Admissible weak factorization systems on extriangulated categories

<abstract><p>Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we first introduce the concept of left Frobenius pairs on an extriangulated category $ \mathcal{C} $, and then establish a bijective correspondence between left Frobenius pairs and certain cotorsion pairs in $ \mathcal{C} $. As an application, some new admissible model structures are established from left Frobenius pairs under certain conditions, which generalizes a result of Hu et al. (J. Algebra 551 (2020) 23–60).</p></abstract>

The notion of extriangulated category was introduced by Nakaoka and Palu giving a simultaneous generalization of exact categories and triangulated categories. Our first aim is to provide an extension to extriangulated categories of Auslander's formula: for some extriangulated category $\mathcal{C}$, there exists a localization sequence $\operatorname{def}\mathcal{C} \to \mod\mathcal{C} \to \operatorname{lex}\mathcal{C}$, where $\operatorname{lex}\mathcal{C}$ denotes the full subcategory of finitely presented left exact functors and $\operatorname{def}\mathcal{C}$ the full subcategory of Auslander's defects. Moreover we provide a connection between the above localization sequence and the Gabriel–Quillen embedding theorem. As an application, we show that the general heart construction of a cotorsion pair $(\mathcal{U}, \mathcal{V})$ in a triangulated category, which was provided by Abe and Nakaoka, is the same as the construction of a localization sequence $\operatorname{def}\mathcal{U} \to \mod\mathcal{U} \to \operatorname{lex}\mathcal{U}$.

It was shown recently that the heart of a twin cotorsion pair on an extriangulated category is semi-abelian. In this article, we consider a special class of hearts of twin cotorsion pairs induced by [Formula: see text]-cluster tilting subcategories in extriangulated categories. We give a necessary and sufficient condition for such hearts to be abelian. In particular, we can also see that such hearts are hereditary. As an application, this generalizes the work by Liu in the exact case, thereby providing new insights into the triangulated case.

We introduce a relative tilting theory in abelian categories and show that this work offers a unified framework of different previous notions of tilting, ranging from Auslander–Solberg relative tilting modules on Artin algebras to infinitely generated tilting modules on arbitrary rings. Furthermore, we see that it presents a tool for developing new tilting theories in categories that can be embedded nicely in an abelian category. In particular, we will show how the tilting theory in exact categories built this way, coincides with tilting objects in extriangulated categories introduced recently. We will review Bazzoni’s tilting characterization, the relative homological dimensions on the induced tilting classes and parametrise certain cotorsion-like pairs by using n-X-tilting classes. As an application, we show how to construct relative tilting classes and cotorsion pairs in Rep(Q,C) (the category of representations of a quiver Q in an abelian category C) from tilting classes in C, where Q is finite-cone-shape.

Let (𝒜, ℬ) be a hereditary cotorsion pair in ModR. We show that the induced cotorsion pair is complete if and only if the induced cotorsion pair is complete. As applications, some covers (precovers) and envelopes (preenvelopes) over certain classes of complexes of modules are obtained.

Recollements from cotorsion pairs

Given a complete hereditary cotorsion pair$(\mathcal{X}, \mathcal{Y})$, we introduce the concept of$(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integern, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤nover a left Noetherian ringR. Similarly, ifRis a left coherent ring in which all flat leftR-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤n. These structures have their analogous in the category of chain complexes.

We work over coherent rings. We consider the class of weakly Ding injective modules. These are the cycles of the exact complexes of FP-injective modules that stay exact when applying a functor Hom ( A , − ) for any FP-injective module A. Throughout the paper, we assume that the class of weakly Ding injective modules, w D I , is closed under extensions. We prove that any weakly Ding injective module is a direct sum of the form F ⊕ D with F FP-injective and with D Ding injective. We also prove that the class of weakly Ding injective modules is the right half of a complete hereditary cotorsion pair. Then we show that w D I is ccovering if and only if it is closed under direct limits.

In this article, the notion of pure hereditary rings is introduced as a nontrivial generalization of the notion of both pure semisimple rings and of hereditary rings. Some properties and examples of pure hereditary rings are given. For a positive integer n, let 𝒫 n be the class of all left R-modules with projective dimension ≤ n. It is shown that over a left pure hereditary ring is a perfect and hereditary cotorsion pair, which gives a partial positive answer to a question of Göbel and Trlifaj. The modules in the class are also studied. Special attention is paid to the case n = 1.