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We show that Krause's recollement exists for any locally coherent Grothendieck category whose derived category is compactly generated. As a source of such categories, we consider the hearts of intermediate and restrictable t -structures in the derived category of a commutative noetherian ring. We show that the induced tilting object over such a heart gives rise to an equivalence between the two Krause's recollements, and in particular, to a singular equivalence. [ABSTRACT FROM AUTHOR]

We distinguish between faint, weak, strong and strict localizations of categories at morphism families and show that this framework captures the different types of derived functors that are considered in the literature. More precisely, we show that Kan and faint derived functors coincide when we use the classical Kan homotopy category, and when we use the Quillen homotopy category, Kan and strong derived functors coincide. Our comparison results are based on the fact that the Kan homotopy category is a weak localization and that the Quillen homotopy category is a strict localization.

Let (S , n) be a commutative noetherian local ring and ω ∈ n be non-zerodivisor. This paper deals with the behavior of the category Mon (ω , P) consisting of all monomorphisms between finitely generated projective S-modules with cokernels annihilated by ω . We introduce a homotopy category H Mon (ω , P) , which is shown to be triangulated. It is proved that this homotopy category embeds into the singularity category of the factor ring R = S / (ω) . As an application, not only the existence of almost split sequences ending at indecomposable non-projective objects of Mon (ω , P) is proved, but also the Auslander–Reiten translation, τ Mon (-) , is completely recognized. Particularly, it will be observed that any non-projective object of Mon (ω , P) with local endomorphism ring is invariant under the square of the Auslander–Reiten translation. [ABSTRACT FROM AUTHOR]

We show that each rigid monoidal category a over a field defines a family of universal tensor categories, which together classify all faithful monoidal functors from a to tensor categories. Each of the universal tensor categories classifies monoidal functors of a given 'homological kernel' and can be realised as a sheaf category, not necessarily on a . This yields a theory of 'local abelian envelopes' which completes the notion of monoidal abelian envelopes. [ABSTRACT FROM AUTHOR]

Recently in [10], it was proved that over any ring R, there exists a complete cotorsion pair (Kp(Flat-R); K(dg-CotF-R)) in K(Flat-R), the homotopy category of complexes of flat R-modules, where Kp(Flat-R) and K(dg-CotF-R) are the homotopy categories raised by flat (or pure) and dgcotorsion complexes of flat R-modules, respectively. This paper aims at recognition of a parallel cotorsion pair in K(Flat-Q), the homotopy category of flat representation of certain quivers Q, where Q may also be infinite. The importance of this result lies in the fact that this homotopy categories do not necessarily raise from the category of modules over some ring. In the other part of this paper, we give a classification of compact objects in K(dg-CotF-Q), the homotopy category of dg-cotorsion complexes of flat representations of certain Q, in terms of the corresponding vertex-complexes

We distinguish between faint, weak, strong and strict localizations of categories at morphism families and show that this framework captures the different types of derived functors that are considered in the literature. More precisely, we show that Kan and faint derived functors coincide when we use the classical Kan homotopy category, and when we use the Quillen homotopy category, Kan and strong derived functors coincide. Our comparison results are based on the fact that the Kan homotopy category is a weak localization and that the Quillen homotopy category is a strict localization. [ABSTRACT FROM AUTHOR]

In a previous paper, we studied the indeterminacy of the value of a derived functor at an object using different definitions of a derived functor and different types of fibrant replacement. In the present work we focus on derived or homotopy limits, which of course depend on the model structure of the diagram category under consideration. The latter is not necessarily unique, which is an additional source of indeterminacy. In the case of homotopy pullbacks, we introduce the concept of full homotopy pullback by identifying the homotopy pullbacks associated with three different model structures of the category of cospan diagrams, thus increasing the number of canonical representatives. Finally, we define generalized representatives or models of homotopy limits and full homotopy pullbacks. The concept of model is a unifying approach that includes the homotopy pullback used by J. Lurie and the homotopy fiber square defined by P. Hirschorn in right proper model categories. Properties of the latter are generalized to models in any model category. [ABSTRACT FROM AUTHOR]

Hyperhomology is applied to give explicit constructions of left or right adjoint functors of some inclusions between unbounded homotopy categories of additive categories arising from cotorsion pairs in abelian categories. These constructions are independent of Brown representability and do not involve cardinality in set theory. Applications are then given to injective cotorsion pairs in abelian categories and some typical cotorsion pairs in module categories of rings. [ABSTRACT FROM AUTHOR]

We investigate the assumptions under which a subclass of flat quasicoherent sheaves on a quasicompact and semiseparated scheme allows us to 'mock' the homotopy category of projective modules. Our methods are based on module-theoretic properties of the subclass of flat modules involved as well as their behaviour with respect to Zariski localizations. As a consequence we get that, for such schemes, the derived category of flat quasicoherent sheaves is equivalent to the derived category of very flat quasicoherent sheaves. If, in addition, the scheme satisfies the resolution property then both derived categories are equivalent to the derived category of infinite-dimensional vector bundles. The equivalences are inferred from a Quillen equivalence between the corresponding models. [ABSTRACT FROM AUTHOR]

In this paper we study N-differential graded categories and their derived categories. First, we introduce modules over an N-differential graded category. Then we show that they form a Frobenius category and that its homotopy category is triangulated. Second, we study the properties of its derived category and give triangle equivalences of Morita type between derived categories of N-differential graded categories. Finally, we show that this derived category is triangle equivalent to the derived category of some ordinary differential graded category. [ABSTRACT FROM AUTHOR]

We show that the category of graphs has the structure of a 2-category with homotopy as the 2-cells. We then develop an explicit description of homotopies for finite graphs, in terms of what we call 'spider moves.' We then create a category by modding out by the 2-cells of our 2-category and use the spider moves to show that for finite graphs, this category is a homotopy category in the sense that it satisfies the universal property for localizing homotopy equivalences. We then show that finite stiff graphs form a skeleton of this homotopy category. [ABSTRACT FROM AUTHOR]

This paper aims at studying the homotopy category of cotorsion flat left modules K (CotF - R) over a ring R. We prove that if R is right coherent, then the homotopy category K (dg - CotF - R) of dg-cotorsion complexes of flat R-modules is compactly generated. This uses firstly the existence of cotorsion flat preenvelopes over such rings and, secondly, the existence of a complete cotorsion pair (K p (Flat - R) , K (dg - CotF - R)) in the homotopy category K (Flat - R) of complexes of flat R-modules, for arbitrary R. In the setting of quasi coherent sheaves over a Noetherian scheme, this cotorsion pair was discovered in the literature. However, we use a more elementary argument that gives this cotorsion pair for arbitrary R. Next we deal with cotorsion flat resolutions of complexes and define and study the notion of cotorsion flat dimension for complexes of flat R-modules. We also obtain an equivalence K (dg - CotF - R) ≈ K (Proj - R) of triangulated categories where K (Proj - R) is the homotopy category of projective R-modules. Combined with the aforementioned result, this recovers a result from Neeman, asserting the compact generation of K (Proj - R) over right coherent R. Also we get the unbounded derived category D (R) of R as a Verdier quotient of K (dg - CotF - R) . [ABSTRACT FROM AUTHOR]

In this paper, we first construct some complete cotorsion pairs on the category ℂ N (𝒢) of unbounded N -complexes of Grothendieck category 𝒢 , from two given cotorsion pairs in 𝒢. Next, as an application, we focus on particular homotopy categories and the existence of adjoint functors between them. These are an N -complex version of the results that were shown by Neeman in the category of ordinary complexes. [ABSTRACT FROM AUTHOR]

We investigate a version of the Green correspondence for categories of complexes, including homotopy categories and derived categories. The correspondence is an equivalence between a category defined over a finite group G and the same for a subgroup H , often the normalizer of a p -subgroup of G. We present a basic formula for deciding when categories of modules or complexes have a Green correspondence and apply it to many examples. In several cases the equivalence is an equivalence of triangulated categories, and in special cases it is an equivalence of tensor triangulated categories. [ABSTRACT FROM AUTHOR]

For an upper triangular matrix ring, an explicit ladder of height 2 of triangle functors between homotopy categories is constructed. Under certain conditions, the author obtains a localization sequence of homotopy categories of acyclic complexes of injective modules. [ABSTRACT FROM AUTHOR]

Let X be a quasi-compact and quasi-separated (not necessarily semi-separated) scheme. The category of all quasi-coherent sheaves of -modules has several different pure derived categories. Recently, categorical pure derived categories of X have been studied in more details. In this work, we focus on the geometrical purity and find a replacement for the geometrical pure derived category of X. [ABSTRACT FROM AUTHOR]

We investigate the behavior of singularity categories and stable categories of Gorenstein projective modules along a morphism of rings. The natural context to approach the problem is via change of rings, that is, the classical adjoint triple between the module categories. In particular, we identify conditions on the change of rings to induce functors between the two singularity categories or the two stable categories of Gorenstein projective modules. Moreover, we study this problem at the level of 'big singularity categories' in the sense of Krause [30]. Along the way we establish an explicit construction of a right adjoint functor between certain homotopy categories. This is achieved by introducing the notion of 0-cocompact objects in triangulated categories and proving a dual version of Bousfield's localization lemma. We provide applications and examples illustrating our main results. [ABSTRACT FROM AUTHOR]

Abstract Let C be an additive category equipped with an automorphism Σ. We show how to obtain n -angulations of (C , Σ) using some particular periodic injective resolutions. We give necessary and sufficient conditions on (C , Σ) admitting an n -angulation. Then we apply these characterizations to explain the standard construction of n -angulated categories and the n -angulated categories arising from some local rings. Moreover, we obtain a class of new examples of n -angulated categories from quasi-periodic selfinjective algebras. [ABSTRACT FROM AUTHOR]

Let Λ be an Artin algebra and M ∈ mod Λ . It is well-known that the Bongartz's Theorem plays an important role in the representation theory of Artin algebras. In this paper, we investigate the relative version of the Bongartz's Theorem in the bounded homotopy category K b (add M) and construct a Bongartz's completion for an add M -relative presilting complex in K b (add M) such that it becomes an add M -relative silting complex. Finally, we give some applications. [ABSTRACT FROM AUTHOR]

Abstract Let A be a Grothendieck category. We prove that there is an infinite regular cardinal λ such that A has λ -pure injective envelopes. Furthermore, we show that any λ -pure acyclic complex in A is a directed union of contractible subcomplexes. This provides an answer to a question which is conjectured by Simson in [14, Conjecture 3.4]. [ABSTRACT FROM AUTHOR]

Given two complete hereditary cotorsion pairs ( Q , R ) and ( Q ′ , R ′ ) in a bicomplete abelian category G such that Q ′ ⊆ Q and Q ∩ R = Q ′ ∩ R ′ , Becker showed that there exists a hereditary abelian model structure M = ( Q , W , R ′ ) on G , where W is a thick subcategory of G . We prove that the homotopy category Ho( M ) of M is triangulated equivalent to the triangulated quotient category D b ( G ) [ Q , R ′ ] ˆ / K b ( Q ′ ∩ R ′ ) , where D b ( G ) [ Q , R ′ ] ˆ is the subcategory of D b ( G ) consisting of all homology bounded complexes with both finite Q dimension and R ′ dimension and K b ( Q ′ ∩ R ′ ) is the bounded homotopy category of Q ′ ∩ R ′ (core) objects. Applications are given in the category of modules. It is shown that the homotopy category of the Gorenstein flat (resp., Ding projective and Gorenstein AC-projective) model structure on the category of modules established by Gillespie and his coauthors can be realized as a certain triangulated quotient category. [ABSTRACT FROM AUTHOR]