From weight structures to (orthogonal) t -structures and back
aa r X i v : . [ m a t h . K T ] J u l From weight structures to (orthogonal) t -structures and back Mikhail V. Bondarko ∗ July 9, 2019
Abstract
In this paper we study t -structures that are "closely related" to weightstructures (on a triangulated category C ). A t -structure couple t =( C t ≤ , C t ≥ ) is said to be right adjacent to a weight structure w =( C w ≤ , C w ≥ ) if C t ≥ = C w ≥ ; if this is the case then t can be uniquelyrecovered from w and vice versa. We prove that if C satisfies the Brownrepresentability property (one may say that this is the case for any "rea-sonable" triangulated category closed with respect to coproducts) then t that is right adjacent to w exists if and only if w is smashing (i.e., co-products respect weight decompositions); in this case the heart Ht is thecategory of those functors Hw op → Ab that respect products (here Hw is the heart of w ). Certainly, the dual to this statement is valid is well,and we discuss its relationship to results of B. Keller and P. Nicolas.We also prove several generalizations and modifications of this re-sult. In particular, we prove that a right adjacent t exists whenever w is a bounded weight structure on a saturated R -linear category C (fora noetherian ring R ). Moreover, we obtain 1-to-1 correspondences be-tween bounded weights structures on C and the classes of those bounded t -structures on it such that Ht has either enough projectives or injectiveswhenever C equals the derived category of perfect complexes D perf ( X ) for X that is regular and proper over Spec R .Furthermore, we generalize the aforementioned existence statement toconstruct (under certain assumptions) a t -structure t on a triangulatedcategory C ′ that is right orthogonal to w ; here C and C ′ are subcategoriesof a common triangulated category D . In particular, if X is proper over Spec R but not necessarily regular then one can take C = D perf ( X ) , C ′ = D bcoh ( X ) or C ′ = D − coh ( X ) , and D = D qc ( X ) . We also study hearts oforthogonal t -structures and their restrictions, and prove some statementson "reconstructing" weight structures from orthogonal t -structures.The main tool of this paper is the notion virtual t -truncations of (co-homological) functors; these are defined in terms of weight structuresand "behave as if they come from t -truncations of representing objects"whether t exists or not. Primary 18E30; Secondary18E40, 18F20, 18G05, 18E10, 16E65. ∗ The main results of the paper were obtained under support of the Russian Science Foun-dation grant no. 16-11-00200. ey words and phrases Triangulated category, weight structure, t -structure, virtual t -truncation, pure functor, coherent sheaves, perfect com-plexes. Contents t -structures 5 t -structure notation . . . . . . . . . . . . . 51.2 Some basics on weight structures . . . . . . . . . . . . . . . . . . 81.3 On orthogonal and adjacent structures . . . . . . . . . . . . . . . 10 t -truncations and their relation to orthogonal t -structures 11 t -truncations and their weight range . . . . . . . . . . . . 122.2 General criteria for the existence of adjacent and orthogonal t -structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 t -structures orthogonal to smashing weight structures 19 t -structures orthogonal to smashing weightstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Weight structures extended from subcategories of compact ob-jects, and orthogonal t -structures . . . . . . . . . . . . . . . . . . 26 t -structures related to saturated categories and coherentsheaves 28 t -structures corresponding to (locally) finite func-tors into R -modules . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 On coherent sheaf examples for the theorem . . . . . . . . . . . . 324.3 Other possible examples related to coherent sheaves . . . . . . . 344.4 On perfect complexes and coherent rings: a reminder . . . . . . . 36 Introduction
This paper is dedicated to the study of those t -structures that are "closelyrelated" to weight structures (on various triangulated categories).Let us recall a bit of history. t -structures on triangulated categories have be-come important tools in homological algebra since their introduction in [BBD82].Respectively, their study and construction is an actual and non-trivial question.Next, in [Pau08] and [Bon10a] a rather similar notion of a weight structure w on a triangulated category C was introduced. Moreover, in ibid. a t -structure t = ( C t ≤ , C t ≥ ) was said to be adjacent to w if C t ≥ = C w ≥ , and certain2xamples of adjacent structures were constructed. Furthermore, in [Bon10b]for a t -structure t on a triangulated category C ′ that is related to C by meansof a duality bi-functor a more general notion of (left) orthogonality of a weightstructure w on C to t was introduced. Also, the relationship between the heartsof adjacent and orthogonal structures was studied in detail.Next, if w is (left or right) adjacent to t then it determines t uniquely and viceversa. Yet the only previously existing way of constructing t if w is given wasto use certain "nice generators" of w (see Definition 3.2.1(5), §4.5 of [Bon10a],Theorem 3.2.2 of [Bon18b], and Proposition 5.3.1 of [Bon16]). However, alreadyin [Bon10a] the notion of virtual t -truncations for (co)homological functors wasintroduced, and it was proved that virtual t -truncations possess several niceproperties. In particular, it was demonstrated that these are closely related to t -structures (whence the name) even though they are defined in terms of weightstructures only.In the current paper we propose a new construction method. We provethat adjacent and orthogonal t -structures can be constructed using virtual t -truncations whenever certain "Brown representability-type" assumptions on C (and C ′ ) are known. Respectively, our results yield the existence of some newfamilies of t -structures.Let us formulate one of these results. For a triangulated category C that is smashing , i.e., closed with respect to (small) coproducts, and a weight structure w on it we will say that w is smashing whenever C w ≥ is closed with respect to C -coproducts (note that C w ≤ is ` -closed automatically). Theorem 0.1 (See Theorem 3.2.3(I)) . Let C be a smashing triangulated cat-egory that satisfies the following Brown representability property: any functor C op → Ab that respects ( C op )-products is representable.Then for a weight structure w on C there exists a t -structure t right adjacentto it if and only if w is smashing. Moreover, the heart of t (if t exists) isequivalent to the category of all those additive functors Hw op → Ab that respectproducts. Note here that (smashing) triangulated categories satisfying the Brown rep-resentability property have recently become very popular in homological algebraand found applications in various areas of mathematics (thanks to the foun-dational results of A. Neeman and others); in particular, this property holdsif either C or C op is compactly generated. Moreover, it is easy to constructvast families of smashing weight structures on C (at least) if C is compactlygenerated; see Remark 3.2.4(1) below. The resulting adjacent t -structures are cosmashing (i.e. C t ≤ is closed with respect to C -products); thus there appearsto be no way to construct them using previously known methods.Certainly the dual to Theorem 0.1 is valid as well. Moreover, if C op sat-isfies the dual Brown representability property and w is both cosmashing and In contrast to ibid. and [Bon10b], in the current paper we use the so-called homologicalconvention for t and w , and say that t is right adjacent to w if C t ≥ = C w ≥ . Besides,D. Paukstello and several other authors use the term "co- t -structure" instead of "weightstructure". These definitions are contained in Definition 5.2.1 below; however, our main exampleshave motivated us to concentrate on the particular case where C and C ′ are subcategories ofa common triangulated category D for most of this paper. Here Hw is the heart of w ; note also that G : Hw op → Ab respects products whenever itconverts Hw -coproducts into products of groups. t -structure t (i.e., C t ≤ = C w ≤ ) restricts to thesubcategory of compact objects of C as well as to all other "levels of smallness"for objects. Combining this statement with an existence of weight structuresTheorem 3.1 of [KeN13] we obtain a statement on t -structures extending The-orem 7.1 of ibid.We also prove certain alternative versions of Theorem 0.1 that can be appliedto "quite small" triangulated categories. Instead of the Brown representabilitycondition for C one can demand it to satisfy the R -saturatedness one instead(see Definition 4.1.1(2) below; this is an " R -linear finite" version of the Brownrepresentability). Then for any bounded w on C there will exist a t -structureright adjacent to it. According to a saturatedness statement from [Nee18a] thisresult can be applied to the derived category D perf ( X ) of perfect complexes ona regular scheme that is proper over the spectrum of a Noetherian ring R (seeProposition 4.2.1(2)). In this case we obtain 1-to-1 correspondences betweenbounded weights structures on C and the classes of those bounded t -structureson it such that the heart Ht of t has either enough projectives or injectives; seeRemark 4.2.2(3).Moreover, we prove some generalizations of this " R -saturated" existenceresult (see Proposition 4.2.1(1) and Theorem 4.1.2); they produce orthogonal t -structures on bounded and bounded above derived categories of coherent sheavesover X for X that is not necessarily regular.Furthermore, we study the question when a (fixed) t -structure t on a tri-angulated category C ′ is right orthogonal to a weight structure w on a certaincategory C . If C is a triangulated subcategory of C ′ and w that is left orthogo-nal to t exists then Ht has enough projectives, and in certain cases this propertyof Ht is sufficient for the existence of w ; see Theorem 5.3.1. Under some otherassumptions we prove the existence of a weight structure w (on a triangulatedcategory C that is "large enough") that is (strictly) left orthogonal to t with C not being a subcategory of C ′ . However, these assumptions on the heart makethese result more difficult to apply than the aforementioned "converse" ones,and the methods of their proofs are less interesting. Remark . The author certainly does not claim that the methods of the cur-rent paper are the most general among the existing methods for constructing t -structures. In particular, if C is a well generated triangulated category (inparticular, this is the case if C is compactly generated or possesses a combina-torial Quillen model; see Proposition 6.10 of [Ros05]) then the recent Theorem2.3 of [Nee18c] gives all those t -structures that are generated by sets of objectsof C in the sense of Definition 3.2.1(6) below; this statement essentially vastlygeneralizes the well-known Theorem A.1 of [AJS03].Now, if C is well generated then is generated by a set of its objects as itsown localizing subcategory (see Definition 3.2.1(3)). Thus for any smashingweight structure w on it Proposition 2.3.2(10) of [Bon18b] essentially says thefollowing: there exists a set P ⊂ C w ≥ such that the class C w ≥ is the smallestcocomplete pre-aisle that is "generated" by P in the sense of [Nee18c, §0] (cf.Discussion 1.15 of ibid.). Hence Theorem 2.3 of [Nee18c] says that there existsa t -structure on C such that C t ≥ = C w ≥ (see Remark 1.1.3(4) below). Henceloc. cit. generalizes the existence of t part of our Theorem 0.1.On the other hand, note that neither loc. cit. nor Theorem A.1 of [AJS03]says anything on the hearts of t -structures. Also, there appears to be no chance4o extend these existence results to R -saturated categories (in any way).Let us now describe the contents of the paper. Some more information ofthis sort may be found in the beginnings of sections.In §1 we give some definitions and conventions, and recall some basics on t -structures and weight structures.In §2 we define virtual t -truncations of functors and prove several nice prop-erties for them. We also relate the existence of orthogonal t -structures to virtual t -truncations; this gives general if and only if criteria for the existence of adja-cent t -structures.In §3 we study smashing triangulated categories along with the existence of t -structures adjacent to weight structures on them and certain restrictions of these t -structures. We also consider weight structures extended from subcategories ofcompact objects.In §4 we study adjacent and orthogonal t -structures on R -linear triangulatedcategories; this includes R -saturated categories and various derived categoriesof (quasi)coherent sheaves.In §5 we try to answer the question whether the results of previous sectionsgive all t -structures that are adjacent to weight structures on the correspondingcategories. So we study criteria ensuring the existence of a weight structure thatis left adjacent or orthogonal to a given t -structure t ; under certain assumptionswe prove that the answer to our question is affirmative.The author is deeply grateful to prof. A. Neeman for calling his attentionto [KeN13] as well as for writing his extremely interesting texts that are crucialfor the current paper, and also to prof. L. Positselski for an online lesson on thecoherence of rings. t -structures In this section we recall the notions of t -structures and weight structures, alongwith orthogonality and adjacency for them.In §1.1 we introduce some categorical notation and recall some basics on t -structures.In §1.2 we recall some of the theory of weight structures.In §1.3 we recall the definitions of adjacent and orthogonal weight and t -structures that are central for this paper. t -structure notation • All products and coproducts in this paper will be small. • Given a category C and X, Y ∈ Obj C we will write C ( X, Y ) for the setof morphisms from X to Y in C . • For categories C ′ , C we write C ′ ⊂ C if C ′ is a full subcategory of C . • Given a category C and X, Y ∈ Obj C , we say that X is a retract of Y if id X can be factored through Y . Certainly, if C is triangulated or abelian, then X is a retract of Y if and only if X is itsdirect summand. A (not necessarily additive) subcategory H of an additive category C issaid to be retraction-closed in C if it contains all retracts of its objects in C . • For any ( C, H ) as above the full subcategory Kar C ( H ) of C whose objectsare all retracts of (finite) direct sums of objects H in C will be calledthe Karoubi-closure of H in C ; note that this subcategory is obviouslyadditive and retraction-closed in C . • We will say that C is Karoubian if any idempotent morphism yields adirect sum decomposition in it. • The symbol C below will always denote some triangulated category; itwill often be endowed with a weight structure w . The symbols C ′ and D will also be used for triangulated categories only. • For any
A, B, C ∈ Obj C we will say that C is an extension of B by A ifthere exists a distinguished triangle A → C → B → A [1] . • A class
P ⊂
Obj C is said to be extension-closed if it is closed with respectto extensions and contains . • We will write hPi for the smallest full retraction-closed triangulated sub-category of C containing P ; we will call hPi the triangulated subcategory densely generated by P (in particular, in the case C = hPi ).Moreover, the smallest strict full triangulated subcategory of C contain-ing P will be called the subcategory strongly generated by P . • The smallest additive retraction-closed extension-closed class of objects of C containing P will be called the envelope of P . • For
X, Y ∈ Obj C we will write X ⊥ Y if C ( X, Y ) = { } .For D, E ⊂ Obj C we write D ⊥ E if X ⊥ Y for all X ∈ D, Y ∈ E .Given D ⊂ Obj C we will write D ⊥ for the class { Y ∈ Obj C : X ⊥ Y ∀ X ∈ D } . Dually, ⊥ D is the class { Y ∈ Obj C : Y ⊥ X ∀ X ∈ D } . • Given f ∈ C ( X, Y ) , where X, Y ∈ Obj C , we will call the third vertex of(any) distinguished triangle X f → Y → Z a cone of f . • Below A will always denote some abelian category; B is an additive cate-gory. • All complexes in this paper will be cohomological.We will write K ( B ) for the homotopy category of complexes over B . Itsfull subcategory of bounded complexes will be denoted by K b ( B ) . Wewill write M = ( M i ) if M i are the terms of the complex M (in thecohomological indexing). Recall that different choices of cones are connected by non-unique isomorphisms. We will say that an additive covariant (resp. contravariant) functor from C into A is homological (resp. cohomological ) if it converts distinguishedtriangles into long exact sequences.For a (co)homological functor H and i ∈ Z we will write H i (resp. H i )for the composition H ◦ [ − i ] .Let us now recall the notion of a t -structure (mainly to fix notation). Definition 1.1.1.
A couple of subclasses C t ≤ , C t ≥ ⊂ Obj C will be said tobe a t -structure t on C if they satisfy the following conditions:(i) C t ≤ and C t ≥ are strict, i.e., contain all objects of C isomorphic to theirelements.(ii) C t ≤ ⊂ C t ≤ [1] and C t ≥ [1] ⊂ C t ≥ .(iii) C t ≥ [1] ⊥ C t ≤ .(iv) For any M ∈ Obj C there exists a t -decomposition distinguished triangle L t M → M → R t M → L t M [1] (1.1.1)such that L t M ∈ C t ≥ , R t M ∈ C t ≤ [ − .2. Ht is the full subcategory of C whose object class is C t =0 = C t ≤ ∩ C t ≥ .We will also give some auxiliary definitions. Definition 1.1.2.
1. For any i ∈ Z we will use the notation C t ≤ i (resp. C t ≥ i )for the class C t ≤ [ i ] (resp. C t ≥ [ i ] ).2. Ht is the full subcategory of C whose object class is C t =0 = C t ≤ ∩ C t ≥ .3. We will say that t is left (resp. right) non-degenerate if ∩ i ∈ Z C t ≥ i = { } (resp. ∩ i ∈ Z C t ≤ i = { } ).Moreover, we will say that t is non-degenerate if it is both left and rightnon-degenerate.4. We will say that t is bounded below if ∪ i ∈ Z C t ≥ i = Obj C .Moreover, we will say that t is bounded if the equality ∪ i ∈ Z C t ≤ i = Obj C isvalid as well.5. Let D be a full triangulated subcategory of C .We will say that t restricts to D whenever the couple t D = ( C t ≤ ∩ Obj
D, C t ≥ ∩ Obj D ) is a t -structure on D . Remark . Let us recall some well-known properties of t -structures (cf. §1.3of [BBD82]).1. The triangle (1.1.1) is canonically and functorially determined by M .Moreover, L t is right adjoint to the embedding C t ≥ → C (if we consider C t ≥ as a full subcategory of C ) and R t is left adjoint to the embedding C t ≤− → C ;respectively, L t and R t are connected with id C by means of canonical naturaltransformations.For any n ∈ Z we will use the notation t ≥ n for the functor [ − n ] ◦ L t ◦ [ n ] ,and t ≤ n = [ − n − ◦ L t ◦ [ n + 1] .2. Ht is an abelian category with short exact sequences corresponding todistinguished triangles in C .Moreover, have a canonical isomorphism of functors L t ◦ [1] ◦ R t ◦ [ − ∼ =[1] ◦ R t ◦ [ − ◦ L t (if we consider these functors as endofunctors of C ). Thiscomposite functor H t actually takes values in Ht ⊂ C , and it is homological ifconsidered this way. 7. We have C t ≥ = ⊥ C t ≤− . Thus t is uniquely determined by C t ≤ .Moreover, the notion of t -structure is self-dual (cf. Proposition 1.2.4(1)below). Hence C t ≤ = ( C t ≥ ) ⊥ , and thus t is uniquely determined by C t ≥ aswell.4. Though in [BBD82] where t -structures were introduced, in the papersof A. Neeman mentioning this notion, and in several preceding papers of theauthor the "cohomological convention" for t -structures was used, in the currenttext we use the homological convention; the reason for this is that it is coher-ent with the homological convention for weight structures (see Remark 1.2.3(3)below). Respectively, our notation C t ≥ corresponds to the class C t ≤ in thecohomological convention. Let us recall some basic definitions of the theory of weight structures.
Definition 1.2.1.
I. A pair of subclasses C w ≤ , C w ≥ ⊂ Obj C will be saidto define a weight structure w on a triangulated category C if they satisfy thefollowing conditions.(i) C w ≤ and C w ≥ are retraction-closed in C (i.e., contain all C -retracts oftheir objects).(ii) Semi-invariance with respect to translations. C w ≤ ⊂ C w ≤ [1] , C w ≥ [1] ⊂ C w ≥ .(iii) Orthogonality. C w ≤ ⊥ C w ≥ [1] .(iv) Weight decompositions .For any M ∈ Obj C there exists a distinguished triangle L w M → M → R w M → L w M [1] such that L w M ∈ C w ≤ and R w M ∈ C w ≥ [1] .We will also need the following definitions. Definition 1.2.2.
Let i, j ∈ Z ; assume that a triangulated category C is en-dowed with a weight structure w .1. The full category Hw ⊂ C whose objects are C w =0 = C w ≥ ∩ C w ≤ iscalled the heart of w .2. C w ≥ i (resp. C w ≤ i , resp. C w = i ) will denote the class C w ≥ [ i ] (resp. C w ≤ [ i ] , resp. C w =0 [ i ] ).3. C [ i,j ] denotes C w ≥ i ∩ C w ≤ j .
4. Let D be a full triangulated subcategory of C .We will say that w restricts to D whenever the couple w D = ( C w ≤ ∩ Obj
D, C w ≥ ∩ Obj D ) is a weight structure on D .Moreover, in this case we will also say that w is an extension of w D . If i > j and M ∈ C [ i,j ] then M ⊥ M by the orthogonality axiom; thus C [ i,j ] = { } .
8. We will say that M is left (resp., right) w -degenerate (or weight-degenerate if the choice of w is clear) if M belongs to ∩ i ∈ Z C w ≥ i (resp. to ∩ i ∈ Z C w ≤ i ).6. We will say that w is left (resp., right) non-degenerate if all left (resp.right) weight-degenerate objects of C are zero.7. We will call ∪ i ∈ Z C w ≥ i (resp. ∪ i ∈ Z C w ≤ i ) the class of w -bounded below (resp., w -bounded above ) objects of C .Moreover, we will say that w is bounded below (resp. bounded above , resp. bounded ) if all objects of C are bounded below (resp. bounded above,resp. are bounded both below and above).8. We will say that a subcategory H ⊂ C is negative (in C ) if Obj H ⊥ ( ∪ i> Obj( H [ i ])) . Remark .
1. A simple (and still useful) example of a weight structurecomes from the stupid filtration on the homotopy categories of cohomologicalcomplexes K ( B ) for an arbitrary additive B (it can also be restricted to boundedcomplexes; see Definition 1.2.2(4)). In this case K ( B ) w st ≤ (resp. K ( B ) w st ≥ )is the class of objects that are homotopy equivalent to complexes concentratedin degrees ≥ (resp. ≤ ); see Remark 1.2.3(1) of [BoS18] for more detail.We will use this notation below. The heart of this weight structure w st isthe Karoubi-closure of B in K ( B ) ; hence it is equivalent to Kar( B ) (see Remark2.1.4(2) of [BoS19]).2. A weight decomposition (of any M ∈ Obj C ) is almost never canonical.Still for any m ∈ Z the axiom (iv) gives the existence of a distinguishedtriangle w ≤ m M → M → w ≥ m +1 M → ( w ≤ m M )[1] (1.2.1)with some w ≤ m M ∈ C w ≤ m and w ≥ m +1 M ∈ C w ≥ m +1 ; we will call it an m -weight decomposition of M .We will often use this notation below (even though w ≥ m +1 M and w ≤ m M are not canonically determined by M ); we will call any possible choice either of w ≥ m +1 M or of w ≤ m M (for any m ∈ Z ) a weight truncation of M . Moreover,when we will write arrows of the type w ≤ m M → M or M → w ≥ m +1 M we willalways assume that they come from some m -weight decomposition of M .3. In the current paper we use the “homological convention” for weight struc-tures; it was previously used in [BoS18], [Bon16], [BoK18], in [BoV19], and in[BoS19], whereas in [KeN13], [Bon10a], and in [Bon10b] the “cohomological con-vention” was used. In the latter convention the roles of C w ≤ and C w ≥ areinterchanged, i.e., one considers C w ≤ = C w ≥ and C w ≥ = C w ≤ . So, a com-plex X ∈ Obj K ( B ) whose only non-zero term is the fifth one (i.e., X = 0 ) hasweight − in the homological convention, and has weight in the cohomologicalconvention. Thus the conventions differ by “signs of weights”; K ( B ) [ i,j ] is theclass of retracts of complexes concentrated in degrees [ − j, − i ] .We also recall that D. Pauksztello has introduced weight structures indepen-dently (in [Pau08]); he called them co-t-structures. Proposition 1.2.4.
Let m ≤ n ∈ Z , M, M ′ ∈ Obj C , g ∈ C ( M, M ′ ) .1. The axiomatics of weight structures is self-dual, i.e., for C ′ = C op (so Obj C ′ = Obj C ) there exists the (opposite) weight structure w ′ for which C ′ w ′ ≤ = C w ≥ and C ′ w ′ ≥ = C w ≤ .9. C w ≥ = ( C w ≤− ) ⊥ and C w ≤ = ⊥ C w ≥ .3. C w ≤ is closed with respect to all (small) coproducts that exist in C .4. C w ≤ , C w ≥ , and C w =0 are additive and extension-closed.5. For any (fixed) m -weight decomposition of M and an n -weight decompo-sition of M ′ (see Remark 1.2.3(2)) g can be extended to a morphism ofthe corresponding distinguished triangles: w ≤ m M c −−−−→ M −−−−→ w ≥ m +1 M y h y g y j w ≤ n M ′ −−−−→ M ′ −−−−→ w ≥ n +1 M ′ (1.2.2)Moreover, if m < n then this extension is unique (provided that the rowsare fixed).6. If A → B → C → A [1] is a C -distinguished triangle and A, C ∈ C w =0 then this distinguished triangle splits; hence B ∼ = A L C ∈ C w =0 .7. If M belongs to C w ≤ (resp. to C w ≥ ) then it is a retract of any choiceof w ≤ M (resp. of w ≥ M ).8. If M ∈ C w ≥ m then w ≤ n M ∈ C [ m,n ] (for any n -weight decomposition of M ).9. The class C [ m,l ] is the extension-closure of ∪ m ≤ j ≤ l C w = j .10. Let v be another weight structure for C ; assume C w ≤ ⊂ C v ≤ and C w ≥ ⊂ C v ≥ . Then w = v (i.e., these inclusions are equalities). Proof.
All of these statements were essentially proved in [Bon10a] (yet pay at-tention to Remark 1.2.3(3) above!).
Now let us give a certain definition of orthogonality for weight and t -structures.Till §5.2 we will only consider a particular case of the general notion introducedin [Bon10b] (see Remark 1.3.4(1) and Definition 5.2.1 below). Definition 1.3.1.
Assume that C and C ′ are (full) triangulated subcategoriesof a triangulated category D , w is a weight structure on C and t is a t -structureon C ′
1. We will say that w is left orthogonal (or left D -orthogonal ) to t or that t is right orthogonal to w whenever C w ≤ ⊥ D C ′ t ≥ and C w ≥ ⊥ D C ′ t ≤− .2. Dually, we will say that w is right orthogonal (or right D -orthogonal ) to t or that t is left orthogonal to w whenever C ′ t ≥ ⊥ D C w ≤ and C ′ t ≤− ⊥ D C w ≥ .3. If C = C ′ = D and w is left or right orthogonal to t we will also say that w is (left or right) adjacent to t .4. We will say that t is strictly right orthogonal to w and w is strictly leftorthogonal to t if C ′ t ≥ = C ⊥ D w ≤ ∩ Obj C ′ and C ′ t ≤− = ⊥ D C w ≥ ∩ Obj C ′ .10 emark . We will mostly treat the case where w is left orthogonal to t .Respectively, we will say that w is orthogonal (resp. adjacent) to t of that t isorthogonal to w to mean that w is left orthogonal (resp. adjacent) to t and t isright orthogonal to w .Let us now relate the latter definition to the notion of adjacent structuresintroduced in [Bon10a]. Proposition 1.3.3.
For C , w , and t as in Definition 1.3.1(3) we have thefollowing: w is (left) adjacent to t if and only if C w ≥ = C t ≥ . Proof. If C w ≥ = C t ≥ then w is (left) adjacent to t immediately from theorthogonality axioms of weight and t -structures (see Definition 1.1.1(iii) andDefinition 1.2.1(iii)). Conversely, if w is adjacent to t then combining the or-thogonality conditions with Proposition 1.2.4(2) and Remark 1.1.3(3) we obtainthat C t ≥ ⊂ C w ≥ and C w ≥ ⊂ C t ≥ . Hence C w ≥ = C t ≥ as desired. Remark .
1. In Definition 2.5.1 of [Bon10b] and Definition 2.3.1 of [Bon18a]orthogonality was defined in terms of dualities of triangulated categories;see Definition 5.2.1(1,2) and Remark 5.2.2(1) below. The reader may eas-ily check that all the statements of this paper that concern orthogonalstructures can be generalized to C and C ′ related by an arbitrary duality Φ : C op × C ′ → A for an abelian category A ; cf. Remark 2.2.5 below.We prefer to avoid dualities in most of this paper due to the reason thatwe don’t have many interesting examples of orthogonal structures in thismore general setting.2. Proposition 1.3.3 says that our definition of adjacent "structures" is es-sentially equivalent to the original Definition 4.4.1 of [Bon10a] (yet cf.Remark 1.2.3(3) and note that the definition of left and right adjacentweight and t -structures in loc. cit. was "symmetric", i.e., w being leftadjacent to t and t being left adjacent to w were synonyms; in contrast,our current convention follows Definition 3.10 of [PoS16]).3. Recall also that the notions of weight and t -structures essentially have acommon generalization; so, both of these yield certain torsion theories asdefined in [IyY08] (this is the same thing as a complete Hom-orthogonalpair in the terms of [PoS16]); see §3 of [BoV19]. Respectively, our def-inition of adjacent structures is a particular case of Definition 2.2(3) ofibid.Note also that certain shifts of the classes C w ≤ , C w ≥ = C t ≥ , and C t ≤ give a suspended TTF triple in the sense of [HMV17, Definition 2.3]. t -truncations and their relation toorthogonal t -structures This section is dedicated to the virtual t -truncations of functors (these comefrom weight structures) and their general relationship with orthogonal t -structures.In §2.1 we recall the definition of virtual t -truncations of (co)homologicalfunctors and study their (easily defined) weight range.In §2.2 we introduce the notion of "reflection" (in the context of Definition1.3.1) and relate the existence of orthogonal t -structures to virtual t -truncations.11 .1 Virtual t -truncations and their weight range We recall the notion of virtual t -truncations for a cohomological functor H : C → A (as defined in §2.5 of [Bon10a] and studied in more detail in §2 of[Bon10b]). These truncations allow us to "slice" H into w -pure pieces (seeRemark 2.1.5(1–2) below). Definition 2.1.1.
Assume that C is endowed with a weight structure w , n ∈ Z ,and A is an abelian category.1. Let H be a cohomological functor from C into A .We define the virtual t -truncation functors τ ≤ n ( H ) (resp. τ ≥ n ( H ) ) by thecorrespondence M Im( H ( w ≤ n +1 M ) → H ( w ≤ n M )); (resp. M Im( H ( w ≥ n M ) → H ( w ≥ n − M )) ); here we take arbitrary choices ofthe corresponding weight truncations of M and connect them using Proposition1.2.4(5) in the case g = id M .2. Let H ′ : C → A be a homological functor. Then we will write τ ≤ n ( H ′ ) for the correspondence M Im( H ′ ( w ≤ n M ) → H ′ ( w ≤ n +1 M )) and τ ≥ n ( H ′ ) = M Im( H ′ ( w ≥ n − M ) → H ′ ( w ≥ n M )) (here we take the same connectingarrows between weight truncations of M as above).3. Assume that C is a full triangulated subcategory of a triangulated cate-gory D . Then for any M ∈ Obj D we will write H M = H M,C (resp. H M = H MC )for the restriction of the functor (co)represented by M to C (thus H M and H M are functors from C into Ab ). Moreover, sometimes we will say that thesefunctors are D -Yoneda ones, and that H M (resp. H M ) is D -(co)represented by M . We recall the main properties of these constructions. Proposition 2.1.2.
In the notation of the previous definition the followingstatements are valid.1. The objects τ ≤ n ( H )( M ) and τ ≥ n ( H )( M ) are C -functorial in M (andessentially do not depend on any choices).2. The functors τ ≤ n ( H ) and τ ≥ n ( H ) are cohomological.3. There exist natural transformations that yield a long exact sequence · · · → τ ≤ n − ( H ) ◦ [ − → τ ≥ n ( H ) → H → τ ≤ n − ( H ) → τ ≥ n ( H ) ◦ [1] → H − → . . . (2.1.1)(i.e., the result of applying this sequence to any object of C is a long exactsequence); the shift of this exact sequence by positions is given by composingthe functors with − [1] .4. Assume that there exists a t -structure t that is right orthogonal to w (forcertain C ′ and D as in Definition 1.3.1). Then for any M ∈ Obj C ′ the functors τ ≥ n ( H M ) and τ ≤ n ( H M ) (where H M is defined in Definition 2.1.1(3)) are D -represented (on C ) by t ≥ n M and t ≤ n M (see Remark 1.1.3(1)), respectively.5. The correspondence τ ≥ n ( H ′ ) gives a well-defined homological functor,and there exists a homological analogue of the long exact sequence (2.1.1).Moreover, if there exists a t -structure t on a triangulated category C ′ thatis left orthogonal to w (with respect to a triangulated category D containing C and C ′ ), A = Ab , and the functor H is D -corepresented by an object N of C ′ ,12hen τ ≥ n ( H ′ ) is D -corepresented by t ≥ n N and τ ≥ n ( H ′ ) is D -corepresented by t ≤ n N .6. For any i ∈ Z we have τ ≤ n + i ( H ) ∼ = τ ≤ n ( H ◦ [ i ]) ◦ [ − i ] and τ ≥ n + i ( H ) ∼ = τ ≥ n ( H ◦ [ i ]) ◦ [ − i ] , and also τ ≤ n + i ( H ′ ) ∼ = τ ≤ n ( H ′ ◦ [ i ]) ◦ [ − i ] and τ ≥ n + i ( H ′ ) ∼ = τ ≥ n ( H ′ ◦ [ i ]) ◦ [ − i ] .7. Let A ′ be an abelian subcategory of an abelian category A , i.e., A ′ is itsfull subcategory that contains the A -kernel and the A -cokernel of any morphismin A ′ ; assume that w restricts to a triangulated subcategory C ′ of C .Then if the restriction of H to C ′ takes it values in A ′ then the same is truefor all its virtual t -truncations. Proof.
Assertions 1-4 are given by Theorem 2.3.1 of [Bon10b] (yet pay attentionto Remark 1.2.3(3); one should also invoke Remark 5.2.2(1) below to obtainassertion 4). Assertion 5 is easily seen to be dual to the previous ones, whereasassertions 6 and 7 follow from our definitions immediately.Now we define weight range and relate it to virtual t -truncations; some ofthese statements will be applied elsewhere (only). Definition 2.1.3.
Let m, n ∈ Z ; let H be as above.Then we will say that H is of weight range ≥ m (resp. ≤ n , resp. [ m, n ] ) ifit annihilates C w ≤ m − (resp. C w ≥ n +1 , resp. both of these classes). Proposition 2.1.4.
1. Adopt the notation and assumptions of Definition1.3.1(1). Then for N ∈ C ′ t ≤ (resp. N ∈ C ′ t ≥ , resp. N ∈ C ′ t =0 ) thecorresponding D -Yoneda functor H N : C op → Ab (see Definition 2.1.1(3))is of weight range ≤ (resp. ≥ , resp. [0 , ).2. For H as in Definition 2.1.1(1) the functor τ ≤ n ( H ) is of weight range ≤ n ,and τ ≥ m ( H ) is of weight range ≥ m .3. We have τ ≤ n ( H ) ∼ = H (resp. τ ≥ m ( H ) ∼ = H ) if and only if H is of weightrange ≤ n (resp. of weight range ≥ m ).4. We have τ ≤ n ( τ ≥ m )( H ) ∼ = τ ≥ m ( τ ≤ n )( H ) .5. If a cohomological functor H ′ from C into A is of weight range ≥ n (resp. of weight range ≤ n − ) then any transformation T : H ′ → H (resp. H → H ′ ) factors through the transformation τ ≥ n ( H ) → H (resp. H → τ ≤ n − ( H ) ) provided by the formula (2.1.1).6. The (not necessarily locally small) category of weight range [0 , cohomo-logical functors from C into A is equivalent to AddFun( Hw op , A ) in theobvious natural way.7. Assume that H is a weight range [0 , cohomological functor from C , and M is a bounded above (resp. below) object of C . Then H i ( M ) = 0 for i ≫ (resp. for i ≪ ).8. If H is of weight range ≥ m then τ ≤ n ( H ) is of weight range [ m, n ] .Dually, if H is of weight range ≤ n then τ ≥ m ( H ) is of weight range [ m, n ] .9. Assume that m > n . Then the only functors of weight range [ m, n ] arezero ones; thus if H is of weight range ≤ n (resp. ≥ m ) then τ ≥ m ( H ) = 0 (resp. τ ≤ n ( H ) = 0 ). 130. If a cohomological functor H ′ (resp. H ′′ ) from C into A is of weightrange ≥ n (resp. of weight range ≤ n − ) then there are no non-zerotransformations from H ′ into H ′′ .11. The (representable) functor H M = C ( − , M ) : C op → Ab if of weight range ≥ m if and only if M ∈ C w ≥ m .12. If H is of weight range ≥ m (resp. ≤ m ) then the morphism H ( w ≥ m M ) → H ( M ) is epimorphic (resp. the morphism H ( M ) → H ( w ≤ m M ) is monomor-phic); here we take arbitrary choices of the corresponding weight decom-positions of M and apply H to the connecting morphisms. Proof.
1. For N ∈ C ′ t ≤ and N ∈ C ′ t ≥ the weight range estimates for the func-tor H N prescribed by the assertion are given by the definition of orthogonality,and to obtain the claim for N ∈ C ′ t =0 one should combine the first two weightrange statements.2. Let M ∈ C w ≥ n +1 . Then we can take w ≤ n ( M ) = 0 . Thus τ ≤ n ( H )( M ) =0 , and we obtain the first part of the assertion. It second part is easily seen tobe dual to the first part.3. This is precisely Theorem 2.3.1(III.2,3) of [Bon10b] (up to change ofnotation); assertion 4 is given by part II.3 of that theorem.5. The two statements in the assertion are easily seen to be dual to eachother; hence it suffices to consider the case where H ′ is of weight range ≥ n .Next, the obvious functoriality of the definition of virtual t -truncations givesthe following commutative square of transformations: τ ≥ n H ′ τ ≥ n T −−−−→ τ ≥ n H y i ′ y i H ′ T −−−−→ H (2.1.2)(cf. (2.1.1)).Applying assertion 3 we obtain that the transformation i ′ is an equivalence.Hence the transformation τ ≥ n T yields the factorization in question.8. Let H be of weight range ≥ m . Then τ ≤ n ( H ) ∼ = τ ≤ n ( τ ≥ m )( H ) ∼ = τ ≥ m ( τ ≤ n )( H ) (according to the two previous assertions). It remains to ap-ply assertion 2 to obtain the first statement in the assertion, whereas its secondpart is easily seen to be the dual of the first part (and certainly can be provedsimilarly).9. For any l ∈ Z and any cohomological H any choice of an l -weight decom-position triangle (cf. (1.2.1)) for M gives a long exact sequence · · · → H (( w ≤ l M )[1]) → H ( w ≥ l +1 M ) → H ( M ) → H ( w ≤ l M ) → H (( w ≥ l +1 M )[ − → . . . (2.1.3)The exactness of this sequence in H ( M ) for l = n immediately gives the firstpart of the assertion. Next, the second part is straightforward from the first onecombined with assertion 8.6. Immediate from Theorem 2.1.2(2) of [Bon18b].7. Straightforward from the definition of weight range.10. According to assertion 5, any transformation in question factors through τ ≥ n ( H ′′ ) ; thus it is zero according to assertion 9.14ssertion 11 is immediate from Proposition 1.2.4(2).Assertion 12 is a straightforward consequence of assertion 11; just apply(2.1.3) for l = m and for l = m − , respectively. Remark .
1. Roughly, the statements above say that virtual t -truncationsof functors behave as if they corresponded to t -truncations of objects in a cer-tain triangulated "category of functors" (whence the name; certainly, anotherjustification of this idea is provided by the existence of orthogonal t -structuresstatements that will be proved below). In particular, one can "slice" any functorof weight range [ m, n ] for m ≤ n into "pieces" of weight [ i, i ] for m ≤ i ≤ n .Now, composing a "slice" of weight range [ i, i ] with [ i ] one obtains a functor ofweight range [0 , .2. So we recall that functors of this type were studied in detail in (§2.1 of)[Bon18b]; they were called pure ones due to the relation to Deligne’s purity (cf.Remark 2.1.3(3–4) of ibid.).3. The author suspects that the connecting transformation in part 5 of ourproposition is actually unique.We also formulate a simple statement for the purpose of applying it in[BoS19]. Proposition 2.1.6.
For M ∈ Obj C the following conditions are equivalent.(i) M ∈ C w ≥ .(ii) H ( M ) = 0 for any cohomological functor H from C into (an abeliancategory) A that is of weight range ≤ − .(iii) ( τ ≤− H N )( M ) = { } for any N ∈ Obj C .(iv) ( τ ≤− H M )( M ) = { } . Proof.
Condition (i) implies condition (ii) by definition; certainly, (iii) = ⇒ (iv).Next, condition (ii) implies condition (iii) according to Proposition 2.1.4(2).Lastly, if ( τ ≤− H M )( M ) = { } then the long exact sequence (2.1.1) yieldsthat ( τ ≥− H M )( M ) surjects onto C ( M, M ) . Hence the morphism id M factorsthrough w ≥ M ; thus M belongs to C w ≥ . t -structures To generalize the criteria below from the "main" case of adjacent structures tocertain orthogonal structures we will need the following definition.
Definition 2.2.1.
Let C and C ′ be (full) triangulated subcategories of a trian-gulated category D (cf. Definition 1.3.1).Then we will say that C reflects C ′ (in the category D ) whenever the D -Yoneda functor C ′ → AddFun( C op , Ab) : M H M (see Definition 2.1.1(3)) isfully faithful.Let us relate this notion to orthogonal structures and their properties. Thereader may note that some of these proofs can be substantially simplified in thecases C = C ′ (and so, for adjacent structures) and C ′ ⊂ C . Proposition 2.2.2.
Adopt the notation of the previous definition.1. If C ′ ⊂ C then C reflects C ′ . 15. If C reflects C ′ and w is a weight structure on C then for the classes C ′ = C ⊥ D w ≥ ∩ Obj C ′ and C ′ = C ⊥ D w ≤− ∩ Obj C ′ we have C ′ [1] ⊥ C ′ .3. If C reflects C ′ and there exists a t -structure t = ( C ′ t ≤ , C ′ t ≥ ) on C ′ that is right orthogonal to w then t is also strictly right orthogonal to w , i.e., t = ( C ′ , C ′ ) . Moreover, the corresponding Yoneda-type functor Ht → AddFun( Hw op , Ab) is fully faithful.
Proof.
Assertion 1 immediately follows from the Yoneda lemma.2. If M belongs to C ⊥ D w ≥ and M ∈ C ⊥ D w ≤ then the D -Yoneda functor H M : C op → Ab is of weight range ≤ and H M is of weight range ≥ (see theobvious Proposition 2.1.4(1)). By Proposition 2.1.4(10) we obtain that thereare no non-zero transformations between H M and H M .Next we assume in addition that M and M are objects of C ′ . Since C reflects C ′ this vanishing of transformations statements implies that M ⊥ M ;thus C ′ [1] ⊥ C ′ indeed.3. Assume that t is orthogonal to w . Then the definition of orthogonalitysays that C ′ t ≤ ⊂ C ′ and C ′ t ≥ ⊂ C ′ . On the other hand, recall that C ′ t ≥ = ⊥ C ′ C ′ t ≤− and C t ≤ = ( C ′ t ≥ ) ⊥ C ′ (see Remark 1.1.3(3)). Since C ′ ⊥ C ′ [1] , weobtain that the converse inclusions are valid as well; thus t is strictly rightorthogonal to w .Next, if M ∈ C ′ t =0 then the definition of orthogonality implies that thefunctor H M : C op → Ab (see Definition 2.1.1(3)) is of weight range [0 , . Thusit suffices to recall that C reflects C ′ and apply Proposition 2.1.4(6). Remark . Our proposition implies that w determines a t -structure that is(right) adjacent to it uniquely, and this t -structure is strictly right orthogonal to w . Certainly, t determines a weight structure that is left adjacent to it uniquelyas well; see Proposition 1.3.3 and Proposition 1.2.4(2). Moreover, this argumenteasily extends to arbitrary Hom-orthogonal pairs (see Definition 3.1 of [PoS16]);in particular, it works for torsion theories (see §2 of [BoV19] and Definition 2.2of [IyY08]).Moreover, the author conjectures that this uniqueness statements are validfor orthogonal torsion theories whenever C reflects C ′ . Proposition 2.2.4.
Assume that C reflects C ′ (see Definition 2.2.1; here weadopt its notation), w is a weight structure on C , M ∈ Obj C ′ , and that forthe functor H M : C op → Ab (see Definition 2.1.1(3)) its virtual t -truncation τ ≥ H M is represented by some object M ≥ of C ′ .Then the following statements are valid.1. M ≥ belongs to C ⊥ D w ≤− ∩ Obj C ′ .2. The natural transformation τ ≥ ( H M ) → H M mentioned in (2.1.1) is in-duced by some f ∈ C ′ ( M ≥ , M ) .3. The object M ≤− = Cone( f ) belongs to C ⊥ D w ≥ ∩ Obj C ′ .4. For M ′ ∈ Obj C ′ the D -representability of the functor τ ≥ H M ′ by anobject of C ′ is equivalent to that of τ ≤− H M ′ .16 roof.
1. It suffices to recall that τ ≥ H M is of weight range ≥ according toProposition 2.1.4(2)).2. This transformation lifts to C ′ since C reflects C ′ .3. For any N ∈ Obj C applying the functor H N = D ( N, − ) to the dis-tinguished triangle M ≥ → M → M ≤− → M ≥ [1] one obtains a long exactsequence that yields the following short one: → Coker( H N ( M ≥ ) h N → H N ( M )) → H N ( M ≤− ) → Ker( H N ( M ≥ [1]) h N → H N ( M [1])) → (2.2.1)So, for any N ∈ C w ≥ we should check that the homomorphism h N is surjectiveand h N is injective.Applying (2.1.1) to the functor H M (in the case n = 0 ) we obtain a longexact sequence of functors · · · → τ ≥ ( H M ) → H M → τ ≤− ( H M ) → τ ≥ ( H M ) ◦ [1] → H M → . . . (2.2.2)Applying this sequence of functors to N we obtain that the surjectivity of h N along with the injectivity of h N is equivalent to τ ≤− ( H M )( N ) = { } . Recallingthat τ ≤− ( H M ) is of weight range ≤ − according to Proposition 2.1.4(2), weconclude the proof.4. If τ ≥ H M ′ is representable then the previous assertions imply the ex-istence of a D -distinguished triangle M ′≥ → M ′ → M ′≤− → M ′≥ [1] with M ′≤− ∈ C ⊥ D w ≥ ∩ Obj C ′ . Then the object M ′≤− D -represents the functor τ ≤− H M ′ according to Theorem 2.3.1(III.4) of [Bon10b] (and so, τ ≤− H M ′ isrepresentable). The proof of the converse implication is similar. Remark .
1. As we have already noted in Remark 1.3.4(1), it is not reallynecessary to assume that C and C ′ lie in some common triangulated category D . However, the author does not now of any examples such that no D exists but C reflects C ′ in the easily defined generalized sense of this notion (cf. Definition5.2.1 below).On the other hand, the main statements needed for the construction oforthogonal t -structures below are Proposition 2.2.2(2) and Proposition 2.2.4(2).The author does not know of any "abstract" conditions on the categories C and C ′ , and the duality Φ : C op × C ′ → A that would allow to generalize our currentproofs of these statements. However, it appears that if C possesses a "model"that allows to define a triangulated derived category C ′ of "reasonable" functors C → A (this is certainly the case when C is a triangulated subcategory of thehomotopy category of a stable model category) then the (corresponding versionof) Proposition 2.2.4(2) is fulfilled. Next, one can probably lift the square (2.1.2)to C ′ to obtain eventually that the corresponding version of Proposition 2.2.2(2)is valid as well.Possibly, the author will study this matter in a succeeding paper.2. The author does not know whether it makes much sense to take A = Ab in the argument that we have just sketched. Note however that we could haveconsidered a duality with values in A = R − Mod throughout section 4 below.Now we are able to prove our main abstract criterion on the existence of anorthogonal t -structure. 17 heorem 2.2.6. Assume that a triangulated subcategory C of D reflects C ′ ⊂ D (see Definition 2.2.1); let w be a weight structure on C .Then the following conditions are equivalent.(i). There exists a t -structure t on C ′ right orthogonal to w .(ii). The functor τ ≥ H M ′ is D -representable by an object of C ′ for anyobject M ′ of C ′ .(iii). The functor τ ≤− H M ′ is D -representable by an object of C ′ for anyobject M ′ of C ′ .(iv). For any object M ′ of C ′ and i ∈ Z the functors τ ≥ i H M ′ and τ ≤ i H M ′ are D -representable by objects of C ′ . Proof.
Condition (i) implies conditions (ii) and (iii) according to Proposition2.1.2(4). Next, conditions (ii) and (iii) are equivalent by Proposition 2.2.4(4).Moreover, conditions (iv) certainly implies conditions (ii) and (iii), whereas thereverse implication easily follows from Proposition 2.1.2(6).It remains to prove that condition (ii) implies (i). So, we should check thatthe couple ( C ′ , C ′ ) , where C ′ = C ⊥ D w ≥ ∩ Obj C ′ and C ′ = C ⊥ D w ≤− ∩ Obj C ′ (cf.Proposition 2.2.2(2,3)) is a t -structure. Now, these classes are certainly closedwith respect to C ′ -isomorphisms, and the "shift" axiom (ii) of Definition 1.1.1is obviously fulfilled as well. Next, the orthogonality axiom (iii) is given byProposition 2.2.2(2).Hence it remains to check the existence of t -decompositions (this is axiom(iv) of t -structures), which immediately follows from Proposition 2.2.4(1–3).Let us also prove (one more) corollary from Proposition 2.2.2. We will applyit in §4.1 for C ′ = C ; note however that in this case one can avoid usingProposition 2.2.2. Proposition 2.2.7.
Assume that C is densely generated by a single object G and w is a bounded weight structure on C .1. Then there exists N ∈ Z such that the class C w ≤ is contained in theenvelope (see §1.1) of { G [ j ] : j < N } and C w ≥ lies in the envelope of { G [ j ] : j > − N } .2. Assume that C ( G, G [ j ]) = { } for j ≪ , C ′ ⊂ C , and there exists a t -structure on C ′ that is right orthogonal to w . Then C ′ t ≥ = C ⊥ D w ≥ ∩ Obj C ′ and there exists N ′ ∈ Z such that C t ≤ ⊃ C w ≤− N ′ ∩ Obj C ′ ; hence t is boundedas well. Proof.
1. Since w is bounded, applying Proposition 1.2.4(9) we reduce ourassertion to the existence of N such that C w =0 lies in the envelope of { G [ j ] : − N < j < N } . Now, Remark 2.3.5(2) of [Bon18b] says that there existsa finite set of G i ∈ C w =0 such that any element of C w =0 is a retract of adirect sum of a (finite) collection of G i . Since the set { G } densely generates C , it remains to choose N such that all of these G i belong to the envelope of { G [ j ] : − N < j < N } .2. Combining Proposition 2.2.2 with Proposition 1.2.4(2) we obtain that C ′ t ≥ = C ⊥ D w ≥ ∩ Obj C ′ indeed. Hence if C t ≤ ⊃ C w ≤− N ′ ∩ Obj C ′ then t isbounded since w is. These objects are the terms of a bounded choice of a weight complex t ( G ) of G . C w ≤− N ′ ⊥ C ′ t ≥ ⊂ C w ≥ . Now, the existence of N ′ satisfyingthis condition is straightforward from our assumption on G along with assertion1. t -structures orthogonal to smashing weightstructures In this section we study the existence of adjacent weight and t -structures intriangulated categories closed with respect to (co)products (these are calledsmashing and cosmashing ones).In §3.1 we consider smashing weight structures (on smashing triangulatedcategories); these are the ones that respect coproducts.In §3.2 we recall the notion of (dual) Brown representability for smashingtriangulated categories, and prove Theorem 0.1, i.e., that a weight structure w on a category satisfying this condition is left adjacent to a t -structure if andonly if w is smashing. Certainly, the dual to this statement is also valid; more-over, if w is both cosmashing and smashing then the left adjacent t -structure t restricts to the subcategory of compact objects of C as well as to all other"levels of smallness" for objects. Combining this statement with an existence ofweight structures theorem from [KeN13] we obtain a statement on t -structuresextending yet another result of ibid.In §3.3 we study extensions of weight structures from subcategories of com-pact objects and the corresponding adjacent t -structures. We will need a few definitions.
Definition 3.1.1.
1. We will say that a triangulated category C is (co)smashing if it is closed with respect to (small) coproducts (resp., products).2. We will say that a weight structure w on C is (co)smashing if C is(co)smashing and the class C w ≥ is closed with respect to C -coproducts(resp., C w ≤ is closed with respect to C -products; cf. Proposition 1.2.4(3)).3. We will say that a t -structure t on C is (co)smashing if C is (co)smashingand the class C t ≤ is closed with respect to C -coproducts (resp., C t ≥ isclosed with respect to C -products; cf. Remark 1.1.3(3)).4. It will be convenient for us to use the following somewhat clumsy termi-nology: a cohomological functor H ′ from C into A will be called a cp functor if it converts all (small) coproducts into A -products.5. For an infinite cardinal α a homological functor H : C → A is said to be α -small if for any family N i , i ∈ I , we have H ( ` i ∈ I N i ) = lim −→ J ⊂ I, J<α H ( ` j ∈ J N j ) (i.e., the obvious morphisms H ( ` j ∈ J N j ) → H ( ` N i ) form a colimit di-agram; note that this colimit is filtered).Let us now prove some properties of these notions and relate them to t -truncations. 19 roposition 3.1.2. Assume that w is a smashing weight structure on C , H : C → A is a homological functor (where A is an abelian category), i ∈ Z , and α is an infinite cardinal. Then the following statements are valid.1. If α ′ ≥ α then any α -small functor is also α ′ -small.2. H is ℵ -small if and only if it respects coproducts.3. The class C w =0 is closed with respect to C -coproducts.4. Coproducts of w -decompositions are weight decompositions as well.5. Assume that A is an AB4* category and a cohomological functor H ′ from C into A is a cp one. Then τ ≥ i ( H ′ ) and τ ≤ i ( H ′ ) are cp functors as well.6. Assume that A is an AB5 category and H is an α -small functor. Thenthe functors τ ≥ i ( H ) and τ ≤ i ( H ) are α -small as well. Proof.
1. Assume that H is an α -small functor; fix an index set I and certain N i ∈ Obj C . Then for any J ⊂ I we have H ( ` j ∈ J N j ) = lim −→ J ′ ⊂ J, J ′ <α H ( ` j ′ ∈ J ′ N j ′ ) .Combining these statements for all J ⊂ I (actually, it suffices to take J = I along with J of cardinality less than α ′ only here) one easily obtains that H ( ` i ∈ I N i ) = lim −→ J ⊂ I, J<α ′ H ( ` j ∈ J N j ) .2. Since H is additive, H ( ` N i ) = lim −→ J ⊂ I, J< ℵ H ( ` j ∈ J N j ) if and onlyif H respects coproducts (since this colimit will not change if one will consideronly those J that consist of a single element only).3. This is an easy consequence of Proposition 1.2.4(2); see Proposition2.3.2(1) of [Bon18b].4. This is an easy consequence of Proposition 1.2.4(3) along with Remark1.2.2 of [Nee01]; it is given by Proposition 2.3.2(3) of [Bon18b].5. According to Proposition 2.1.2(6), it suffices to verify that the functors τ ≥ ( H ′ ) and τ ≤− ( H ′ ) are cp ones for any cp functor H ′ . For a family { M i } of objects of C we choose certain − and − -weight decompositions for all M i (see Remark 1.2.3(2)). According to the previous assertion, their coproductsgive a − and a − -weight decomposition of ` M i , respectively. Moreover,one can certainly obtain the unique morphisms w ≤− ( ` M i ) → w ≤− ( ` M i ) and w ≥− ( ` M i ) → w ≥ ( ` M i ) compatible with these decomposition trian-gles (see Proposition 1.2.4(5) and Definition 2.1.1(1)) as the coproducts ofthe corresponding morphisms for M i . Applying our assumptions on H ′ and A we obtain that τ ≥ ( H ′ )( ` M i ) ∼ = Q τ ≥ ( H ′ )( M i ) and τ ≤− ( H ′ )( ` M i ) ∼ = Q τ ≥− ( H ′ )( M i ) .Similarly, to prove assertion 6 it suffices to verify that the functors τ ≥ ( H ) and τ ≤− ( H ) are α -small whenever H is. One takes the same weight decom-positions along with their coproducts corresponding to all subsets J of I ofcardinality less than α . Since the colimits in question are filtered ones, theAB5 assumption on A allows to compute lim −→ J ⊂ I, J<α
Im( H ( ` j ∈ J w ≤− N j ) → H ( ` j ∈ J w ≤− N j )) and lim −→ J ⊂ I, J<α
Im( H ( ` j ∈ J w ≥− N j ) → H ( ` j ∈ J w ≥− N j )) as the corresponding images of colimits to obtain the statement in question. Remark .
1. The current version of Proposition 3.1.2(5) is sufficient for thepurposes of this paper; yet the following modification of this statement is quiteuseful also (and will be applied elsewhere).20o, assume that α is a regular cardinal (i.e., it cannot be presented as a sumof less then α cardinals that are less than α ), the category C is closed withrespect to coproducts of cardinality less then α , w is a weight structure on C such that C w ≥ is closed with respect to C -coproducts of cardinality less then α , H ′ is a cohomological functor from C into A that converts coproducts ofthis sort into products, and products of cardinality less then α are exact on A .Then the obvious modification of the proof of Proposition 3.1.2(5) yields thatall virtual t -truncations of H ′ also convert C -coproducts of cardinality less then α into products.2. The author suspects that is suffices to assume that A is an AB4 categoryin part 6 of our proposition. At least, this is easily seen to be the case for α = ℵ ,i.e., for functors that respect coproducts (cf. parts 2 and 5 of the proposition).However, below we will only need the A = Ab case of the statement. t -structures orthogonal to smash-ing weight structures To formulate the main results of this section we will need some more definitions.
Definition 3.2.1.
Let C be a smashing triangulated category, P is a subclassof Obj C , C ′ is an arbitrary triangulated category, and P ′ ⊂ Obj C ′ .1. We will say that C satisfies the Brown representability property wheneverany cp functor from C into Ab is representable.Dually, we will say that C ′ satisfies the dual Brown representability prop-erty if C ′ is smashing and any functor from C ′ into Ab that respectsproducts is corepresentable (i.e., if C ′ op satisfies the Brown representabil-ity assumption).2. For an infinite cardinal α an object M of C is said to be α -small (in C ) ifthe functor H M = C ( M, − ) : C → Ab is α -small (see Definition 3.1.1(5)).Moreover, ℵ -small objects of C (corresponding to functors that respectcoproducts) will also said to be compact .3. We will say that a full strict triangulated subcategory D ⊂ C is localizing whenever it is closed with respect to C -coproducts. Respectively, we willcall the smallest localizing subcategory of C that contains a given class P ⊂
Obj C the localizing subcategory of C generated by P .4. We will say that P compactly generates C and that C is compactly gen-erated if P generates C as its own localizing subcategory and P is a set of compact objects of C .Moreover, we will say that a subcategory C of C compactly generates C whenever C is essentially small and (any) its small skeleton compactlygenerates C .5. We will say that P ′ generates a weight structure w on C ′ whenever C ′ w ≥ =( ∪ i> P ′ [ − i ]) ⊥ .6. We will say that P ′ generates a t -structure t on C ′ whenever C ′ t ≤ =( ∪ i> P ′ [ i ]) ⊥ . 21 emark .
1. Recall that C satisfies both the Brown representability prop-erty and its dual whenever it is compactly generated; see Proposition 8.4.1,Proposition 8.4.2, Theorem 8.6.1, and Remark 6.4.5 of [Nee01]. Moreover, theBrown representability property is fulfilled whenever C is just ℵ -perfectly gen-erated (see Definition 8.1.4 and Theorem 8.3.3 of ibid.).Recall also that any triangulated category possessing a combinatorial (Quillen)model satisfies the dual Brown representability property; see §0 of [Nee08] (thestatement is given by the combination of Theorems 0.17 and 0.14 of ibid.).Furthermore, the abundance of smashing triangulated categories that satisfythe Brown representability property has motivated the author not to considerorthogonal t -structures on C ′ = C in Theorem 3.2.3(I) below (in contrast toTheorem 4.1.2(II); note however that one can easily formulate and prove thecorresponding modification of Theorem 3.2.3(I)).2. The easy Lemma 4.1.4 of [Nee01] says that for any infinite cardinal α theclass of α -small objects gives a (full strict) triangulated subcategory C ( α ) of C .Moreover, if α is regular (see Remark 3.1.3(1)) then C ( α ) is closed with respectto C -coproducts of less than α objects; see Lemma 4.1.5 of ibid.On the other hand, an object M of C is α -small if and only if any morphismfrom M into ` i ∈ I N i factors through ` j ∈ J N j for some J ⊂ I of cardinalityless than α . Now, take M = ` i ∈ I M i where all M i are non-zero and I is ofcardinality α . Then id M does not possess a factorization through any ` j ∈ J M j for J < α ; hence M is not α -small. These observations demonstrate that thefiltration of C by C ( α ) is "often non-trivial". Note moreover that any object of C is α -small for some cardinal α if C is well-generated , whereas this is the casewhenever C possesses a combinatorial model (by Proposition 6.10 of [Ros05];cf. part 1 of this remark).3. A class P ′ as above is easily seen to determine weight and t -structures itgenerates on C ′ (if any) completely; see either of Proposition 2.4(1) (along with§3) of [BoV19] or Remark 1.1.3(3) and Proposition 1.2.4(2) above.Now we prove our first "practical" existence of t -structures results; see Def-initions 3.1.1 and 3.2.1 for the notions mentioned in our theorem. Theorem 3.2.3.
Let w be a weight structure on C .I. Assume that C satisfies the Brown representability property (and so, C issmashing).Then there exists a t -structure t r right adjacent to w if and only if w issmashing. Moreover, t r is cosmashing (if exists; see Definition 3.1.1(3)) and itsheart is equivalent to the category of those additive functors Hw op → Ab thatrespect products.II. Assume that w is cosmashing and C satisfies the dual Brown repre-sentability property.1. Then there exists a smashing t -structure t l left adjacent to w and Ht l is equivalent to the category of those additive functors Hw → Ab that respectproducts.2. Assume that w is also smashing. Then for any infinite cardinal α theweight structure t l given by the previous assertion restricts to the subcate-gory C ( α ) of C (see Remark 3.2.2(2) and Definition 1.1.2(5)); this restricted t -structure t ( α ) is the only t -structure on C ( α ) that is left orthogonal to w .22 roof. I. The "only if" assertion is essentially given by Proposition 2.4(6) of[BoV19] (cf. §3. of ibid.; the statement is also very easy for itself).Conversely, assume that w is smashing. According to Proposition 2.2.2(1) wecan apply Theorem 2.2.6 (in the case D = C ′ = C ) to obtain that the existenceof t r is equivalent to the representability of τ ≤ H M ′ for any representable functor H M ′ . Next, Proposition 3.1.2(5) says that τ ≤ H M ′ is a cp functor since H M ′ is.Hence τ ≤ H M ′ is representable by the Brown representability assumption, andwe obtain that t r exists indeed.Next, the category C is cosmashing according to Proposition 8.4.6 of [Nee01](since it satisfies the Brown representability property). Moreover, t r = ( C , C ) ,where C = C ⊥ w ≥ and C = C ⊥ w ≤− according to Proposition 2.2.2. Hence theclass C t r ≥ is closed with respect to C -coproducts; thus t r is cosmashing aswell.Lastly, since t r = ( C , C ) , the class C t =0 equals ( C w ≥ ∪ C w ≤− ) ⊥ ; hence Ht can be calculated using Proposition 2.3.2(8) of [Bon18b] (see also Remark2.1.3(2) of ibid. and Proposition 2.1.4(6) above).II.1. This is just the categorical dual to assertion I.2. The uniqueness of a t -structure on C ( α ) that is left orthogonal to w is givenby (the dual to) Proposition 2.2.2. Next, for any object M of C ( α ) the functor H M is α -small by the definition of C ( α ) . Now, for M ′ = t l ≥ M Proposition2.1.2(5) says that H M ′ ∼ = τ ≥ H M . Hence the functor H M ′ is α -small as wellaccording to Proposition 3.1.2(6), and we obtain that M ′ is an object of C ( α ) .Thus M has a t l -decomposition whose components are objects of C ( α ) ; therefore t l restricts to C ( α ) indeed. Remark .
1. Now let us discuss examples to Theorem 3.2.3(I).According to Theorem 5 of [Pau12], any set P of compact objects of C generates a (unique) smashing weight structure (see Definition 3.2.1(5) and Re-mark 3.2.2(3)). Moreover, Theorem 4.5(2) of [PoS16] and Theorem 4.2.1(1,2) of[Bon16] (we will mention these statements in the the proof of Corollary 3.2.5(2)below) enable one to check whether two weight structures obtained this way aredistinct. Thus one may say that there are lots of smashing weight structures on C whenever there are "plenty" of compact objects in it (see Theorem 4.15 of[PoS16] for a certain justification of this claim for derived categories of commu-tative rings). Thus part I of our theorem yields a rich collection of t -structures,and the author does not know of any other methods that give all of them (cf.Remark 0.2).2. Recall (from Proposition 3.4(4) of [BoV19]) that "shift-stable" weightstructures are in one-to-one correspondence with exact embeddings i : L → C possessing right adjoints. Hence applying our theorem in this case we obtain thefollowing: if i possesses a right adjoint respecting coproducts and C satisfies theBrown representability property then for the full triangulated subcategory R of C with Obj R = L ⊥ the embedding R → C possesses a right adjoint as well.Thus R is admissible in C in the sense of [BoK89] and the embedding R → C may be completed to a gluing datum (cf. [BBD82, §1.4] or [Nee01, §9.2]).So we re-prove Corollary 2.4 of [NiS09].3. The author suspects that the heart of the restricted t -structure t ( α ) inpart II.2 of our theorem can be computed similarly to the hearts in assertionsI and II.1, and so using the theory of w -pure functors as developed in §2 of23Bon18b].Let us now verify that Theorem 3.1 of [KeN13] (that essentially generalizesTheorem 3.2 of [Pau08]) gives an example for the setting of Theorem 3.2.3(II.2),and study the corresponding structures in detail. Corollary 3.2.5.
Let A be an essentially small abelian semi-simple subcategoryof the subcategory C ( ℵ ) of C that generates C as its own localizing subcategory,and assume that Obj A ⊥ C ∪ i< Obj A [ i ] .1. Then there exist a smashing and cosmashing weight structure w and a t -structure t on C that are generated by Obj A , and t is left adjacent to w .2. t restricts to the subcategory C ( α ) (see Remark 3.2.2(2)) for any infinitecardinal α . Moreover, in the corresponding couple t ( ℵ ) = ( C ( ℵ ) t ( ℵ ≤ , C ( ℵ ) t ( ℵ ≥ ) the class C ( ℵ ) t ( ℵ ≤ (resp. C ( ℵ ) t ( ℵ ≥ ) is the envelope of ∪ i ≤ Obj A [ i ] (resp. of ∪ i ≥ Obj A [ i ] ) in C (see §1.1). Proof.
1. Since A is semi-simple, the category AddFun( A, Ab) is semi-simpleas well. Thus we can apply Theorem 3.1 of [KeN13] to obtain the existenceof a weight structure w on C such that C w ≥ = ( ∪ i> P [ − i ]) ⊥ (i.e. w is gen-erated by P = Obj A ) and C w ≤ = ( ∪ i> P [ i ]) ⊥ . Since the category C iscompactly generated by A , it satisfies the dual Brown representability property(by the aforementioned Theorem 8.6.1 and Remark 6.4.5 of [Nee01]). Next, w is obviously smashing and cosmashing. Applying Theorem 3.2.3(II.1), we ob-tain the existence of a smashing t -structure t that is left adjacent to w . Since C w ≤ = C t ≤ , we obtain that t is generated by P (as a t -structure) as well.2. t restricts to the subcategory C ( α ) for any infinite cardinal α accord-ing to part II.2 of Theorem 3.2.3. Thus it remains to prove that the classes C t ≤ ∩ Obj C ( ℵ ) = C w ≤ ∩ Obj C ( ℵ ) and C t ≥ ∩ Obj C ( ℵ ) are the envelopesin question. The latter statement an easy consequence of Theorem 4.2.1(2) of[Bon16] applied to w and t respectively (see Remark 4.2.2(1,2) of ibid.; notealso that Theorem 4.5(1,2) of [PoS16] gives this statement in the case under theassumption that C is a "stable derivator" triangulated category). Remark .
1. Thus we obtain a serious generalization of the existence ofa (certain) t -structure on C ( ℵ ) part of [KeN13, Theorem 7.1].Note also that the assumption that C is compactly generated by A does notappear to be necessary in Theorem 3.1 of ibid.; thus is may be omitted inour corollary as well. However, in this case it is probably more interestingto look at the localizing subcategory C ′ of C generated by Obj A instead.We note that the embedding i : C ′ → C possesses an exact right adjoint F (since C ′ satisfies the Brown representability property; see Theorem8.4.4 and Lemma 5.3.6 of ibid.), and F allows to recover w and t fromtheir restrictions to C ′ (whose existence is given by the present form ofCorollary 3.2.5) via Propositions 3.2(5) and 3.4(3) of [BoV19].2. Now we try to study the question which t -structures on C ( ℵ ) extend toexamples for our corollary.So, assume that C is an arbitrary triangulated category and t ′ is a t -structure on C ( ℵ ) , and take A ′ = Ht ′ . Then we have Obj A ′ ⊥ ( ∪ i> Obj A ′ [ − i ]) by the orthogonality axiom for t ′ .24hus any essentially small abelian subcategory A of Ht ′ whose objects aresemi-simple satisfies all the assumptions of our corollary except the onethat A compactly generates C . Hence we can apply our corollary to thelocalizing subcategory C ′ of C generated by Obj A .3. Now assume in addition that C is compactly generated, t ′ is bounded, and Ht ′ is a length category (cf. Theorem 7.1 of [KeN13]). Then the category C ( ℵ ) is essentially small according to Lemma 4.4.5 of [Nee01]; hence Ht ′ is essentially small as well, and we can take A to be its subcategory ofsemi-simple objects.Since t ′ is bounded and Ht ′ is a length category, the category C ( ℵ ) isdensely generated by Obj A ; hence A is easily seen to generate C as itsown localizing subcategory. Thus one can apply Corollary 3.2.5 to thissetting. Moreover, it is easily seen that our assumptions on t ′ (combinedwith part 2 of our corollary) imply that the corresponding t -structure t ℵ coincides with t ′ .4. It would be interesting to find which assumptions on a general t -structure t ′ on C ( ℵ ) ensure that t ′ "extends" to a t -structure on C and a weightstructure w that is right adjacent to t .Suppose that C is compactly generated (or at least that C ( ℵ ) is essentiallysmall). Then it appears to be quite reasonable to consider the weightstructure w that is generated (in C ) by C ( ℵ ) t ′ ≥ (see Definition 3.2.1(5)and Remark 3.2.4(1)) and the t -structure t that is generated by C ( ℵ ) t ′ ≤ (see Definition 3.2.1(6); the existence of t ′ is provided by Theorem A.1 of[AJS03]). However, it is not clear how to check whether t is left adjacentto w in the general case.5. Let us now describe an example that demonstrates that certain additionalassumptions are necessary to ensure that this t is left adjacent to w . Possi-bly, for this purpose it suffices to impose certain conditions on Ht (for t asabove), but it would certainly be more interesting to seek for formulationsin terms of Ht ′ .So, we take R to be a left semi-hereditary ring (see Definition 4.4.1(3)below or §0.3 of [Coh85]) that is not noetherian; in particular, one can take R to be any non-noetherian valuation ring (see Proposition 4.4.2(1)). Wewill consider left R -modules only, and set C = D ( R ) , i.e., C is the derivedcategory of (left) R -modules. Then C ( ℵ ) is the subcategory of perfectcomplexes in D ( R ) (a well-known fact; see Proposition 4.4.2(6) below), i.e.,its objects are quasi-isomorphic to bounded complexes of finitely generatedprojective R -modules.Now we claim that the canonical t -structure t on C (i.e., t -truncations arethe canonical truncations of complexes, and t -homology is the " R -module"one) restricts to C ( ℵ ) . Indeed, objects of C ( ℵ ) are t -bounded; hence itsuffices to verify that the cohomologies of perfect complexes are perfectthemselves. Now, these cohomology modules are finitely presented (seeDefinition 4.4.1(1) and Proposition 4.4.2(2,3) below; cf. also Theorem A.9of [Coh85]), whereas for any finitely presented R -module M one can take25 surjection R n → M , and its kernel is projective and finitely presentedaccording to Proposition 4.4.2(4).We will write t ′ for the restriction of t to C ( ℵ ) . It remains to verify for P = Obj Ht ′ that C w ≥ = ( ∪ i> P [ − i ]) ⊥ ) and C w ≤ = ( ∪ i> P [ i ]) ⊥ isnot a weight structure. Assume the opposite; then Hw obviously containsinjective R -modules (placed in degree ) and is closed with respect to C -coproducts. Since R is not noetherian, the Bass-Papp Theorem 1.1 of[Bas62] implies that Hw also contains a non-injective R -module M . Thenwe take an embedding M → I of M into an injective module, and considerthe corresponding distinguished triangle M → I → N → M [1] . Since N is an R -module (put in degree ) as well, we obtain N ∈ C w ≥ . Hence N ⊥ M [1] and we obtain that M is a retract of I . Thus is an injective R -module, and we obtain a contradiction.More generally, it is sufficient to assume that R is any coherent ring suchthat any finitely presented module over it possesses a bounded projectiveresolution; see Proposition 4.4.2(5) below. The author suspects that ringsof the form R ′ [ x , x , . . . , x ] , where R ′ is commutative semi-hereditary,give examples of these assumptions. t -structures Now we prove that weight structures extend from subcategories of compactobjects to the localizing subcategories they generate. The (proof of the) firstpart of the following theorem is quite similar to the corresponding argumentsin §2 of [BoS19]. Note also that one can certainly assume that E is compactlygenerated and thus equals D . Theorem 3.3.1.
Let E be a smashing triangulated category, C = E ( ℵ ) , D isthe localizing subcategory of E generated by Obj C , and w is a weight structureon C .I.1. Then w extends uniquely to a smashing weight structure w D on D , i.e., D w D ≤ ∩ Obj C = C w ≤ and D w D ≥ ∩ Obj C = C w ≥ .2. D w D ≤ (resp. D w D ≥ ) is the smallest extension-closed subclass of Obj D that is closed with respect to D -coproducts and contains C w ≤ (resp. C w ≥ ).3. D w =0 consists of all retract of all (small) D -coproducts of elements of C w =0 .II. Assume in addition that the category C is essentially small.1. Then there exists a t -structure t D right adjacent to w D (on D ); it is rightorthogonal to w .2. t D is strictly right orthogonal to w ; hence t D is both smashing andcosmashing.3. Ht D is equivalent (in the obvious way) to the category AddFun( Hw op , Ab) .4. For an infinite cardinal β consider the full subcategory D β of D thatconsists of those N such that D ( M, N ) < β for all M ∈ Obj C .Then this subcategory is triangulated, t D restricts to it, and the heart ofthis restriction t β is naturally equivalent to the category of those functors from Hw op into Ab whose values are of cardinality less than β .26 roof. I.1,2. We will write C and C for the classes of objects described inassertion I.2. Let us prove that ( C , C ) is a weight structure on D indeed.Firstly, axiom (ii) of Definition 1.2.1 (for w ) easily implies that C ⊂ C [1] and C [1] ⊂ C . Combining this statement with Proposition 5.1.1(II.2) belowwe obtain that C and C are retraction-closed in D .Next, the compactness of the elements of C w ≤ in D implies that the class C w ≤ ⊥ is closed with respect to coproducts. Since it is also extension-closed andcontains C w ≥ by the axiom (iii) of Definition 1.2.1, this orthogonal contains C [1] , i.e., C w ≤ ⊥ C [1] . Thus C w ≤ ⊂ ⊥ C [1] , and since the latter class isclosed with respect to coproducts and extension, we obtain that C ⊥ C [1] (cf.the proof of [BoK18, Lemma 1.1.1(2)]).Let us now prove the existence of weight decompositions, i.e., for the set E of those M ∈ Obj D such that there exists a distinguished triangle LM → M → RM → LM [1] with LM ∈ C and RM ∈ C [1] we should prove E = Obj D .Now, E certainly contains Obj C , and Proposition 5.1.1(I, II.1) below impliesthat it is also extension-closed and closed with respect to coproducts. Hence E = Obj D , and we obtain that w D = ( C , C ) is a weight structure on D indeed. Certainly, this weight structure is smashing.Lastly, assume that v is a smashing weight structure on D such that C w ≤ ⊂ D v ≤ and C w ≥ ⊂ D v ≥ . Then we obviously have D w D ≤ ⊂ D v ≤ and D w D ≥ ⊂ D v ≥ , and applying Proposition 1.2.4(10) we obtain v = w D .3. Denote our candidate for D w D =0 by C . Firstly we note that C ⊂ D w D =0 since the latter class is closed with respect to coproducts according to Proposi-tion 3.1.2(3).Applying Proposition 1.2.4(6) we obtain that C is extension-closed (since thisclass is certainly additive); certainly, it is also closed with respect to coproducts.Next we apply Proposition 5.1.1(I, II.1) once again to obtain that the classof extensions of elements of D w D ≥ by that of C is extension-closed and closedwith respect to coproducts; hence this class coincides with D w D ≥ . Thus for any M ∈ D w D =0 there exists its weight decomposition LM → M → RM → LM [1] with LM ∈ C . Since M is a retract of LM according to Proposition 1.2.4(7),we obtain that M ∈ C .II.1. The category D is compactly generated by C in this case; hence D satisfies the Brown representability property (see Remark 3.2.2(1)). Next, w D is smashing; thus a t -structure t D adjacent to it exists according to Theorem3.2.3(I). Certainly, t D is also orthogonal to w .2. t D is cosmashing according to Theorem 3.2.3(I).Next, Proposition 2.2.2 implies that t D is strictly right orthogonal to w D ,i.e., D t D ≤ = D w D ≥ ⊥ and D t D ≥ = D w D ≤− ⊥ . Since for any object N of D the class ⊥ D N is closed with respect to coproducts and extensions, assertionI.2 implies that D w D ≥ ⊥ = C w ≥ ⊥ and D w D ≤− ⊥ = C w ≤− ⊥ . Lastly, theelements of C w ≥ are compact in D ; hence t D is smashing as well.3. According to Theorem 3.2.3(I), the category Ht D is equivalent to thecategory of those functors from Hw opD into Ab that respect products. Thus itremains to apply the description of Hw D provided by assertion I.3.4. The proof relies on the following obvious observation that will be denotedby (*): the category of abelian groups of cardinality less than β is a (full) exactabelian subcategory of Ab . It immediately implies that the category D β istriangulated. 27o prove that t restricts to D β we argue similarly to the proof of Theorem3.2.3(II.2). For an object N of D β the values of the corresponding functor H N : C op → Ab are of cardinality less than β by the definition of D β , and combining(*) with Proposition 2.1.2(7) we obtain that the functor τ ≥ H N possesses thisproperty as well. Since for N ′ = t D, ≥ N we have τ ≥ H N ∼ = H N ′ , N ′ is an objectof D β . Thus N has a t D -decomposition whose components are objects of D β ,i.e., t D restricts to D β .It remains to calculate the heart of the t -structure t β obtained. Applyingassertion II.3 we obtain that it suffices to verify the following: a w -pure functor C op → Ab has values of cardinality less than β if and only if the values of itsrestriction to Hw satisfy this property. This is immediate from Lemma 2.1.4 of[Bon18b] along with (*). Remark .
1. The restriction of our theorem to the case where w is boundedand C is essentially small was essentially established in §4.5 of [Bon10a]; cf.Theorem 3.2.2 of [Bon18b] and Remark 2.3.2(2) of [BoS19] for some more detail.2. Similarly to Corollary 2.3.1(2) of ibid., for any regular cardinal α theweight structure w restricts to the smallest subcategory of D that contains C and is closed with respect to coproducts of cardinality less than α . Moreover,this filtration of D may be easily completed to a filtration indexed by all infinitecardinals, cf. loc. cit.Moreover, part I of our theorem along with Remark 3.2.6(5) demonstratethat it is "easier to extend weight structures from compact objects than t -ones".3. Note that one can easily obtain plenty of examples for our theorem suchthat w is unbounded.Indeed, can obtain lots of unbounded weight structures on essentially smalltriangulated categories using (say) the previous part of this remark. Next, itappears that any "reasonable" essentially small triangulated category C is thesubcategory of compact objects in some smashing triangulated category D ; cf.Remark 5.4.3(1) and Proposition 5.5.2 of [Bon16] (that relies on [FaI07]; oneshould dualize it and pass to a subcategory).4. Theorem B of [Kra01] relates the filtration of D by the subcategories D β to the so-called β -compactness filtration (as introduced in [Nee01]). t -structures related to saturated categoriesand coherent sheaves In §§4.1–4.3 we will treat R -linear categories and functors. we will always assumethat R is an associative commutative unital coherent (see Definition 4.4.1 below)ring; moreover, R will be Noetherian in §§4.2–4.3.We start §4.1 from recalling (from [Nee18a] and preceding papers on thesubject) the definition of R -saturated categories and (locally) finite R -linearfunctors. Next we prove the existence of a t -structure adjacent to a boundedweight structure on a R -saturated category C ; we also generalize this statementto the case of orthogonal structures.In §4.2 we describe rich families of "geometric" examples to these statements;one takes C and C ′ to be the derived categories of perfect complexes and ofbounded (above) complexes of coherent sheaves over X , where X is proper over Spec R ; we also apply duality if X is regular (or Gorenstein).28n §4.3 we discuss whether one can obtain interesting results "starting from" C = D bcoh ( X ) (instead of C = D perf ( X ) ).In §4.4 we recall certain definitions and statements related to coherent ringsand perfect complexes. t -structures corresponding to (locally)finite functors into R -modules To help the reader we recall that all Noetherian rings are coherent, and if R is Noetherian then an R -module is finitely presented if and only if it is finitelygenerated. Moreover, the restriction of Proposition 4.4.2 below to the case ofNoetherian rings is very well-known. We mention coherent rings in this sectionjust for the sake of generality. Since in the most interesting of the currentlyavailable examples for our statements the ring R is Noetherian, the reader mayrestrict herself to the Noetherian ring case. Definition 4.1.1.
Let C be R -linear category.1. We will say that an R -linear cohomological functor H from C into R − Mod is (anti) locally finite whenever for any M ∈ Obj C the R -module H ( M ) isfinitely presented and ( H ( M [ − i ]) =) H i ( M ) = { } for i ≪ (resp. for i ≫ )Moreover, we will say that H is finite if it is both locally and anti-locallyfinite.2. We will say that C is R -saturated if the representable functors from C into R − Mod are exactly all the finite ones.3. The symbol
Fun R ( C, D ) will denote the (possibly, big) category of R -linear (additive) functors from C into D whenever C and D are R -linear cate-gories.4. We will write R − mod for the category of finitely presented R -modules(see Definition 4.4.11) below).The following statement should be understood as the conjunction of threeits versions. To obtain the "finite" version one should ignore all adjectives inbrackets, to obtain the "locally finite" version one should take the first adjectivesin all the brackets into account, and one should take the second adjectives toget the "anti-locally finite" version. This rule should also be applied to some ofthe sentences in the proof. Theorem 4.1.2.
Assume that C and C ′ are full R -linear triangulated subcate-gories of an R -linear triangulated category D (recall that R is a coherent ring), C is endowed with a bounded (resp. bounded below, resp. bounded above)weight structure w , and a subcategory C ′ of C is characterized by the followingcondition: for N ∈ Obj D the D -Yoneda-functor H N : C op → R − Mod (seeDefinition 2.1.1(3)) is finite (resp. locally finite, resp. anti-locally finite) if andonly if N is an object of C ′ .I. Then all virtual t -truncations of finite (resp. locally finite, resp. anti-locally finite) functors C op → R − Mod are finite (resp. locally finite, resp.anti-locally finite) as well.II. Assume that finite (resp. locally finite, resp. anti-locally finite) functors C op → R − Mod are precisely the functors of the form H N for N ∈ Obj C ′ , andthat C reflects C ′ in D (see Definition 2.2.1).29. Then C ′ is a triangulated subcategory of D and there exists a (unique) t -structure t on C ′ that is strictly right orthogonal to w .2. The obvious Yoneda-type functor from the category Ht into the sub-category Fun R ( Hw op , R − mod) of Fun R ( Hw op , R − Mod) is an equivalence ofcategories.III. Assume that D is smashing, C equals D ( ℵ ) and compactly generates D (thus C is essentially small).1. Then the t -structure t D on D provided by Theorem 3.3.1(II) restricts to C ′ (i.e., t = (( C w ≥ ) ⊥ D ∩ Obj C ′ , ( C w ≤− ) ⊥ D ∩ Obj C ′ ) is a t -structure on C ′ ).2. The obvious Yoneda-type functor from Ht into the category Fun R ( Hw op , R − mod) ⊂ Fun R ( Hw op , R − Mod) is an equivalence of categories.
Proof.
I. Recall that virtual t -truncations of cohomological functors are coho-mological. Moreover, virtual t -truncations of R -linear functors are obviously R -linear.Now, let H be a functor whose values are finitely presented R -modules.Since the category R − mod is an abelian subcategory of R − Mod according toProposition 4.4.2(3) below, the values of virtual t -truncations of H are finitelypresented as well according to Proposition 2.1.2(7). Next, for any k ∈ Z and any cohomological functor H from C the functor τ ≤ k ( H ) is of weight range ≤ k according to Proposition 2.1.4(2). Assume that w is bounded below; then for any fixed k and M ∈ Obj C we have τ ≤ k ( H )( M [ j ]) =0 if j is large enough. Hence if H takes values in R − mod then the functor τ ≤ k ( H ) is locally finite. Applying the long exact sequence (2.1.1) we obtain thatthe functor τ ≥ k +1 ( H ) is locally finite if H is.Moreover, the corresponding boundedness statement for the case where w isbounded above and H is anti-locally finite is dual to the "locally finite" versionthat we have just verified.Lastly, combining the locally finite and the anti-locally finite cases of ourassertion one immediately obtains its finite case.II.1. Since R − mod is an exact abelian subcategory of R − Mod (see Propo-sition 4.4.2(1)), C ′ is easily seen to be a triangulated subcategory of D .Next, combining our assumptions with assertion I we obtain that virtual t -truncations of functors of the type H N for N ∈ Obj C ′ are D -represented byobjects of C ′ as well. Thus we can combine Theorem 2.2.6 with Proposition2.2.2(3) to obtain the result.2. Certainly, restricting locally or anti-locally finite functors from C to Hw yields additive functors from Hw op into R − mod . This restriction gives anembedding of Ht into Fun R ( Hw op , R − mod) according to Proposition 2.2.2(3).Lastly, assume that A belongs to Fun R ( Hw op , R − mod) . It remains tocheck that the corresponding w -pure functor H A : C op → R − Mod provided byProposition 2.1.4(6) (cf. also Remark 2.1.5(2)) is (anti) locally finite whenever w is bounded below (resp. above). Actually, it is not necessary to assume that R is coherent to prove part I of our theorem.Indeed, for any M ∈ Obj C and n ∈ Z we can complete the morphisms w ≤ n M → w ≤ n +1 M and w ≥ n − M → w ≥ n M to distinguished triangles to obtain that the values of τ ≤ n ( H ) and τ ≥ n ( H ) (see Definition 2.1.1(1)) are quotients of finitely presented R -modules by some imagesof modules of this type. Thus the values of virtual t -truncations of H are finitely presented R -modules (see Lemma A.8 of [Coh85]).On the other hand, it appears that the subcategory C ′ of C does not have to be triangulatedif R is not coherent. R − mod is an abelian subcategory of R − Mod once again and combining it with Lemma 2.1.4 of [Bon18b] we obtainthat the values of H A are finitely presented as well. Since H A is of weight range [0 , and w is bounded, H A is (anti) locally finite immediately from Proposition2.1.4(7).III.1. Since w is orthogonal to t D , Proposition 2.1.2(4) says that for any N ∈ Obj D the functors D -represented by its t D -truncations on C are thecorresponding virtual t -truncations of the functor H N . Thus for any M ∈ Obj C ′ the objects t D, ≤ k M and t D, ≥ k M are objects of C ′ as well according to assertionI; hence t D restricts to C ′ indeed.2. The full faithfulness of the functor Ht → Fun R ( Hw op , R − Mod) is im-mediate from Theorem 3.3.1(II.3). Moreover, this functor obviously factorsthrough
Fun R ( Hw op , R − mod) , and arguing similarly to the proof of assertionII.2 we obtain the equivalence in question. Remark .
1. In the examples in §4.2 and §4.3 below the ring R willalways be noetherian. So let us demonstrate that non-noetherian coherentrings are also actual (at least) in the context of parts I and III of ourtheorem.Similarly to Remark 3.2.6(5), we take R to be an arbitrary (not necessar-ily noetherian) coherent ring which we have to assume to be commutativehere, set D = D ( R ) ; then C = D ( ℵ ) is the subcategory of perfect com-plexes (see Proposition 4.4.2(6) below).Hence C is equivalent to K b (Proj fin R ) , where Proj fin R is the category offinitely generated projective R -modules, and we set w to be the "stupid"weight structure whose heart (essentially) equals Proj fin R ; see Remark1.2.3(1). Since w is bounded, we can apply any of the versions of (partsI and III) of our theorem if we take C ′ to be equal to the category C ′ i ⊂ D for ≤ i ≤ ; here C ′ corresponds to finite cohomological functorsfrom C into R − Mod , C ′ corresponds to locally finite functors, and C corresponds to anti-locally finite ones. Moreover, it is easily seen that C ′ ∼ = D b ( R − mod) , C ′ ∼ = D − ( R − mod) , and C ′ ∼ = D + ( R − mod) .Certainly, the corresponding t -structures are the canonical ones.This example is certainly closely related to Proposition 4.2.1(1) below; weput it here just to demonstrate that it may be actual to consider the casewhere R is coherent but not noetherian.2. The question whether C = C ′ in the notation that we have just intro-duced is well-known to be equivalent to the regularity of R . The reader isrecommended to look at the paper [KhS19] for an interesting discussionof this matter (in the language of adjacent structures) in the more generalsetting of modules over ring spectra.3. In §4.2 we will discuss interesting "geometric" examples for our theorem.Unfortunately, this does not include any examples for the locally finiteand anti-locally finite versions of part II of the theorem.It is also worth noting that bigger families of examples can be obtainedby means of Theorem 0.3 of [Nee18a] and Theorem 0.3 of [Nee18b].31. The finite presentation of values of functors and the vanishing conditionscan certainly be treated separately. So, we could have replaced the cate-gory R − mod of finitely presented R -modules by any other exact subcat-egory of R − Mod in our definitions and formulations. In particular, onecan consider certain "levels of C -smallness" of objects of D (cf. Theorem3.2.3(II.2)).5. Certainly, one can take R = Z ; this allows to apply parts I and III of ourtheorem to arbitrary triangulated categories. Corollary 4.1.4.
Assume that C is R -saturated (where R is a coherent ring)and w is a bounded weight structure on it. Then the following statements arevalid.1. For any i ∈ Z and M ∈ Obj C the functors τ ≤ i ( H M ) and τ ≥ i ( H M ) arerepresentable, where H M = C ( − , M ) .2. There exists a t -structure right adjacent to w . Moreover, t is bounded if C is densely generated by a single object G .3. Its heart Ht is naturally equivalent to Fun R ( Hw op , R − mod) . Proof.
Putting C ′ = D = C in Theorem 4.1.2(II) we obtain everything exceptthe boundedness of t in assertion 2.Now, assume that C is densely generated by { G } . Since the functor H G = C ( − , G ) is finite, C ( G, G [ j ]) = { } for almost all i ∈ Z . Thus we can applyProposition 2.2.7(2) for C ′ = C to obtain that t is bounded. Now assume that R is a (commutative unital) noetherian ring. Proposition 4.2.1.
Let X be a scheme that is proper over Spec R .1. Take D = D qc ( X ) (the unbounded derived category of quasi-coherentsheaves on X ), C = D perf ( X ) (the triangulated category of perfect complexes ofcoherent sheaves on X ), and C ′ = D bcoh ( X ) (resp. C ′ = D − coh ( X ) ; here D bcoh ( X ) is the bounded derived category of coherent sheaves on X , and D − coh ( X ) is itsbounded above version).Then C ∼ = C op , and the "finite versions" of those assumptions of Theo-rem 4.1.2(I-III) that do not mention w (resp. the "locally finite versions" ofthe corresponding assumptions of Theorem 4.1.2(I,III)) are fulfilled for thesecategories.2. Assume that X is a regular scheme (that is proper over Spec R ). Thenthe category C = D perf ( X ) equals C ′ = D bcoh ( X ) ; thus it is R -saturated.Moreover, C is densely generated by a single object G . Proof.
1. Since R is noetherian, X is a noetherian scheme; thus it is well knownthat objects of C compactly generate D and C ∼ = C op . Thus it remains to applyCorollary 0.5 of [Nee18a]; note here that the set of R -linear transformationsbetween two R -linear functors between R -linear categories coincides with theset of transformations between the underlying additive functors.2. It is well known that in this case D perf ( X ) = D bcoh ( X ) indeed; thus C is R -saturated. Moreover, C is densely generated by a single object according toTheorem 0.5 of [Nee17] (see also Remark 0.6 of loc. cit.).32 emark .
1. Thus for X and C as in part 2 of the proposition both leftand right adjacent t -structures to any bounded weight structure exist.Moreover, any weight structure w on D perf ( X ) (for X that is proper over Spec R ) gives a smashing t -structure on D qc ( X ) that is right orthogonalto t . This t restricts to D − coh ( X ) if w is bounded below, and (also) restrictsto D bcoh ( X ) if w is bounded.2. More generally, if X is a Gorenstein (but not necessarily regular) schemethen the coherent duality functor D X gives an equivalence D bcoh ( X ) op → D bcoh ( X ) that restricts to an equivalence D perf ( X ) op → D perf ( X ) . Hencefor any bounded weight structure w on D perf ( X ) there also exists a leftorthogonal weight structure on D bcoh ( X ) .3. Let us discuss the question which t -structures do possess left adjacentweight structures (in the setting of our proposition).Suppose that we start with a t -structure t on C = C ′ . The easy Theorem5.3.1(I.1) below implies that there are enough projectives in Ht if thereexists a weight structure w that is left adjacent to t . Moreover, part IV ofthat theorem says that if this condition is fulfilled, t is bounded, and C op is R -saturated then w does exist.Now we combine part 2 of our of our proposition with Corollary 4.1.4(2) toobtain the following: if X is a regular scheme that is proper over Spec R and C = D perf ( X ) then there exists a 1-to-1 correspondence betweenbounded weights structures on C and (their right adjacent) bounded t -structures such that Ht has enough projectives. Since C is self-dual we alsoobtain a 1-to-1 correspondence between bounded weights structures on C and bounded t -structures such that Ht has enough injectives. Certainly,these two statements can be combined to obtain a rather curious corre-spondence between those bounded t -structures such that Ht has enoughprojectives and those bounded t -structures such that Ht has enough in-jectives. Note also that there exists plenty of examples of bounded weightstructures on C whenever X = P n (Spec R ) (and R is a regular ring); seepart 5 of this remark.We will discuss a similar problem for orthogonal structures in Remark4.3.2(4) below.4. The author also conjectures that all non-degenerate weight structures and t -structures on D perf ( X ) are bounded in this case.Note here that one can easily obtain non-trivial degenerate weight struc-tures and t -structures D perf ( X ) by gluing (at least) if X is a projectivespace (say, over a field); cf. part 5 this remark. Certainly, degenerateweight and t -structures are not bounded.5. To make our proposition "practical" one needs to have some boundedweight structures on a triangulated category C as in our proposition.Unfortunately, no "simple" general constructing methods similar to thatprovided by Theorem 5 of [Pau12] (see Remark 3.2.4(1)) are available inthis setting. However, one can glue weight structures (see Remark 3.2.4(2)and Theorem 8.2.3 of [Bon10a]). That is, if a triangulated category C is33lued from certain D and E in the sense of [BBD82, §1.4] then any pair of(bounded) weight structures gives a "compatible" weight structure on E .Note here that one can shift weight structures on D and E in the obviousway; thus a single pair of weight structures on D and E gives a familyof weight structures on C indexed by Z × Z (whereas shifting a weightstructure of this sort corresponds to adding ( i, i ) to these parameters).Now let us pass to more concrete examples; cf. also Remark 4.1.3(1). Onecan construct a rich family of bounded weight structures on D perf ( X ) atleast in the case where X = P n for some n > (one can probably take anyregular base ring R here) by means of gluing. More generally, it sufficesto assume that D perf ( X ) possesses a full exceptional collection of objects.Since one can "shift weight structures on components", one can obtainplenty of non-trivial weight structures and t -structures on D perf ( X ) .Another important statement is that bounded weight structures C are inone-to-one correspondence with those additive retraction-closed subcate-gories B of C such that B strongly generates C and B is negative in C ;see Proposition 5.1.2 below. Thus one may look for negative subcategoriesin D perf ( X ) to obtain examples of weight structures.6. It appears that the first result in the direction of Proposition 4.2.1(2) wasTheorem 2.14 of [BoK89] where the case of a smooth projective varietyover a field was considered. However, if X is singular then one has to take C ′ = C ; the corresponding statements were only recently established byNeeman, and they motivated our Definition 2.2.1 along with those resultsof this paper that depend on it.Recall also that Theorem 4.3.4 of [BVd03] gives a certain a "non-commutativegeometric" example of an R -saturated category (for R being a field).7. The author suspects that some of the statements in our theorem can begeneralized to the case where R is (not not necessarily noetherian itselfbut) coherent and can be presented as a "flat enough" direct limit ofnoetherian rings. Once again, R is a noetherian ring. We recall some more results of Neeman andD. Murfet. Proposition 4.3.1.
Let X be a scheme that is proper over Spec R ; take C = D bcoh ( X ) op .1. Assume in addition that regular alterations (see Remark 4.3.2(3) below)exist for all integral closed subschemes of X ; take the categories D = D qc ( X ) op and C ′ = D perf ( X ) op (resp. C ′ = D − coh ( X ) op ).Then the "finite versions" of those assumptions of Theorem 4.1.2(I,II) thatdo not mention w (resp. the "anti-locally finite versions" of the correspondingassumptions of Theorem 4.1.2(I)) are fulfilled for our ( C, C ′ , D ) .2. Take D to be the mock homotopy category K m (Proj X ) of projectivesover X as defined in [Mur07, Definition 3.3]; see Remark 4.3.2(2) below formore detail. Then D is compactly generated by (the image with respect to34 full embedding of) C , and C essentially equals the subcategory of compactobjects of D . Proof.
1. This is most of Theorem 0.2 of [Nee18b].2. See Theorems 4.10 and 7.4 of [Mur07].
Remark .
1. Now let us discuss the relation of these statements to themain subject of the paper.The main problem is that if X is not regular then it is well-known thatthere exist objects M and N of C op = D bcoh ( X ) such that C op ( M, N [ i ]) = { } for arbitrarily large values of i (this is an easy consequence of theSerre criterion; cf. Proposition 4.2.1(1) and Remark 4.1.3(2)); hence therecannot exist any bounded weight structures on C . Thus one has no chanceto apply the finite version of Theorem 4.1.2 in this setting.The author does not know whether there can exist bounded below weightstructures on C in this case. On the other hand, if X = Spec R thenit is easily seen that that the stupid weight structure on the category K − (Proj R ) of bounded above complexes of R -modules restricts to itssubcategory C op = D bcoh ( X ) ; certainly, the corresponding weight structureon C is bounded above. The author suspects that this example can beextended at least to the case X = P n (Spec R ) for any n ≥ ; cf. Remark4.2.2(5).2. One can obtain a certain t -structure on a subcategory of D = K m (Proj X ) by means of Theorem 4.1.2(III) if one takes the subcategory C ′ of D corresponding to those anti-locally finite functors from C = D bcoh ( X ) op that are represented by objects of D and chooses a bounded above weightstructure on C .On the other hand, the embedding D bcoh ( X ) op → K m (Proj X ) is given bya rather non-trivial construction from §7 of [Mur07]. So, the author doesnot currently know how to compute this category C ′ . Note however thatthe aforementioned example of a bounded below weight structure on C (in the case X = Spec R ) yields that our theory in non-vacuous in thissetting.3. So, Theorem 4.1 does not allow us to obtain any orthogonal t -structuresfrom Proposition 4.3.1(1). However, we will soon explain that that thelatter proposition is useful for "recovering" left orthogonal weight struc-tures.Thus it is worth recalling that alterations were introduced in [dJo96];they generalize Hironaka’s resolutions of singularities. Since the latterexist for arbitrary quasi-excellent Spec Q -schemes according to Theorem1.1 of [Tem08], part 1 of our proposition can be applies whenever R is anquasi-excellent noetherian Q -algebra. Moreover, regular alterations of allintegral (closed) subschemes of X exist whenever X is of finite type overa scheme S that is quasi-excellent of dimension at most ; see Theorem1.2.5 of [Tem17].4. Similarly to Remark 4.2.2(3), let us now discuss to which extent the or-thogonality relation between weight structures and certain t -structures inbijective in the setting of Proposition 4.2.1(1).35o, assume that X is proper over Spec R . If w is a bounded weightstructure on C = D perf ( X ) then there exists an orthogonal t -structure t on C ′ = D bcoh ( X ) such that Ht ∼ = Fun R ( Hw op , R − mod) ; see Remark4.2.2(1).Conversely, assume that t is a bounded t -structure on C ′ such that Ht is equivalent to the category of R -linear functors from an R -linear cate-gory H into R − mod , and regular alterations exist for all integral closedsubschemes of X . Combining Proposition 4.3.1(1) with Proposition 5.3.3below we obtain that there exists a bounded weight structure w ′′ on asubcategory C ′′ of C that is strictly orthogonal to t . Moreover, if t isactually orthogonal to a bounded weight structure w on C then we obtain C ′′ = C and w ′′ = w here; see Remark 5.3.4.So the only obstacle for obtaining a one-to-one correspondence for these C and C ′ (and under our assumptions on X ) is that we do not knowwhether t that is orthogonal to a bounded weight structure on C as aboveis necessarily bounded. Possibly, this question is related to Theorem 0.15of [Nee17]. We will recall some basics on finitely presented modules and coherent rings.Below we will only consider left R -modules, where R is associative unital ring;moreover, recall that in §4.1 we assume that R is commutative. Definition 4.4.1.
1. We will say that a (left) R -module M is finitely pre-sented if there exists an exact sequence P → P → M → of R -modules,where P i are finitely generated R -projective.2. We will say that R is (left) coherent if any finitely generated left ideal of R is finitely presented.3. Moreover, R is (left) semi-hereditary if any finitely generated left ideal of R is projective.4. We will use the notation D ( R ) for the derived category of (left) R modules.Moreover, we will write D perf ( R ) for the full subcategory of D ( R ) of per-fect complexes , i.e. its objects are quasi-isomorphic to bounded complexesof finitely generated projective R -modules.Now we recall some basic properties of these notions. Proposition 4.4.2.
1. Any valuation ring is semi-hereditary.2. All semi-hereditary and (left) noetherian rings are coherent.3. If R is coherent then any finitely generated submodule of a finitely pre-sented module is finitely presented, and finitely presented modules forman exact abelian subcategory of R − Mod .4. If R is semi-hereditary and P is a finitely generated projective R -modulethen any finitely generated submodule of P is projective.36. If R is coherent and a finitely presented R -module M has an R -projectiveresolution of length n then it also possesses a projective resolution of length n whose terms are finitely presented R -modules.6. D ( R ) is compactly generated by { R } (considered as a left module overitself and put in degree as a complex), and D ( R ) ( ℵ ) = D perf ( R ) . Proof.
All of these statements appear to be rather well-known. Moreover, asser-tions 1 and 2 are obvious (note that any finitely generated projective R -moduleis finitely presented); assertion 3 is immediate from Theorem 2.4 of [Swa18]along with Theorem A.9 of [Coh85] (where right modules over a ring R wereconsidered) and assertion 4 is straightforward from Corollary 0.3.3 of ibid.5. We recall that if → N → P → Q → is an exact sequence of R -modules, P is projective, and Q is of projective dimension at most j for some j > (i.e., Q has a projective resolution of length j ) then N is of projectivedimension at most j − . Hence the following easy inductive argument gives thestatement: if n = 0 then M is projective and finitely presented itself; otherwisewe can take an exact sequence → N → P → M → with P being finitelypresented projective to obtain that N is of projective dimension at most n − and also finitely presented according to assertion 3.6. D ( R ) is obviously compactly generated by { R } . Applying Lemma 4.4.5of [Nee01] we obtain that D ( R ) ( ℵ ) equals h{ R }i (i.e., the subcategory of C densely generated by { R } ). Thus it remains to note that D perf ( R ) is a fullstrict triangulated subcategory of D ( R ) that obviously lies in h{ R }i (look at thedistinguished triangles corresponding to stupid truncations of complexes), andto obtain the equality in question we apply the fact that D perf ( R ) is Karoubian(that is well-known and also follows immediately from Proposition 5.1.2(2) be-low). In this section we study the question when a t -structure possesses a left adjacentor orthogonal weight structure. So, in certain cases we are able to recover w from a right adjacent t -structure t that can be constructed using the results ofprevious sections.For this purpose in §5.1 we recall some statements related to the constructionof weight structures.In §5.2 we recall the (aforementioned) general definition of duality betweentwo triangulated categories and construct an interesting family of examples.In §5.3 we prove that the existence of a left adjacent weight structure isclosely related to the existence of enough projectives in the heart of t . We alsoprove the existence of a left orthogonal weight structure in a context related toTheorem 4.1.2 and Proposition 4.2.1(2). Proposition 5.1.1.
Assume that A and B are extension-closed classes of ob-jects of C . 37. Assume that A ⊥ B [1] . Then the class A ⋆ B of all extensions of elementsof B by elements of A is extension-closed as well.II. Assume in addition that C is smashing, and A and B are closed withrespect to C -coproducts.1. Then A ⋆ B is closed with respect to C -coproducts as well.2. Assume that A is closed either with respect to [ − or with respect to [1] .Then A is retraction-closed in C . Proof.
All of these statements are rather easy.Assertions I and II.1 immediately follow from Proposition 2.1.1 of [BoS19].Assertion II.2 is a straightforward consequence of Corollary 2.1.3(2) of ibid. (seeRemark 2.1.4(4) of ibid.)We also recall a generalization of a well-known existence of weight structuresresult from [Bon10a].
Proposition 5.1.2.
Let B be an additive negative subcategory (see Definition1.2.2(8)) of a triangulated category C such that C is densely generated by Obj B .1. Then the envelopes (see §1.1) C w ≤ and C w ≥ of the classes ∪ i ≤ Obj B [ i ] and ∪ i ≥ Obj B [ i ] , respectively, give a bounded weight structure w on C , and Hw equals Kar C ( B ) .2. Obj Kar C ( B ) strongly generates C . Moreover, C w ≤ (resp. C w ≥ ) is theextension-closure of ∪ i ≤ Obj Kar C ( B )[ i ] (resp. of ∪ i ≥ Obj Kar C ( B )[ i ] ). Proof.
This is just (most of) Corollary 2.1.2 of [BoS18].
Let us now recall the definition of duality.
Definition 5.2.1.
Let A be an abelian category.1. We will call a (covariant) bi-functor Φ : C op × C ′ → A a duality if it isbi-additive, homological with respect to both arguments, and is equipped witha (bi)natural bi-additive transformation Φ( − , − ) ∼ = Φ( − [1] , − [1]) .2. Suppose that C is endowed with a weight structure w , C ′ is endowedwith a t -structure t . Then we will say that w is left orthogonal to t and t is right orthogonal to w with respect to Φ if the following orthogonality conditionis fulfilled: Φ( X, Y ) = 0 if X ∈ C w ≤ and Y ∈ C ′ t ≥ or if X ∈ C w ≥ and Y ∈ C ′ t ≤− .3. Assume that t is right orthogonal to w with respect to Φ .Then we will also say that t is − -orthogonal (resp. + -orthogonal ) to w (withrespect to Φ ) if for any Y ∈ Obj C ′ we have Y ∈ C ′ t ≤− (resp. Y ∈ C ′ t ≥ )whenever Φ( X, Y ) = 0 for all X ∈ C w ≥ (resp. X ∈ C w ≤ ).Moreover, we will say that t is strictly right orthogonal to w and w is strictlyleft orthogonal to t if t is both − and + -orthogonal to w .4. We will write P t for the class of those X ∈ Obj C such that Φ( X, Y ) = 0 for all Y ∈ C ′ t ≤− ∪ C ′ t ≥ (cf. §3 of [NSZ19]); P ′ t = ⊥ C ′ ( C ′ t ≤− ∪ C ′ t ≥ ) . Remark .
1. If C and C ′ are triangulated subcategories of a triangulatedcategory D then the restriction of the bi-functor D ( − , − ) to C op × C ′ is obviouslya duality. Thus Definition 5.2.1 is compatible with Definition 1.3.1.38ore generally, if i : C → D and i ′ : C ′ → D are arbitrary exact functorsthen D ( i ( − ) , i ′ ( − )) is a duality as well. Moreover, this duality is nice in thesense of Definition 2.5.1(2) of [Bon10b]. However, the notion of niceness is notrelevant for the purposes of the current paper.2. Certainly, Φ = 0 is a duality, and any w and t on the correspondingcategories are orthogonal with respect to it. Thus a certain strictness conditionappears to be quite actual (at least) in the setting of general dualities.The importance of this notion is also illustrated by Proposition 5.2.3(II.4)below (note that it is applied in Theorem 5.3.1(I.3)).Now let us study the relation between Hw and Ht . Proposition 5.2.3.
Assume that t is right orthogonal to w with respect to Φ .I.1. Then C w =0 ⊂ P t . Moreover, for any object M ∈ P t the functor Φ( M, − ) restricts to an exact functor E M : Ht → A , and we have Φ( M, − ) ∼ = E M ◦ H t .2. Assume in addition that t is − or + -orthogonal to w (with respect to Φ ).Then { E M : M ∈ C w =0 } is a conservative collection of functors Ht → A (cf.Remark 5.2.4 below).3. Conversely, assume that functors of the type E M for M ∈ C w =0 form aconservative collection and t is right (resp. left) non-degenerate. Then t is − (resp. + ) right orthogonal to w .II. Assume that C ⊂ C ′ and Φ is the restriction of C ′ ( − , − ) to C op × C ′ ,i.e., t is right orthogonal to w in C ′ .1. Then C w ≥ = C ′ t ≥ ∩ Obj C , C w ≤ = ⊥ C ′ t ≥ ∩ Obj C , and C w =0 = P t .2. For any P ∈ P t we have natural isomorphisms of functors C ′ ( P, − ) ∼ = C ′ ( P, H t ( − )) ∼ = Ht ( H t ( P ) , H t ( − )); the first of them is induced by the transformations id C ′ → t ≤ and H t → t ≤ (see Remark 1.1.3(1)).3. Assume that t is cosmashing, C equals C ′ and satisfies the dual Brownrepresentability property. Then H t gives an equivalence of (the full subcategoryof C given by) P t with the subcategory of projective objects of Ht .4. If t is + -orthogonal to w then C ′ t ≥ is closed with respect to C ′ -products. Proof.
I.1. C w =0 ⊂ P t immediately from our definitions.If → A → A → A → is a short exact sequence in Ht then A → A → A → A [1] is well-known to be a distinguished triangle. Applying the functor Φ( M, − ) to it and recalling the definition of P t we obtain an exact sequence M, A [ − → E M ( A ) → E M ( A ) → E M ( A ) → Φ( M, A [1]) = 0 ;hence E M is exact indeed. Moreover, the functors Φ( M, − ) and E M ◦ H t arehomological and annihilate both C ′ t ≤− and C ′ t ≥ ; hence they are isomorphic.2. Since all of these functors are exact, it suffices to verify that for anynon-zero N ∈ C ′ t =0 there exists M ∈ C w =0 such that Φ( M, N ) = 0 . Now,if Φ( M, N ) = 0 for all M ∈ C w =0 then Theorem 2.1.2(2) of [Bon18b] easilyimplies that Φ( M ′ , N ) = 0 for any M ′ ∈ Obj C . Combining this statementwith either − or + -orthogonality of t to w we immediately obtain N = 0 (i.e.,a contradiction).3. If t is right (resp. left) non-degenerate, it suffices to verify that H ti ( N ) = 0 whenever i < (resp. i > ), N ∈ Obj C ′ , and Φ( M, N ) = 0 for all M ∈ C w = i .For this purpose it is certainly sufficient to check that H t ( N ) = 0 whenever Φ( M, N ) = 0 for all M ∈ C w =0 . The latter statement is immediate from our39ssumption on C w =0 along with the isomorphism Φ( M, − ) ∼ = E M ◦ H t providedby assertion I.1.II.1. The argument is rather similar to the proof of Proposition 2.2.2 (cf.Remark 2.2.3). We will use the notation ( C , C ) for the couple ( ⊥ C ′ t ≥ ∩ Obj
C, C ′ t ≥ ∩ Obj C ) .Since w is left orthogonal to t , we have C w ≤ ⊂ C and C w ≥ ⊂ C . Next, C ⊥ C [1] , and applying Proposition 1.2.4(2) we also obtain inverse inclusions.Thus w = ( C , C ) indeed; hence C w =0 = C ∩ C = P t .2. All of these statements are rather easy; they are given by Lemma 2(1) of[NSZ19].3. We should prove that any projective object P of Ht "lifts" to P t .Now, the functor H t respects products according to the easy Lemma 1.4 of[Nee18a] (applied in the dual form; cf. also Proposition 3.4(2) of [BoV19]); thusthe composition G P = Ht ( P , − ) ◦ H t : C → Ab respects products as well.Moreover, G P is obviously homological functor. Thus it is corepresentable bysome P ∈ Obj C that certainly belongs to P t , and it remains to apply theprevious assertion.4. Obvious. Remark . Since the functors of the type E M that we consider in part I ofour proposition are exact (on Ht ), the conservativity of { E M : M ∈ C w =0 } isfulfilled if and only if for any non-zero N ∈ C ′ t =0 there exists M ∈ C w =0 suchthat E M ( N ) = 0 .To describe an interesting family of orthogonal structures (for Ht that is notnecessarily semi-simple) we need the following definition that is closely relatedto Definition D.1.13 of [B-VK16] (see Remark 5.2.7(1) below). Definition 5.2.5.
For an abelian category A we will say that an increasingfamily of full strict abelian subcategories A ≥ i ⊂ A , i ∈ Z , give a nice splitfiltration for A if ∪ i ∈ Z Obj A ≥ i = Obj A , ∩ i ∈ Z Obj A ≥ i = { } , and there existexact left adjoints W ≥ i to the embeddings A ≤ i ⊂ A . Proposition 5.2.6.
Adopt the assumptions of Definition 5.2.5.1. Then A ≥ i are actually Serre subcategories of A .Moreover, the localization functors A ≥ i /A ≥ i +1 → A i possess exact left ad-joints l i . So we will assume that A i are subcategories of A ; for an object N of A and any j ∈ Z we will write N j for l j ( W ≥ i M ) .Furthermore, Obj A i ⊥ Obj A j whenever i = j .2. Assume that all the categories A i are (abelian) semi-simple. Then thecategory A ′ of semi-simple objects of A consists of finite coproducts of objectsof various A i .3. Let C ′ be a triangulated category endowed with a non-degenerate (seeDefinition 1.1.2(3)) t -structure t such that Ht = A ; take C = K b ( A ′ ) .Then the pairing Φ : C op × C ′ → Ab that sends ( M, N ) into L i ∈ Z ( L j ∈ Z H i ( M ) j , H ti ( N ) j ) ,where H ∗ ( M ) is the homology of the complex M and H t ∗ ( N ) is the t -homologyof N , is a duality. Moreover, t is strictly right orthogonal to the stupid weightstructure w on C with respect to Φ . Proof.
1. The exactness of W ≥ i easily implies that A ≥ i are Serre subcategoriesof A indeed (cf. Remark D.1.20 of [B-VK16]). Hence the existence and exactness40f l i is given by Lemma D.1.15 of ibid. (applied in the dual form; see Remark5.2.7(1)).2. We should prove that simple objects of A are precisely the elements of ∪ Obj A i . This is immediate from Proposition D.1.16(1) of ibid.3. The choice of the isomorphism Φ( − , − ) ∼ = Φ( − [1] , − [1]) is obvious. Next, C is semi-simple; hence any its object is a sum of shifts of objects of A i . Com-bining this fact with the exactness of all l j ◦ W ≥ j : N N j along with thesemi-simplicity of A j we obtain that the functor Φ( M, − ) is homological for any M ∈ Obj C . Moreover, the semi-simplicity of C means that any distinguishedtriangle in it is a sum of rotations of triangles of the form M → M → → M [1] ;this observation easily implies that the functor Φ( − , N ) is cohomological for anyobject N of C ′ .Furthermore, t is obviously right orthogonal to w with respect to Φ . Since t is non-degenerate, to verify strictness it suffices to check that for any non-zero N ∈ C ′ t =0 there exists M ∈ C w =0 such that Φ( M, N ) = 0 . The latter statementis an easy consequence of the definition of Φ ; see Proposition D.1.16(3) of loc.cit. Remark .
1. The difference of our definition 5.2.5 from the definition of a weight filtration in D.1.14 of ibid. is that we reverse the arrows and change thesign of inequalities (for A ≤ i ).2. Assume that endomorphism rings of simple objects of A are commuta-tive (and so, they are fields). Then one can easily "replace" the duality Φ inProposition 5.2.6(3) by a duality Φ ′ : C op × C → A ′ such that Φ ′ ( M, N ) = M whenever N = M are simple objects of A ′ ⊂ A and Φ ′ ( M, N ) = 0 if M and N are non-isomorphic simple objects of A ′ .3. Certainly, we could have taken C = K ( A ′ ) instead of C = K b ( A ′ ) inProposition 5.2.6(3); one can also take any intermediate triangulated categoryhere. Now we study the question which weight structures are adjacent to weight struc-tures; yet in certain cases we are only able to construct a weight structure on asubcategory of the corresponding category.
Theorem 5.3.1.
Let t be a t -structure on C ′ .I. Assume that there exists a weight structure w on a triangulated category C ⊂ C ′ that is left orthogonal to w (in C ′ ; here we use Definition 1.3.1(1)) and C ′ t =0 ⊂ Obj C .1. Then there are enough projectives in Ht , for any M ∈ C ′ t =0 there existsan Ht -epimorphism from H t ( P ) into M for some P ∈ C w =0 , and the functor H t induces an equivalence of Kar( Hw ) with the category of projective objectsof Ht .2. Moreover, Hw is equivalent to the latter category whenever the class Obj C is retraction-closed in C ′ and C ′ is Karoubian.3. Assume in addition that t is left non-degenerate. Then t is + -orthogonalto w ; hence C ′ t ≤ is closed with respect to C ′ -products.41I. Assume that there are enough projectives in Ht and for any projectiveobject P ′ of Ht there exists P ∈ P ′ t (see Definition 5.2.1(4)) along with an Ht -epimorphism H t ( P ) → P ′ .1. Then the full subcategory C of C ′ whose object class equals ∪ i ∈ Z C ′ t ≥ i istriangulated, and there exists a weight structure w on C such that C w ≥ = C ′ t ≥ and t is − -orthogonal to w .2. Furthermore, one can extend (see Definition 1.2.2(4)) w as above to aweight structure w ′ on C ′ that is left adjacent to t whenever any of the followingadditional assumptions is fulfilled:a. t is bounded below (see Definition 1.1.2(4)).b. There exists an integer n such that C t ≤ ⊥ C t ≥ n .III. Assume in addition that C ′ satisfies the dual Brown representabilityproperty (see Definition 3.2.1(1)), t is cosmashing and Ht has enough projec-tives. Then the category C ′ is smashing, there exists a weight structure ˜ w on thelocalizing subcategory ˜ C of C ′ that is generated by C ′ t ≤ such that C ˜ w ≤ = C ′ t ≤ ,and H ˜ w is equivalent to the subcategory of projective objects of Ht .IV. Assume that R is a commutative unital coherent ring, the category C ′ op is R -saturated, t is bounded, and Ht has enough projectives. Then there existsa weight structure w ′ on C ′ that is left adjacent to t . Proof.
I.1. Fix M ∈ C t =0 and consider its w -decomposition P p → M → w ≥ M → P [1] . Since M ∈ C ′ t =0 , Proposition 5.2.3(II.1) implies that P belongsto C w ≥ ; hence P belongs to C w =0 according to Proposition 1.2.4(8)). Next, P ∈ C w ≥ ⊂ C ′ t ≥ (see Proposition 5.2.3(II.1)); hence the object P = t ≤ P equals H t ( P ) (see Remark 1.1.3). Therefore P is projective in Ht according toProposition 5.2.3(II.2).Next, the adjunction property for the functor t ≤ (see Remark 1.1.3(1))implies that p factors through the t -decomposition morphism P → P . Now wecheck that the corresponding morphism P → M is an Ht -epimorphism. Thisis certainly equivalent to its cone C belonging to C t ≥ . The octahedral axiomof triangulated categories gives a distinguished triangle ( t ≥ P )[1] → w ≥ M → C → ( t ≥ P )[2] ; it yields the assertion in question since w ≥ M ∈ C w ≥ ⊂ C ′ t ≥ and the class C t ≥ is extension-closed. Thus we obtain that Ht has enoughprojectives.Now, the category of projective objects of Ht is certainly Karoubian. Aswe have just verified, for any projective object Q of Ht there exists an Ht -epimorphism H t ( S ) → Q for some S ∈ C w =0 . Since H t ( S ) is projective in Ht according to Proposition 5.2.3(II.2), this epimorphism splits, i.e., Q equalsthe image of some idempotent endomorphism of H t ( S ) . Applying Proposition5.2.3(II.2) once again and lifting this endomorphism to Hw we obtain that Kar( Hw ) is equivalent to the category of projective objects of Ht indeed.2. Since C w =0 is retraction-closed in C ′ , Hw is Karoubian as well. Hence Hw ∼ = Kar( Hw ) in this case and we obtain the result in question.3. If t is + -orthogonal to w then C ′ t ≤ is closed with respect to C ′ -productsaccording to Proposition 5.2.3(II.4).Applying Proposition 5.2.3(I.3) we obtain that it remains to verify that thefunctors of the form E M for M ∈ C w =0 give a conservative family of functors Ht → Ab . Now, for any object N of Ht our assumptions give the existence of aprojective object P of Ht that surjects onto it. Moreover, applying Proposition5.2.3(II.2) we obtain the existence of P ∈ C w =0 and a morphism h from P such42hat H t ( h ) is isomorphic to this surjection P → N . Hence E P ( N ) = 0 if N isnon-zero, and we obtain the conservativity in question (see Remark 5.2.4).II.1. C is triangulated since the functor H t : C ′ → Ht is homological. Nextwe take C = ⊥ C ′ t ≥ ∩ Obj C , C = C t ≥ , and prove that ( C , C ) is a weightstructure on C (cf. Proposition 5.2.3(II.1)).The only non-trivial axiom check here is the existence of w -decompositionsfor all objects of C . Let us verify the existence of a w -decomposition for any M ∈ C t ≥ i by induction on i . The statement is obvious for i > since M ∈ C w ≥ = C t ≥ and we can take a "trivial" weight decomposition → M → M → .Now assume that existence of w -decompositions is known for any M ∈ C t ≥ j for some j ∈ Z . We should verify the existence of weight decomposition of anelement N of C t ≥ j − . Certainly, N is an extension of N ′ [ − j −
1] = H t ( N [ j +1])[ − j − by t ≥ j N (see Remark 1.1.3(1) for the notation). Since the latterobject possesses a weight decomposition, Proposition 5.1.1(I) (with A = C and B = C [1] ) allows us to verify the existence of a weight decomposition of N ′ [ − j − (instead of N ). Our assumptions imply that there exists an epimorphism H t ( P ) → N ′ with P ∈ P ′ t = C ∩ C . Then a cone C of the correspondingcomposed morphism P → N ′ is easily seen to belong to C ′ t ≥ . Since both P and C possess weight decompositions, applying Proposition 5.1.1(I) once againwe obtain the statement in questions.Lastly, t is − -orthogonal to w immediately from Remark 1.1.3(3).2. If the assumption a is fulfilled then we can just take w ′ = w since C obviously equals C ′ .Now suppose that assumption b is fulfilled. Similarly to the previous proof,it suffices to verify that for the couple w ′ = ( ⊥ C ′ t ≥ , C ′ t ≥ ) the corresponding w ′ -decompositions exist for all objects of C ′ .Since w ′ is an extension of w , assertion II.1 gives the existence of w ′ -decompositions for all elements of C ′ t ≥ − n . Next, our orthogonality assumptionon t yields that C ′ t ≤ − n ⊂ C w ≤ ; hence one can take trivial w ′ -decompositionsfor elements of C ′ t ≤ − n . It remains to note that Obj C ′ = C ′ t ≥ − n ⋆ C ′ t ≤ − n =Obj C ′ by axiom (iv) of weight structures, and apply Proposition 5.1.1(I) onceagain.III. C ′ is smashing according to Proposition 8.4.6 of [Nee01] (applied inthe dual form). Applying Proposition 5.2.3(II.2–3) we obtain that C ′ and t satisfy the assumptions of assertion II.1. We argue similarly to its proof andverify that the corresponding ( ˜ C , ˜ C ) give a weight structure ˜ w on ˜ C . Onceagain, for this purpose it suffices to verify that the class ˜ C = ˜ C ⋆ ˜ C [1] equals Obj ˜ C . Immediately from assertion II, ˜ C contains C ′ t ≥ j for all j ∈ Z . Moreover, ˜ C is extension-closed and closed with respect to C ′ -coproducts according toProposition 5.1.1(I, II.1); hence ˜ C equals Obj ˜ C indeed.Lastly, Obj ˜ C is retraction-closed in C ′ and C ′ is Karoubian according toProposition 5.1.1(II.2); hence H ˜ w is equivalent to the subcategory of projectiveobjects of Ht according to assertion I.2.IV. We want to apply assertion II.1 in this case; to check its assumptions itcertainly suffices to verify that for any projective object P ′ of Ht there exists P ∈ P ′ t such P ′ ∼ = H t ( P ) . According to Proposition 5.2.3(II.2), the latter statementis equivalent to the corepresentability of the functor G P = Ht ( P , − ) ◦ H t : C → R − Mod (cf. the proof of Proposition 5.2.3(II.3)). Now, the values of G P R -modules (since morphisms between any two objects of C ; here we also apply Proposition 4.4.2(3)). Since t is bounded, G P is finite inthe sense of Definition 4.1.1(1), and we obtain the corepresentability in question(see Definition 4.1.1(2)).Lastly, the corresponding category C equals C ′ since t is bounded (cf. as-sertion II.2; one can also apply its formulation directly). Since the resultingweight structure w ′ = w is left orthogonal to t , it is also left adjacent to it (seeDefinition 1.3.1(3)). Remark .
1. Certainly, part III of our theorem becomes more interestingin the case C = C ′ .2. Moreover, parts I and III of our theorem can be considered as a certaincomplement to Theorem 3.2.3(I). So we obtain that the class of t -structures rightadjacent to smashing weight ones is "closely related" to the one of cosmashing t -structures such that Ht has enough projectives.3. Similarly, parts I, II, and IV of our theorem complement Corollary4.1.4(2,3); see Remark 4.2.2(3).It is also an interesting question whether for a general R -saturated category C a weight structure w that is left adjacent to a t -structure t is bounded if andonly if t is.4. The condition C t ≤ ⊥ C t ≥ n for n ≫ (see part II.2 of our theorem) is anatural generalization of the finiteness of the cohomological dimension condition(for an abelian category).5. In parts II and III of our theorem the "starting points" for constructingweight decompositions were the classes C ′ t ≥ and P ′ t . Now, it is easily seen thatone can construct a weight structure starting from P t ⊂ Obj C if C op reflects C ′ op (in a certain D op or with respect to a duality) using Proposition 5.1.2.Certainly, one will also need certain corepresentability assumptions to provethat P t is large enough in some sense (if it is). Proposition 5.3.3.
Assume that R is a commutative unital coherent ring, C and C ′ are R -linear triangulated subcategories of an R -linear triangulatedcategory D , the corresponding functor M H MC ′ gives an equivalence of C op with the category of finite homological functors from C ′ into R − Mod (i.e., thefunctors of the type H M are finite as R -linear functors from C ′ op into R − Mod ),and t is a bounded t -structure on C ′ such that Ht is equivalent to the categoryof R -linear functors from an R -linear category H into R − mod .Then there exists a bounded weight structure w ′′ on the subcategory C ′′ = h P t i of C such that Hw ′′ = P t ⊃ H and w ′′ is strictly left orthogonal to t . Proof.
First we prove that P t gives a negative subcategory of C . So we fix A, B ∈ P t , i ≥ , and prove that A ⊥ B [ i ] by an argument somewhat similar tothe proof of Proposition 2.1.4(5).So T be a transformation H B [ i ] C ′ → H AC ′ . We can obviously complete it to acommutative square H B [ i ] C ′ ◦ t ≥ i −−−−→ H AC ′ ◦ t ≥ i y T ′ y H B [ i ] C ′ T −−−−→ H AC ′ T ′ is an isomorphism and H AC ′ ◦ t ≥ i = 0 , we obtain T = 0 .Applying Proposition 5.1.2 we obtain the existence of a bounded weightstructure w ′′ on the category C ′′ = h P t i such that C ′′ w ′′ =0 = Kar C P t = P t .Next, the description of C ′′ w ′′ ≤ and C ′′ w ′′ ≥ provided by this proposition easilyimplies that w ′′ is left orthogonal to t .Now, t is non-degenerate since it is bounded. Applying Proposition 5.2.3(I.3)we obtain that to finish the proof and verify strong orthogonality it suffices toverify that any object M of H lifts to an element of P t . Now we argue similarlyto the proof of Proposition 5.2.3(II.3). 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