Locally type $\text{FP}_n$ and $n$-coherent categories

Abstract

We study finiteness conditions in Grothendieck categories by introducing the concepts of objects of type FPn and studying their closure properties with respect to short exact sequences. This allows us to propose a notion of locally type FPn categories as a generalization of locally finitely generated and locally finitely presented categories. We also define and study the injective objects that are Ext-orthogonal to the class of objects of type FPn, called FPn-injective objects, which will be the right half of a complete cotorsion pair.As a generalization of the category of modules over an n-coherent ring, we present the concept of n-coherent categories, which also recovers the notions of locally noetherian and locally coherent categories for n = 0, 1. Such categories will provide a setting in which the FPn-injective cotorsion pair is hereditary, and where it is possible to construct (pre)covers by FPn-injective objects. Moreover, we see how n-coherent categories provide a suitable framework for a nice theory of Gorenstein homological algebra with respect to the class of FPn-injective modules. We define Gorenstein FPn-injective objects and construct two different model category structures (one abelian and the other one exact) in which these Gorenstein objects are the fibrant objects.

Torsion and torsion-free classes from objects of finite type in Grothendieck categories

“…More specifically, one of our main results shows that I n is a torsion class if, and only if, every object of type FP n has projective dimension at most 1 (see Corollary 2.12). Furthermore, we complement this equivalence by showing that, if in addition G is a locally type FP n -category in the sense of [5], or if G has a projective generator, then I n is a torsion class if, and only if, it is a 1-tilting class (see Theorem 2.13).…”

“…The definition of FP n -injective R-modules over any ring with identity is rather standard. However, the authors in [5] have been able to extend it to any Grothendieck categories establishing basic results.…”

“…The previous serves as a starting point to motivate the present article. Our main purpose will be to take one step further and establish necessary and sufficient conditions under which I n , the class of FP n -injective objects in G (recently introduced in [5]), is a torsion class. This class is formed by those objects M ∈ G which are injective relative to the class FP n of objects of type FP n .…”

“…Organization We shall begin this article presenting some preliminary notions, such as (co)torsion pairs and 1-(co)tilting objects. Section 2 will be devoted to the study of the class I n of FP n -injective objects in a Grothendieck category G. We first recall from [5] the concept of objects of type FP n . We then recall the notion of FP n -injective objects, and prove some of its properties.…”

“…From Remark 4.10, we know that G is 1-coherent, as well. So by[5,], FP 1 (G) is closed under kernels of epimorphisms. So FP 1 (G) is abelian.…”