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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 21. Выпуск 3.
УДК 539.3 DOI 10.22405/2226-8383-2020-21-3-29-38
Ядра весовых структур — это в точности слабо идемпотентно
полные категории
М. В. Бондарко, С. В. Востоков
Михаил Владимирович Бондарко — доктор физико-математических наук, профессор РАН, Санкт-Петербургский государственный университет (г. Санкт-Петербург). e-mail: m.bondarko@spbu.ru
Сергей Владимирович Востоков — доктор физико-математических наук, профессор, Санкт-Петербургский государственный университет (г. Санкт-Петербург). e-mail: s.vostokov@spbu.ru
Аннотация
Мы доказываем, что аддитивные категории, являющиеся ядрами весовых структур — это в точности все слабо идемпотентно полные категории, т. е, категории, в которых расгце-пимые мономорфизмы соответствуют прямым слагаемым. Мы также приводим ряд условий, эквивалентных слабой идемпотентной полноте (часть их полностью нова), и обсуждаем слабо идемпотентные пополнения аддитивных категорий.
Ключевые слова: слабо идемпотентно полная категория, идемпотентное пополнение, слабое ретракт-замыкание, триангулированная категория, весовая структура, ядро.
Библиография: 15 названий. Для цитирования:
М. В. Бондарко, С. В. Востоков. Ядра весовых структур — это в точности слабо идемпотентно полные категории // Чебышевский сборник, 2020, т. 21, вып. 3, с. 29-38.
CHEBYSHEVSKII SBORNIK Vol. 21. No. 3.
UDC 539.3 DOI 10.22405/2226-8383-2020-21-3-29-38
The hearts of weight structures are the weakly idempotent
complete categories1
M. V. Bondarko, S. V. Vostokov
Mikhail Vladimirovich Bondarko — Doctor of Physical and Mathematical Sciences, RAS Professor, Saint Petersburg State University (Saint-Petersburg). e-mail: m.bondarko@spbu.ru
Sergei Vladimirovich Vostokov — Doctor of Physical and Mathematical Sciences, Professor, Saint Petersburg State University (Saint-Petersburg). e-mail: s.vostokov@spbu.ru
1This work was funded by the Russian Science Foundation under grant no. 16-11-00200.
Abstract
This paper proves that additive categories that occur as hearts of weight structures are precisely the weakly idem,potent complete categories, that is, the categories where all split monomorphisms give direct sum decompositions. The work also gives several other conditions equivalent to weak idempotent completeness (some of them are completely new) and discusses weak idempotent completions of additive categories.
Keywords: weakly idempotent complete category, idempotent completion, weak retraction-closure, triangulated category, weight structure, heart.
Bibliography: 15 titles. For citation:
M. V. Bondarko, S. V. Vostokov, 2020, "The hearts of weight structures are the weakly idempotent complete categories", Chebyshevskii sbornik, vol. 21, no. 3, pp. 29-38.
1. Introduction
The goal of this note is to study additive categories that can occur as hearts of weight structures on triangulated categories.2 Actually, an answer to this question can be extracted from Theorem 4.3.2(1,11) of [3]; yet the corresponding calculation of hearts does not contain all the detail. For this reason, in the current paper we study the corresponding weakly idempotent complete (additive) categories and weak idem,potent completions in detail. Another notion important for this paper is the weak retraction-closure (of a subcategory; see Definition 1).
Let us briefly describe the contents of the paper. An additive category B^ is said to be weakly idempotent complete if any 5-split monomorphism gives a direct sum decomposition (see Definition 2(1)); obviously, any idempotent complete category is weakly idempotent complete. B^ is said to be weakly retraction-closed in B^ D R if for a ^'-isomorphism Y = X 0 Z the object Z belongs to Obj 5 whenever X and Y do. In §3 we prove that B^ is weakly idempotent complete if and only if it is weakly retraction-closed in a (weakly) idempotent complete category B'' D B_. These conditions are equivalent to the existence of C 5 and an idempotent complete B'' D 5 such that B^ equals the corresponding weak retraction-closure of 5'' in B^. Moreover, the weak retraction-closure of B^ in the idempotent completion Kar(5) gives a canonical weak idempotent completion wKar(5) of and we prove that the universality of the Kar-construction also yields that of the wKar-one. Furthermore, 5 is weakly idempotent complete if and only if any contractible bounded ^-complex splits (into a direct sum of isomorphisms; see Proposition 1(7)).
In §4 we recall some basics on weight structures. Recall that these are given by classes C_w^0 and C_w^0 of objects of a triangulated category C; the heart Hw of w is the additive subcategory Qw^o n C-w^o- The aforementioned Theorem 4.3.2(1,11) of [3] (along with the somewhat stronger Corollary 2.1.2 of [5]) gives an almost complete characterization of bounded weight structures. Loc. cit. implies that any (additive) connective subcategory 5 of C_ gives a canonical bounded weight structure w on the smallest strictly full triangulated subcategory ^ of C that contains and Hw consists of _D-retracts of objects of Now, Theorem 1 implies that Hw is equivalent to wKar(5); thus Hw is equivalent to 5 whenever B^ is weakly idempotent complete. Moreover, the results of §3 easily imply that weakly idempotent complete categories are precisely the ones that occur as weakly retraction-closed subcategories of triangulated categories; they are also the categories equivalent to hearts of (bounded) weight structures. Furthermore, we prove that a full embedding B_ ^ B^ induces an equivalenee of Kb(B) with Kb(B^) if and onlv if B'' is essentially a subcategory of wKar(£), and ^ ) if this is the case.
2Recall that weight structures are certain "cousins"of i-structures (see Remark 4 below) that were introduced in [3] and [8]; in the latter paper they were called co-t-structures. Weight structures have several interesting applications to representation theory, motives, and algebraic topology; see [5] for some references.
2. On additive categories and (weak) retraction-closures
All categories and functors (including embedding ones) in this paper will be additive.
• Given a category C and X,Y e Obj C we will write C(X, Y) for the set of morphisms from X to F in C.
• For categories C' and C we write C' C C if C' is a full subcategory of C.
• Given a category ^^d X,Y e Obj C, we say that X is a retract of F if idx can be factored through F.3
• A class of objects D in (an additive category) R is said to be retraction-closed in B_ if it contains all 5-retracts of its elements.
• For any (B_, .D) as above we will write Kar# (D) for the dass of all 5-retracts of elements of D.
• We will say that R is idempotent complete if any idempotent endomorphism gives a direct sum decomposition in it; cf. Definition 1.2 of [1].
• The idempotent completion Kar(5) (no lower index) of R is the category of "formal images" of idempotents in Respectively, its objects are the pairs (B,p) for B e Obj R, p e R(B, B), p2 = p, and the morphisms are given by the formula
Kar(R)((X,p), (X',p>)) = {f e R(X, X') : p' o f = f o p = f}.
The correspondence B ^ (B, id#) (for B e Obj fully embeds R into Kar(5), and it is well known that Kar(5) is essentially the smallest idempotent complete category containing B: sgg Proposition 1.3 of ibid.
Now we will give definitions that appear to be (more or less) new.
Definition 1. Let R! be an (additive) subcategory of B.
1. We will write wKarg(R') for the full subcategory of R whose objects are those Z e Obj R such that there exist X,Y e Obj B^ with X 0 Z = Y. We will call wKarg(R') the weak retraction-closure of R'' in R.
2. We will say that B'' is weakly retraction-closed in R if wKarg(B[) = R.
Below we will need the following simple statements.
Lemma 1. Let R^ be a subcategory of R.
If R! is retraction-closed in R then it is also weakly retraction-closed in R.
2. wKarg(B[) is weakly retraction-closed in R.
Proof. 1. Obvious.
2. For an object Z of B and X,Y e ObjwKar^(B') such that X 0 Z = Y we should prove that Z is an object of wKar#(5') as well. Now we recall Definition 1(1) and choose XuX2,YuY2 e Obj B! such that X 0 Xx = I^d Y 0 Yi = Y2. Then Z 0№ © Yi) = Y 0 Yi 0 Xi = Y2 0 Xv Since bo th X2 0 Yi and Y2 0 Xi are objects of g, we obtain the result. □
3Clearly, if С is triangulated then X is a retract of Y if and only if X is its direct summand.
3. On weakly idempotent complete categories
Let us give some more definitions. Throughout this paper B_ will be an (additive) category.
Definition 2. 1. We will say that B is weakly idempotent complete if any split B_-mono-morphism i : X - Y (that is, idx equals p o i for some p e B_(Y, X)) is isomorphic to the monomorphism idx © 0 : X - X 0 Z for some object Z of B_.
2. Assume that ^ is essentially small. Then the split Grothendieck group Kadd(B) is the abelian group whose generators are the isomorphism classes of objects of B_, and the relations are of the form [B] = [A] + [C] for a 11 A, B,C e Obj B such that B = A 0 C.
Now we prove that this definition is equivalent to several other ones.
Proposition 1. The following assumptions on B are equivalent.
1. B is weakly idem,potent complete.
2. B is weakly retraction-closed in any (additive) category B'' containing B as a strictly full subcategory.
3. B'' is a weakly retraction-closed subcategory of some weakly idem,potent complete category B! ■
4- The obvious embedding of B into the category wKar(5) = wKarKar(B)(B) (see Definition 1 (1)) is an equivalence.
5. B is equivalent to the category wKar(5") for some (additive) category Bl ■
6. There exist additive categories B'' C B C B' such that Bl is idem,potent complete and B = wKar^/ (5'').
1. If a bounded B-complex is contractible (i.e., it is zero in Kb(B)) then it splits, that is, it has the form 0idWi [—i] for some N1 e Obj B-
PROOF. Obviously, condition 1 implies condition 2, and condition 4 implies condition 5. Next, replacing B by its isomorphism-closure in Kar(5) we obtain that condition 2 implies condition
4. Moreover, if B is equivalent to the category wKar(5") then we can replace Bl and Kar(5") by equivalent categories so that B!' C B C B_', B! is equivalent to Kar(5''), and B_ is a strict subcategory of Bl - Hence condition 5 implies condition 6.
Next, applying Lemma 1(2) we obtain that condition 5 implies condition 3; note that Bl is weakly idempotent complete since it is idempotent complete.
Now assume that Bl is a weakly retraction-closed subcategory of a weakly idempotent complete category B!- We should prove that any split monomorphism i : X - Y is isomorphic to the monomorphism idx 0 0 : X - X 0 Z for some object Z of B- Since Bl is weakly idempotent complete, we obtain that Z as desired exists in the category B! D B- Since B_ is a weakly retraction-closed subcategory of Bl and = Y, we obtain Y e Objhence B is weakly idempotent
complete indeed.
Lastly we prove the equivalence of conditions 1 and 7. If p o i = idx for some 5-morphisms X - Y -- X then the complex
• ••- o - X - Y id-=>°P Y - X - 0 - ...
is easily seen to be split in K(KarB)', hence it is zero in K(B) as well. If it is also split in K(B) then ^d p come from a isomorphism Y = thus condition 7 follows from condition 1.
Let us establish the converse implication by the induction on the essential length of a complex M = (Ml); that is, we look for the minimal I ^ 0 such that the terms M1 are zero fori<m and i > n, where n — m = I. Contractible complexes (over an arbitrary additive category) obviously splits if its essentiall length is at most 1. Now, assume that M is contractible of length I ^ 2, and all contractible complexes of length less than I split. Clearly, the contracting homotopv provides a factorization of idMm through the boundarv dm : Mm ^ Mm+i, Hence the complex M is
isomorphic to Cone(idMm)[—1 — m] 0 M', where M' is of length I — 1. Obviously, M' is contractible
□
Remark 1. 1. The notions of a weakly retraction-closed subcategory and of the weak retraction-closure are obviously self-dual.
Hence conditions 2, 4, 5, 6 of our proposition are self-dual as well; this is also true for condition 7 (that is, these assumptions are fulfilled for R if and only if they are valid for R°p). Thus the notion of weak idem,potent completeness is self-dual. Hence weak idem,potent completions can also be characterized by the duals of conditions 1 and 3 in Proposition 1. In particular, we obtain Lemma 7.1 of [6].
2. We will call the category wKar(5) = wKarKar(B) the weak idem,potent completion of R following Rem,ark 7.8 of [6]. We will justify this terminology and also prove and extend the claim, made in loc. cit. in Corollary 1 below.
3. Let R be an (associative unital) ring. Let us describe certain categories that fulfil the assumptions of Proposition 1(6).
Take B!_' to be the category of free left finitely generated R-modules and B!_ to be the category of all left R-modules. Then the corresponding category R = wKar(5'') is just, the category of finitely generated stably free left R-modules,4
This example demonstrates that weakly idem,potent, complete categories do not have to be idem,potent, complete and gives a nice example of weak idem,potent, completions (along with weak retraction-closures).
4-. The argument used in the proof of the implication (1) (7) easily implies that any bounded above or below contractible R-complex splits as well.
On the other hand, Proposition 10.9 of [6] says that arbitrary (unbounded) contractible B_-complexes split, if and only if B is idem,potent, complete.
These statements (along with our arguments above) are closely related to Rem,ark, 1.12 of [9]. Corollary 1. Let, F : Ri ^ B2 be an additive functor.
1. Then there exists a natural "idem,potent, complete version"Kar(F) : Kar(5i) ^ Kar(^2) that restricts to a functor wKar(F) : wKar(5i) ^ wKar(^2).
2. Consequently, if B2 is (weakly) idem,potent, complete then F extends to an additive functor from Kar(^i) (resp. from wKar(B_i)) into B2.
3. Assume that R is essentially small. Then Kar(5) also is, and wKar(5) consists of those M e ObjKar(^) such that the class of M in K0(Kar(B)) (see Definition 2(2)) belongs to the image of the obvious homomorphism Kadd(B) ^ Kgdd(Kar(5)).
Proof. 1. It is easily seen that F yields a canonical additive functor Kar(F) that sends (B,p) for B e Obj Bi, p e Bi(B, B), p2 = p into (F(B), f (p)) indeed.
Next, if X 0 Z = F in Kar^) then Kar(F)(X) 0 Kar(F)(Z) = Kar(F)(Y). Thus if an object Z of Kar(5i) ^^togs to wKar(5i) then Kar(F)(Z) belongs to wKar(^2) indeed.
4The authors are deeply grateful to Vladimir Sosnilo for this nice observation.
2. If is idempotent complete then it is equivalent to the category Kar(B2); hence one can modify Kar(F) to obtain the extension in question.
Similarly, if B_2 is weakly idempotent complete then it is equivalent to the category wKar(^2) (see condition 4 in Proposition 1); thus one can modify wKar(F) to obtain the result.
3. The essential smallness of Kar(5) obviously follows from that of B_-
Next we note that the definition of K0(Kar(B_)) immediately implies the following: we have [M] = [N1] — [N2] to some objects Ni of B_ (being more precise, here we consider the objects (Ni, idWi ) of Kar(B)) whenever there exists B e ObjKar(B) such th at M 05 0 N2 = M 0 B. Since B is a retract of an object of this is equivalent to the existence of B' e Obj B_ such that M 0 B' 0 N2 = ^ 0 ^5'. Our assertion fol lows immediately. □
Remark 2. Let us now relate the terminology in the current paper to that in earlier ones.
It appears that the term "weakly idem,potent complete "for a category B_ was introduced in [6, Definition 7.2]. In [7] (probably, this is where this notion was originally introduced) it was said that retracts have complements (in B)-,5 whereas in Definition 1.11 of [9] it, was said that B_ is s em, i-saturated. Most of the conditions in Proposition 1 and Theorem 1 were not mentioned in these papers.
Recall also that weak idempotent completions were called small envelopes in Definition 4.3.1(3) of [3] and semi-saturations in §1.12.1 of of [9].
4. Weight structures: short reminder
Let us start from the definition of a weight structure (note however that the only axiom of weight structures that we will mention explicitly in this text is the axiom (i)). The symbol C_ in this paper will always denote some triangulated category.
It will be convenient for us to use the following notation below: for D,E C Obj C_ we will write D ± £ if C(X, Y) = {0} for all X e D and Y e E.
Definition 3. I. A couple (C_w^o,C_w^o) of classes of objects of C_ will be said to give a weight structure w on C_ if the following conditions are fulfilled.
(i) ÇW^Q and C_w^0 are retraction-closed in Ç (i.e., contain all C_-ret,ract,s of their objects).
(ii) Semi-invariance with respect to translations.
C [1] an d Q-w^0[1] C
(Hi) Orthogonality.
^ Qw^o [1]-
(iv) Weight decompositions.
For any M e Obj C_ there exists a distinguished triangle
LwM ^ M ^ RwMM[1] such that Lw M e Q_w^0 an d Rw M e C_w ^0[1\.
We will also need the following definitions.
Definition 4. Assume that a triangulated category Q is endowed with, a weight structure w, i e Z.
1. The full category Hw C Ç whose objects are C_w=0 = C_w^0 nc ,n <0 is called the heart of w.
Cw>i (resp. Cw<i, resp. C_w=i) will denote the class C_w^0[i] (resp. resp. Qw=0[i]).
5Recall that the main Proposition of ibid, says that weakly idempotent complete categories closed with respect to countable coproducts are idempotent complete.
3. We will say that (C, w) is bounded and C is a bounded weighted category if Obj C =
= UiezQ.w>i = U iezCw<:i.
4- Let D be a full triangulated subcategory of Q.
We will say that w restricts to R whenever the couple w= (C_w^0 n Obj R, C_w>0 n Obj D) is a weight structure on D.
5. We will say that the subcategory R C C is connective (in Q) if Obj H_ ± (U>0 Obj(^[i]))-6
6. The smallest strictly full triangulated subcategory of C containing R will be called the subcategory strongly generated by Rin C_.
1. We will say that a class V C Obj C is extension-closed if V contains 0 and for any C_-distinguished triangle A ^ C ^ B ^ A[1] the object B belongs to V whenever (both) A and C do.
Remark 3. 1. A simple (and still quite useful) example of a weight structure comes from the stupid filtration on the hom,ot,opy category of cohomological complexes K(B') for an arbitrary additive R; it can also be restricted to the subcategory Kb(B) of bounded complexes (see Definition 4(4)). In this case K(B)Wst^0 (resp. K(B)Wst^0) is the class of complexes that are hom,ot,opy equivalent to complexes concentrated in degrees ^ 0 (resp. ^ 0); see Rem,ark 1.2.3(1) of [5] for more detail.
The heart of the weight structure wst is the retraction-closure of Rin K(B'); hence it is equivalent, to Kar(5) (since both K-(R) and K+(B) are idem,potent, complete).
The restriction ofwstto Kb(B) will be denoted by wbst; in Theorem 1 below we will demonstrate that its heart Hwbst is equivalent to wKar(5).
2. In this note we use the "homological convention" for weight structures. This is the convention used by several papers of the first author (including [5] and [4]). However, in [3] the so-called
[1] —1 is one of the reasons for us not to cite ibid, below; another one is that the exposition of the theory of weight complexes (that we will apply in the proof of Theorem 1) in §3 of [3] is rather inaccurate.
Let us now recall the relation of connective subcategories to weight structures.
Proposition 2. Let C_be a triangulated category.
I. Assume that w is a weight structure on C_.
1. Then the classes C_w^0, C_w^0, andC_w=0 are extension-closed; consequently, they are additive.
2. Let v be another weight structure for Q; suppose that C_w^0 C C_v<0 and C_w^0 C Then w = v (i.e., the inclusions are equalities).
II. Under the assumptions of Definition 4(5) there exists a unique weight structure wb^ on the category D = {B} whose heart contains B_. Moreover, this weight structure is bounded and DWr=0 =Karc (Obj B).
Proof. 1.1. See Proposition 1.2.4(3) and Remark 1.2.3(4) of [5].
2. This is Proposition 1.2.4(7) of loc. cit.
□
6In earlier texts of the first author connective subcategories were called negative ones. Moreover, in several papers (mostly, on representation theory and related matters) a connective subcategory satisfying certain additional assumptions was said to be silting; this notion generalizes the one of tilting.
5. On hearts of weight structures
Theorem 1. The following assumptions on (an additive category) R are equivalent as well.
1. R is weakly idem,potent complete.
2. There exists a triangulated category C such that R is its weakly retraction-closed subcategory.
3. R is equivalent to the heart of a weight structure.
4- R is equivalent to the heart of a bounded weight structure.
5. B is equivalent to the heart Hwbst of the weight strudure wbst on the category Kb(R) (see Remark 3(1)).
6. For any category R'' such that the embedding R'' ^ wKar(5'') factors through a fully faithful functor B ^ wKar(5'') and Rf' is connective in a triangulated category C_ (see Definition 4(5)), there exists a unique weight structure w on the triangulated subcategory D of C_ strongly generated by B'' such that the heart Hw is naturally equivalent to R (that is, Hw contains R'' and the embedding B!' ^ Hw factors through an equivalence of B with Hw ).
proof. Clearly, condition 5 implies condition 4, and 4 implies 3. Next, we can take R'' = R in condition 6. Since R is connective in the category C_ = D = Kb(R) and strongly generates it, we obtain that condition 6 implies condition 5.
Now, axiom (i) of Definition 3 implies that Hw is retraction-closed in C_ (note that it is an additive subcategory by Proposition 2(1.1)). Thus condition 3 implies condition 2.
Furthermore, any triangulated category is easily seen to be weakly idempotent complete since for X and Y as in Definiton 2(1) we have Y = X 0 Cone(X ^ Y). Thus condition 2 implies that R is weakly idempotent complete (i.e., that condition 1 is fulfilled); see Proposition 1(3).
Thus it remains to verify that any weakly idempotent complete category R fulfils condition
6. The existence and the uniqueness of a weight structure w on D such that B" C Hw follows immediately from Proposition 2(11); we also obtain the existence of a fully faithful functor Hw ^ Kar(5''), whereas the latter category is clearly equivalent to Kar(5). Moreover, Hw is weakly idempotent complete (recall that we have just proved that our condition 3 implies condition 1); hence Corollary 1 implies that the embedding R ^ R'' factors through a full embedding of R into Hw.
Since R is weakly idempotent complete, it remains to verify that for any M e Dw=0 there exist objects X and Y of R such that = Y. We will deduce this statement from the existence
of splittings of contractible complexes in Kb(Hw) \ for this purpose we invoke the theory of (weak) weight complex functors as provided by Proposition 1.3.4 of [4].
Part 6 of loc. cit. associates to M its weight complex t(M) e Obj K(Hw). Parts 4 and 10 of loc. cit. imply that t(M) = M (in the homotopv category K(Hw)). On the other hand, parts 4 and 9 easily yield that t(M) is homotopv equivalent to a complex N e Obj Kb(Rf') C K(Hw). Hence there exists a K(Hw)^^^^hism f : M ^ N such that Cone(/) is contractible. Since Cone(/) e Obj Kb(Hw) and we already proved that Hw is weakly idempotent complete, we obtain that Cone(/) splits in Kb(Hw).Now, if Ni e Obj R!' are the terms of N, then this splitting yields M 0ieZ N2i-i = 0ieZ N2i. ^to concludes the proof. □
Remark 4. Weight structures are well known to be closely related to t-stru,ctures (as introduced in §1.3 of [2]). However, the properties of weight structures are significantly distinct from that of t-strudures. Recall in particular that the hearts of t-structures are precisely the abelian categories. Hence there are plenty of additive categories that are hearts of some weight structures and cannot occur as hearts of t-structures; cf. Rem,ark 1(3).
The following statement gives one more characterization of weak idempotent completions as well as certain Grothendieck group isomorphisms.
Corollary 2. 1. If В С В' then the corresponding embedding Kb(B) ^ Kb(B') is an equivalence if and only if the embedding В ^ wKar(^) factors through a fully faithful functor from, В'[ in to wKar(5).
2. Consequently, if В'' is essentially small and В С В'' С wKar(5) then the obvious homomorphism K0dd(В) ^ K0dd(B') (see Definition 2(2)) is bijective.
proof. 1. Assume that the embedding Kb(B_) ^ Kb(B') is an equivalence. Then we can assume that the stupid weight structure on Кb(B') (see Remark 3(1)) gives a weight structure v on С = Кb(B_). Since the classes C_v^0 and C_v^0 are closed with respect to isomorphisms (see the axiom (i) in Definition 3), we obtain С^ъ С C_v^0 and C_wъ С C_v^0. Thus wbt = v according to Proposition 2(1.2). Since Hwbt is equivalent to wKar(5) (see condition 5 in Theorem 1), this clearly gives a fully faithful functor from Bl into wKar(^).
Now let us prove the converse implication. We should prove that D = Кb(B'), where D^ is the closure of Кb(B) in Kb(B') with respect to isomorphisms, if В' is equivalent to a subcategory of wKar(5^w, D is a strictly full subcategory of Кb(Б') that essentially contains wKar(5) (see condition 6 in Theorem 1). Since Kb(B'') is clearly strongly generated (see Definition 4(6)) bv Б', we easily obtain the equality in question.
2. According to assertion 1, the embedding Kb(B) ^ Kb(B'[) is an equivalence in this case. Thus it suffices to recall that for any essentially small (additive) category A the group K$dd(A) can be computed as a certain triangulated Grothendieck group of the category Кb(A)-, see Definition 2 and Theorem 1 of [10]. □
Remark 5. 1. One can also prove that a category B^ D Bis essentially a subcategory of Kar(B) if and only if К (В) = К (В'); this is also equivalent to К+(B) = К + (Б') and К-(Б) = К- (Б'). То prove the "if "implications here one can apply stupid weight structure arguments similar to the one in the proof of Corollary 2, and one can invoke either Rem,ark 3.3.2(2) of [4] or Rem,ark 1.12.4 of [9] to obtain the converse implications.
2. This observation along with Proposition 1(7), Proposition 10.9 of [6] (see Rem,ark 1(4)), and Rem,ark 3 justifies the following vague claim,: Kar(B) is the "extension"of В corresponding to unbounded B-complexes, wheras wKar(5) "corresponds to "Kb(B).
3. One can also prove Corollary 2(2) more explicitly; cf. Corollary 1(3).
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5. Bondarko M.V., Sosnilo V.A., On constructing weight structures and extending them to idempotent extensions// Homology, Homotopv and Appl., vol. 20(1), 2018, 37-57.
6. Biihler Т., Exact categories// Expo. Math. 28 (2010), 1-69.
7. Frevd P., Splitting homotopv idempotents, in: Proceedings of the Conference on Categorical Algebra, La Jolla, CA, 1965, Springer, New York, 1966, 173-176.
8. Pauksztello D., Compact cochain objects in triangulated categories and co-t-structures// Central European Journal of Mathematics, vol. 6(1), 2008, 25-42.
9. Neeman A., The derived category of an exact category// J. Algebra 135 (2), 1990, 388-394.
10. Rose D., A note on the Grothendieck group of an additive category// Vestn. Chelvab. Gos. Univ., 17(3), 2015, 135-139.
REFERENCES
1. Balmer P., Schlichting M., 2001, "Idempotent completion of triangulated categories", Journal of Algebra, 236, no. 2, 819-834.
2. Beilinson A., Bernstein J., Deligne P., 1982, "Faisceaux pervers" Asterisque, 100, 5-171.
3. Bondarko M. V., 2010, "Weight structures vs. ¿-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)", J. of K-theory, v. 6(3), 387-504, see also http: //arxiv.org/abs/0704.4003
4. Bondarko M. V., 2018, "On weight complexes, pure functors, and detecting weights", preprint, https://arxiv.org/abs/1812.11952
5. Bondarko M. V., Sosnilo V. A., 2018, "On constructing weight structures and extending them to idempotent extensions", Homology, Homotopy and Appl, vol. 20(1), 37-57.
6. Biihler T., 2010, "Exact categories", Expo. Math. 28, 1-69.
7. Frevd P., 1965, "Splitting homotopy idempotents", Proceedings of the Conference on Categorical Algebra, La Jolla, CA, Springer, New York, 1966, 173-176.
8. Pauksztello D., 2008, "Compact cochain objects in triangulated categories and co-t-structures", Central European Journal of Mathematics, vol. 6(1), 25-42.
9. Neeman A., 1990, "The derived category of an exact category", J. Algebra 135 (2), 388-394.
10. Rose D., 2015, "A note on the Grothendieck group of an additive category", Vestn. Chelyab. Gos. Univ., 17(3), 135-139.
Получено 29.06.2020 г.
Принято в печать 22.10.2020 г.
