩ U [ − be the heart of thet-structure. Assume that C is a bounded cosilting complex. Then H is a Grothendieck category[5], and there is a realization functor real C : D b ( H ) → D b ( R ) between the bounded derivedcategories [31]. Then the cosilting object C is cotilting if and only if real C is an equivalence [31,Corollary 5.2]. Lemma 5.10. If C is a cotilting object in D ( R ) then C m is a cotilting object for any m ∈ mSpec( R ) . Furthermore, if C is pure-injective then Q m ∈ mSpec( R ) C m is a cotilting object in D ( R ) equivalent to C .Proof. By Lemma 5.7, the cosilting object C is equivalent to D = Q m ∈ mSpec( R ) C m , whichmeans that Prod( C ) = Prod( D ) . Therefore, Prod( C m ) ⊆ ⊥ < C , which by the usual adjunctionargument translates to Prod( C m ) ⊆ ⊥ < C m in D ( D m ) .If C is pure-injective then Q m ∈ mSpec( R ) C m is cosilting in D ( R ) by Lemma 5.7. Then it remainsto note that the equivalence between cosilting objects clearly preserves the cotilting property.Indeed, if C ′ and C ′′ are two equivalent cosilting objects then Prod( C ) = Prod( C ′ ) , and therefore Prod( C ) ⊆ ⊥ < C if and only if Prod( C ′ ) ⊆ ⊥ < C ′ . (cid:3) However, it is not true in general that if C m is a cotilting object for any m ∈ mSpec( R ) thenthe cosilting object C has to be cotilting as demonstrated in the following example. Example 5.11. Let k be a field and R = k ω be the countably infinite product of k . Recallthat R is a von Neumann regular ring, and therefore in particular every simple R -module isinjective. Let m be any maximal ideal of R such that Hom R ( R/ m , R ) = 0 . Such maximal idealsare plentiful — recall that maximal ideals of R are in a natural bijection with ultrafilters on ω and the desired property of m is satisfied if and only if the corresponding ultrafilter is notprincipal. We also set M = mSpec( R ) \ { m } .Put C = R/ m [0] ⊕ Q n ∈M R/ n [ − and we claim that C is a cosilting object in D ( R ) . Since C isa product of two shifted stalk complexes of injective R -modules, we infer that C is pure-injectiveobject of D ( R ) . By the injectivity, the orthogonal V = ⊥ > C is determined on cohomologyand can be easily computed: V = { X ∈ D ≥ | H ( X ) ∈ F } , where F = { M ∈ Mod - R | Hom R ( M, Q n ∈M R/ n ) = 0 } . It is easy to see that F = Add( R/ m ) = Prod( R/ m ) . It followsthat C ∈ V and that V is product-closed. By Lemma 5.5, C is a cosilting object.Next we show that C is not a cotilting object. From Hom R ( R/ m , R ) = 0 it follows thatthe intersection of all maximal ideals belonging to M is zero. Therefore, Q n ∈M R/ n contains acopy of R as a submodule. Since R/ m is injective, this yields Hom R ( Q n ∈M R/ n , R/ m ) = 0 , andtherefore there is a non-zero map C → C [ − in D ( R ) , witnessing that C is not cotilting. inally, let n be a maximal ideal. Then the colocalization C n is equal either to R/ m [0] in case n = m or to R/ n [ − in case n = m . In either case, C n is a shift of the injective cogenerator ofthe category of vector spaces over the field R/ n = R n , and so C n is a cotilting object in D ( R n ) .5.1. Cofinite type and compatible families of Thomason filtrations. We start by gen-eralizing [3, Theorem 3.8] and characterize the Thomason filtrations which are induced by acosilting object. Proposition 5.12. Let R be a commutative ring. There is a bijective correspondence betweenthe following families:(i) equivalence classes of cosilting objects of cofinite type in D ( R ) , and(ii) non-degenerate Thomason filtrations X = ( X n | n ∈ Z ) on Spec( R ) .The correspondence assigns to a cosilting object C of cofinite type the Thomason filtrationassociated to the compactly generated t-structure induced by C via Theorem 2.3.Proof. Let ( U , V ) be a compactly generated t-structure in D ( R ) corresponding to a Thomasonfiltration X . By [26, Theorem 4.6], the t-structure ( U , V ) is induced by a cosilting object if andonly if it is non-degenerate. So the result holds by the proof Theorem 3.15. (cid:3) Lemma 5.13. The bijection of Proposition 3.11 restricts to a bijection ( Non-degenerate Thomasonfiltrations X of Spec( R ) ) − ←−→ ( Compatible families { X ( m ) | m ∈ mSpec( R ) } of non-degenerate Thomason filtrations ) . Proof. The condition ( † ) (see §3.2) ensures that X n = S m ∈ mSpec( R ) X ( m ) ∗ n and X ( m ) ∗ n = X n ∩ Spec( R m ) ∗ for each n ∈ Z . This already implies that S n ∈ Z X n = Spec( R ) if and only if S n ∈ Z X ( m ) n = Spec( R m ) for each m ∈ mSpec( R ) as well as that T n ∈ Z X n = ∅ if and onlyif T n ∈ Z X ( m ) n = ∅ for each m ∈ mSpec( R ) . (cid:3) Definition 5.14. Let C ( m ) be a cosilting object of cofinite type in D ( R m ) for each m ∈ mSpec( R ) corresponding to a non-degenerate Thomason filtration X ( m ) in Spec( R m ) . The family { C ( m ) | m ∈ mSpec( R ) } is said to be compatible if the family { X ( m ) | m ∈ mSpec( R ) } is compatible.We say that two compatible families { C ( m ) | m ∈ mSpec( R ) } and { D ( m ) | m ∈ mSpec( R ) } of cosilting objects of cofinite type are equivalent if the cosilting objects C ( m ) and D ( m ) areequivalent for each m ∈ mSpec( R ) . Theorem 5.15. There is a bijection Cosilting objects C in D ( R ) of cofinite typeup to equivalence − ←−→ Compatible families { C ( m ) | m ∈ mSpec( R ) } of cosilting objects of cofinite typeup to equivalence induced by the assignment C 7→ { C m | m ∈ mSpec( R ) } and { C ( m ) | m ∈ mSpec( R ) } 7→ Y m ∈ mSpec( R ) C ( m ) . roof. The assignment C 7→ { C m | m ∈ mSpec( R ) } clearly preserves the appropriate equivalenceclasses and so is well-defined. Since C is equivalent to the cosilting object Q m C m in D ( R ) byLemma 5.7, the assignment is injective on equivalence classes. Let { C ( m ) | m ∈ mSpec( R ) } be a compatible family of cosilting objects of cofinite type which by the definition correspondsto a compatible family { X ( m ) | mSpec( R ) } of non-degenerate Thomason filtrations. Let X be the corresponding non-degenerate Thomason filtration on Spec( R ) via Lemma 5.13, whichfurther corresponds to a cosilting object C in D ( R ) via Proposition 5.12. Let ( U , V ) be the t-structure induced by C , then C m induces the t-structure ( U m , V m ) by Lemma 5.6 and Lemma 3.4.By Proposition 3.13(ii) we see that ( U m , V m ) corresponds to the Thomason filtration X ( m ) viaProposition 3.11. Then ( U m , V m ) is the t-structure induced by C ( m ) and therefore the cosiltingobjects C m and C ( m ) are equivalent in D ( R m ) for each m ∈ mSpec( R ) . Finally, let us usethis to show that D = Q m ∈ mSpec( R ) C ( m ) is a cosilting object in D ( R ) which is equivalent to C . We have Prod( C m ) = Prod( C ( m )) for any m ∈ mSpec( R ) . By Lemma 5.7, C is equiva-lent to Q m ∈ mSpec( R ) C m , and so Prod( C ) = Prod( Q m ∈ mSpec( R ) C m ) . Together, we showed that Prod( C ) = Prod( Q m ∈ mSpec( R ) C m ) = Prod( Q m ∈ mSpec( R ) C ( m )) = Prod( D ) . From this, it fol-lows easily that for any X ∈ D ( R ) and i ∈ Z we have Hom D ( R ) ( X, C [ i ]) = 0 if and only if Hom D ( R ) ( X, D [ i ]) = 0 , and so ( ⊥ ≤ C, ⊥ > C ) = ( ⊥ ≤ D, ⊥ > D ) , establishing that D is a cosiltingobject equivalent to C . (cid:3) We also have this auxiliary result using the local-global criterion for compact generation fromSection 3. Proposition 5.16. Let C be a pure-injective cosilting object in D ( R ) . Then C is of cofinite typeif and only if C m is of cofinite type for each m ∈ mSpec( R ) .Proof. Follows directly from Proposition 3.13. (cid:3) We say that a cosilting object C is n -term for some n ≥ if C is isomorphic in D ( R ) to acomplex of injective R -modules concentrated in degrees , , . . . , n − . Corollary 5.17. Let C be a cosilting object in D ( R ) and n ≥ . Then C is n -term if and onlyif C m is n -term for any m ∈ mSpec( R ) . In particular, the bijection of Theorem 5.15 restricts forany n ≥ to a bijection n -term cosilting objects C in D ( R ) of cofinite typeup to equivalence − ←−→ Compatible families { C ( m ) | m ∈ mSpec( R ) } of n -term cosilting objects of cofinite typeup to equivalence . Proof. Let us assume that C is already a complex of injective R -modules concentrated in degrees , , . . . , n − . In particular, C m = R Hom R ( R m , C ) ∼ = Hom R ( R m , C ) . As Hom R ( R m , − ) sendsinjective R -modules to injective R m -modules, this establishes that C m is n -term.For the converse, recall from Lemma 5.7 that C is equivalent to Q m ∈ mSpec( R ) C m . Since anyinjective R m -module is also injective as an R -module, we conclude that C is n -term providedthat C m is n -term for any m ∈ mSpec( R ) . (cid:3) orollary 5.18. If R is noetherian, we have bijections ( Pure-injective cosilting objects C in D ( R ) up to equivalence ) − ←−→ Compatible families { C ( m ) | m ∈ mSpec( R ) } of pure-injective cosilting objectsup to equivalence and ( n -term cosilting objects C in D ( R ) up to equivalence ) − ←−→ Compatible families { C ( m ) | m ∈ mSpec( R ) } of n -term cosilting objectsup to equivalence . Proof. Follows immediately by recalling Theorem 5.4 and Theorem 5.3. (cid:3) Recall that an R -module C is n -cotilting if and only if it is an n -term cosilting complex whenconsidered as an object D ( R ) by taking its stalk complex in degree zero. Then our correspondencealso restricts to the one of [38]. Recall from Remark 3.10 that our notions of equivalence andcompatible condition on families restrict perfectly well to the notions used in [38]. Corollary 5.19. If R is noetherian, we have bijections for any n > ( n -cotilting modules C in Mod - R up to equivalence ) − ←−→ ( Compatible families { C ( m ) | m ∈ mSpec( R ) } of n -cotilting modules up to equivalence ) . Proof. The only non-trivial task is to show that if C is an n -cotilting module then R Hom( R m , C ) is isomorphic to the R -module Hom R ( R m , C ) in D ( R ) . For this, it is sufficient to show that Ext iR ( R m , C ) = 0 for all i > . But this follows from the well-known fact that any cotilting classin Mod - R contains all flat R -modules. (cid:3) Cosilting modules over noetherian rings. Given an R -module M , we use Cogen( M ) to denote the subcategory of all R -modules cogenerated by M , that is, all R -modules admitting amonomorphism into an arbitrary direct product of copies of M . Given a map Q η → Q betweeninjective R -modules we define a subcategory B η = { M ∈ Mod - R | Hom R ( M, η ) is an epimorphism } . We say that an R -module C is a cosilting module if there is an injective copresentation → C → Q η → Q such that B η = Cogen( C ) . We say that two cosilting modules C, C ′ are equivalent if they induce the same cosilting class, that is, Cogen( C ) = Cogen( C ′ ) . It is well-known thatcosilting modules C and C ′ are equivalent if and only if Prod( C ) = Prod( C ′ ) .The cosilting modules were introduced a module-theoretic shadows of 2-term cosilting com-plexes in [14], dualizing results of [4]. More precisely, by an argument dual to that of [4, Theorem4.11], we have the following result. Theorem 5.20. Let R be a ring. Then there is a bijection ( -term cosilting complexesup to equivalence ) − ←−→ ( cosilting R -modulesup to equivalence ) . This correspondence assigns to a -term cosilting complex σ the R -module H ( σ ) . ow let R be a commutative noetherian ring. Then the equivalence classes of cosilting modulesare in natural bijection with Thomason sets [2]. Let → C → Q η → Q be an injectivecopresentation for an R -module C . If C is cosilting with respect to η then it will induce aunique torsion pair ( ⊥ C, Cogen( C )) up to equivalence, see [14, Corollary 3.5], where ⊥ C = { M ∈ Mod - R | Hom R ( M, C ) = 0 } . We know that over a commutative ring R , there is abijective correspondence between hereditary torsion pairs of finite type, and Thomason subsetsof Spec( R ) , see Theorem 2.4. Note that the cosilting class Cogen( C ) is a definable class in Mod - R , so it is closed under direct limits. Therefore, since R is noetherian, ( ⊥ C, Cogen( C )) is ahereditary torsion pair of finite type, see [2, Lemma 4.2]. The Thomason subset correspondingto ( ⊥ C, Cogen( C )) is Y = S { V ( I ) | I f.g. ideal such that R/I ∈ ⊥ C } .In Proposition 5.22, we show that this Thomason set coincides with the one obtained fromthe cosilting module by passing to the 2-term cosilting complex with Theorem 5.20, and thenextracting the zero term Thomason set of the filtration associated via Proposition 5.12. Definition 5.21. Let C ( m ) be a cosilting R m -module for each m ∈ mSpec( R ) correspondingto a Thomason subset X ( m ) in Spec( R m ) . The family { C ( m ) | m ∈ mSpec( R ) } is said to be compatible if the family { X ( m ) | m ∈ mSpec( R ) } is compatible. We say that two compatiblefamilies { C ( m ) | m ∈ mSpec( R ) } and { D ( m ) | m ∈ mSpec( R ) } of cosilting objects of cofinite typeare equivalent if the cosilting modules C ( m ) and D ( m ) are equivalent for each m ∈ mSpec( R ) . Proposition 5.22. Let R be a commutative noetherian ring. Then there is a bijection Compatible families { σ ( m ) | m ∈ mSpec( R ) } of -term cosilting complexesup to equivalence − ←−→ Compatible families { C ( m ) | m ∈ mSpec( R ) } of cosilting R m -modulesup to equivalence . This correspondence assigns to a -term cosilting complex σ ( m ) the R -module H ( σ ( m )) .Proof. The assignment σ ( m ) → H ( σ ( m )) clearly preserves the equivalence and so is well-defined.By Theorem 5.20, it remains to prove that the class { σ ( m ) | m ∈ mSpec( R ) } is compatible ifand only if { C ( m ) | m ∈ mSpec( R ) } is compatible. Let X ( m ) = ( X ( m ) n | n ∈ Z ) be the non-degenerate Thomason filtration of Spec( R m ) corresponding to σ ( m ) for each m ∈ mSpec( R ) .Recall that the family { X ( m ) | m ∈ mSpec( R ) } is compatible if { X ( m ) n | m ∈ mSpec( R ) } is acompatible family of Thomason subsets for each n ∈ Z . Let Y ( m ) be the Thomason subset of Spec( R m ) corresponding to C ( m ) for each m ∈ mSpec( R ) .First, we claim that X ( m ) = Y ( m ) . In fact, by [20, Theorem 5.1], we have X ( m ) = S { V ( I m ) | I is an ideal of R such that R m /I m ∈ ⊥ ≤ σ ( m ) } , Y ( m ) = S { V ( J m ) | J is an ideal of R such that R m /J m ∈ ⊥ H ( σ ( m )) } .Since σ ( m ) is a -term cosilting complex concentrated in degree 0 and 1, we see that H i ( σ ( m )) = 0 for all i < . Then the claim follows by Hom D ( R m ) ( R m /I m , σ ( m )[ < and the isomorphism Hom D ( R m ) ( R m /I m , σ ( m )) ∼ = Hom R m ( R m /I m , H ( σ ( m ))) . oreover, one can check that X ( m ) n = S { V ( I m ) | I is an ideal of R such that R m /I m [ − n ] ∈ ⊥ ≤ σ ( m ) } = Spec( R m ) for all n < and X ( m ) n = X ( m ) n +1 = · · · for all n ≥ . On theother hand, since the Thomason filtration X ( m ) is non-degenerate, that is, T n ∈ Z X ( m ) n = ∅ and S n ∈ Z X ( m ) n = Spec( R ) , we see that X ( m ) n = ∅ for all n ≥ , and X ( m ) n = Spec( R ) forall n < . Therefore, we easily get that { X ( m ) | m ∈ mSpec( R ) } is compatible if and only if { Y ( m ) | m ∈ mSpec( R ) } is compatible. (cid:3) Corollary 5.23. Let R be a commutative noetherian ring. Then there is a bijection ( cosilting R -modulesup to equivalence ) − ←−→ ( Compatible families { C ( m ) | m ∈ mSpec( R ) } of cosilting R m -modules up to equivalence ) induced by the assignment C 7→ { C m | m ∈ mSpec( R ) } and { C ( m ) | m ∈ mSpec( R ) } 7→ Y m ∈ mSpec( R ) C ( m ) . Proof. By Corollary 5.18, we know that there are bijections between -term cosilting complexes C in D ( R ) up to equivalence and compatible families { C ( m ) | m ∈ mSpec( R ) } of -term cosiltingcomplexes up to equivalence. Thus the result follows by Theorem 5.20 and Proposition 5.22. (cid:3) Silting objects Let us fix an injective cogenerator W in Mod - R , and denote by ( − ) + the duality functor ( − ) + = R Hom R ( − , W ) : D ( R ) → D ( R ) . The following result extends the well-known explicitduality between n -tilting R -modules and n -cotilting R -modules of cofinite type. Theorem 6.1. [3, Theorem 3.3, Theorem 3.8] Let us consider the assignment Φ : T T + onobjects of D ( R ) . Then:(i) Φ induces an injective map from the set of equivalence classes of silting objects in D ( R ) of finite type to cosilting objects in D ( R ) of cofinite type.(ii) Φ induces a bijective map from the set of equivalence classes of bounded silting complexesin D ( R ) to bounded cosilting complexes in D ( R ) of cofinite type.(iii) If R is commutative noetherian then Φ induces an bijective map from the set of equivalenceclasses of silting objects in D ( R ) of finite type to pure-injective cosilting objects in D ( R ) . Definition 6.2. Let T ( m ) be a silting object of finite type in D ( R m ) for each m ∈ mSpec( R ) , C ( m ) = T ( m ) + the cosilting object of cofinite type obtained via Theorem 6.1, which furthercorresponds to a non-degenerate Thomason filtration X ( m ) in Spec( R m ) via Proposition 5.12.The family { T ( m ) | m ∈ mSpec( R ) } is said to be compatible if the family { X ( m ) | m ∈ mSpec( R ) } is compatible. We say that two compatible families { T ( m ) | m ∈ mSpec( R ) } and { U ( m ) | m ∈ mSpec( R ) } of silting objects of finite type are equivalent if the silting objects T ( m ) and U ( m ) are equivalent for each m ∈ mSpec( R ) . emma 6.3. Let T ∈ D ( R ) be a silting object and p ∈ Spec( R ) . Then T p is a silting object in D ( R p ) . If T is of finite type in D ( R ) then so is T p in D ( R p ) .Proof. The proof is completely dual to that of Lemma 5.6. The second claim is straightforwardto check. (cid:3) A silting object T in D ( R ) is n -term for some n ≥ if T is isomorphic in D ( R ) to a com-plex of projective R -modules concentrated in degrees − ( n − , − ( n − , . . . , − , . In view ofTheorem 6.1, if T is an n -term silting object then T + is clearly an n -term cosilting object of D ( R ) .For any maximal ideal m , let us denote ( − ) + m = R Hom R m ( − , W m ) and note that W m is aninjective cogenerator for the category Mod - R m . By the adjunction, we have for any X ∈ D ( R ) the simple formula ( X + ) m = ( X m ) + m . Theorem 6.4. For any n ≥ , there is a bijection ( n -term silting objects T in D ( R ) up to equivalence ) − ←−→ ( Compatible families { T ( m ) | m ∈ mSpec( R ) } of n -term silting objects up to equivalence ) . If R is commutative noetherian, there is also a bijection Silting objects T in D ( R ) of finite typeup to equivalence − ←−→ Compatible families { T ( m ) | m ∈ mSpec( R ) } of silting objects of finite typeup to equivalence . Both bijections are induced by the assignment T 7→ { T m | m ∈ mSpec( R ) } .Proof. Let T be a silting object in D ( R ) , m ∈ mSpec( R ) , C = T + and let { C ( m ) | m ∈ mSpec( R ) } be the compatible family of cosilting objects corresponding to C via Proposition 5.12.Note that we can choose C ( m ) = C m for all m ∈ mSpec( R ) . By the formula above, we see that C ( m ) = ( T + ) m = ( T m ) + m . Together with Lemma 6.3, this yields that { T m | mSpec( R ) } is acompatible family of silting objects and so both the assignment is well-defined. By Theorem 6.1,the assignment ( − ) + is injective on equivalence classes on silting objects, and therefore so is theassignment T 7→ { T m | m ∈ mSpec( R ) } .It remains to show that the assignment T 7→ { T m | m ∈ mSpec( R ) } is surjective. Let { T m | m ∈ mSpec( R ) } be a compatible family of silting objects of finite type and let C ( m ) = T ( m ) + = T ( m ) + m for each m ∈ mSpec( R ) . Then { C ( m ) | m ∈ mSpec( R ) } is a compatible family ofcosilting objects of cofinite type, so there is a corresponding cosilting object C ∈ D ( R ) viaTheorem 5.15. If either C is n -term, or R is commutative noetherian then Theorem 6.1 yieldsa silting object T of finite type such that T + is equivalent to C as a cosilting object. Nowfor reach m ∈ mSpec( R ) , ( T m ) + m = ( T + ) m is a cosilting object in D ( R m ) equivalent to C ( m ) .Therefore, Theorem 6.1 again implies that T m is equivalent to T ( m ) as silting object for each m ∈ mSpec( R ) . (cid:3) Remark 6.5. In view of Theorem 5.15, it would be tempting to express the converse assignmentof the bijection of Theorem 6.4 in terms of the coproduct L m ∈ mSpec( R ) T ( m ) . However, givena silting object T the coproduct L m ∈ mSpec( R ) T m is often not a silting object anymore, see [38,Remark 2.9]. 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Zhu) Department of Mathematics, Nanjing University, Nanjing 210093, China Email address : [email protected]@hotmail.com