By the Telescope Conjecture for Module Categories, we mean the following claim: “Let R be any ring and (A,B) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then (A,B) is of finite type.” We prove a modification of this conjecture with the word ‘finite’ replaced by ‘countable’. We show that a hereditary cotorsion pair (A,B) of modules over an arbitrary ring R is...

The purpose of this paper is to describe a connection between model categories, a structure invented by algebraic topologists that allows one to introduce the ideas of homotopy theory to situations far removed from topological spaces, and cotorsion pairs, an algebraic notion that simultaneously generalizes the notion of projective and injective objects. In brief, a model category structure on a...

In the paper of Keller and Reiten, it was shown that the quotient of a triangulated category (with some conditions) by a cluster tilting subcategory becomes an abelian category. After that, Koenig and Zhu showed in detail, how the abelian structure is given on this quotient category, in a more abstract setting. On the other hand, as is well known since 1980s, the heart of any tstructure is abel...

Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is ExtR(G,G) = 0 holds and follow Schultz [22] to call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples for splitters. Are there others? Answering an open problem by Schultz [22] we will show that there are more splitters, in fact we are able to prescribe the...

The theory of finitely generated relative (co)tilting modules has been established in the 1980s by Auslander and Solberg, infinitely tilting have recently studied many authors context Gorenstein homological algebra. In this work, we build on developing “Gorenstein approximations” employing these approximations to study classes their associated cotorsion pairs. As applications our results, discu...

Rooted quivers are quivers that do not contain A∞ ≡ ··· → • → • as a subquiver. The existence of flat covers and cotorsion envelopes for representations of these quivers have been studied by Enochs et al. The main goal of this paper is to prove that flat covers and cotorsion envelopes exist for representations of A∞. We first characterize finitely generated projective representations of A∞. We ...

Let $\mathscr{C}$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. We introduce the notions left Frobenius pairs, ($n$-)cotorsion pairs and (weak) Auslander-Buchweitz contexts respect to in $\mathscr{C}$. show how construct cotorsion pais from $n$-cotorsion establish one-to-one correspondence between contexts. also study relation certain that pairs. These work ...

In this paper, the notion of cotorsion classes is introduced into higher homological algebra . Our results motivate definition, showing that n -cotorsion satisfies usual properties one could expect. particular, a version Wakamatsu's Lemma proved. At last section, connections with wide subcategories are studied.