Locally type \(\mathrm{FP}_n\) and \(n\)-coherent categories. (English) Zbl 1508.18003
The first and third author [J. Pure Appl. Algebra 221, No. 6, 1249–1267 (2017; Zbl 1362.18019)] described a nice interplay between modules of type \(\mathrm{FP}_{n}\)and \(n\)-coherentrings in terms of closure properties. The principal objective in this paper is to present and study the concept of \(n\)-coherent categories as a general framework for the study of finiteness conditions of objects, based mainly in the proposal of the concept of locally type \(\mathrm{FP}_{n}\) categories and \(n\)-coherent objects, as generalizations of locally f-initely generated and locally finitely presented categories, and of noetherian and coherent objects [B. Stenström, Rings of quotients. An introduction to methods of ring theory. Berlin-Heidelberg-New York: Springer-Verlag (1975; Zbl 0296.16001; J. Stovicek, “On purity and applications to coderived and singularity categories”, Preprint, arXiv:1412.1615]. The main result is Theorem 5.5, where several characterizations of \(n\)-coherent categories are given. One of these characterizations is given in terms of the existence of a hereditary small cotorsion theory generated by the class of objects of type \(\mathrm{FP}_{n}\). Theorem 5.5 also generalizes the results in [D. Bravo and M. A. Pérez, J. Pure Appl. Algebra 221, No. 6, 1249–1267 (2017; Zbl 1362.18019)] concerning modules of type \(\mathrm{FP}_{n}\), \(\mathrm{FP}_{n}\)-injective modules and \(n\)-coherent rings to the more general context of Grothendieck categories.
The synopsis of the paper goes as follows.
The synopsis of the paper goes as follows.
- § 2
- is concerned with categorical and homological preliminaries.
- § 3
- presents the concept of objects of type \(\mathrm{FP}_{n}\)in a Grothendieck category, studying several closure properties along with some alternative descriptions under some extra assumption in the ground category. The authors also define locally type \(\mathrm{FP}_{n}\) categories as a formal setting for the existence of objects of type \(\mathrm{FP}_{n}\).
- § 4
- investigates injectivity relative to objects of type \(\mathrm{FP}_{n}\). The authors define the class \(\mathcal{FP}_{n}\)-Inj of \(\mathrm{FP}_{n}\)-injective objects, showing that this class is the right half of a complete cotorsion pair \(\left( ^{\bot_{1}}\left( \mathcal{FP} _{n}\text{-Inj}\right) ,\mathcal{FP}_{n}\text{-Inj}\right) \) cogenerated by a set in any locally type \(\mathrm{FP}_{n}\)category.
- § 5
- is devoted to \(n\)-coherent categories. One of the principal results in this section is that the previous cotorsion pair is hereditary iff the ground category is \(n\)-coherent. Another important result holding in \(n\)-coherent categories is that \(\mathcal{FP}_{n}\)-Inj is a covering class.
- § 6
- defines the Gorenstein \(\mathrm{FP}_{n}\)-injective objects, constructing model structures such that they form the class of fibrant objects.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18C35 | Accessible and locally presentable categories |
| 18A25 | Functor categories, comma categories |
| 18E10 | Abelian categories, Grothendieck categories |
| 18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
| 18G15 | Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) |
| 18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |
| 18N40 | Homotopical algebra, Quillen model categories, derivators |
Keywords:
objects of type \(\mathrm{FP}_n\); \(\mathrm{FP}_n\)-injective objects; locally type \(\mathrm{FP}_n\) categories; \(n\)-coherent categoriesReferences:
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