Generalized periodicity theorems

Abstract

Let R be a ring and S be a class of strongly finitely presented (FP ∞ ) R-modules closed under extensions, direct summands, and syzygies. Let (A, B) be the (hereditary complete) cotorsion pair generated by S in Mod-R, and let (C, D) be the (also hereditary complete) cotorsion pair in which C = lim − → A = lim − → S. We show that any A-periodic module in C belongs to A, and any D-periodic module in B belongs to D. Further generalizations of both results are obtained, so that we get a common generalization of the flat/projective and fp-projective periodicity theorems, as well as a common generalization of the fp-injective/injective and cotorsion periodicity theorems. Both are applicable to modules over an arbitrary ring, and in fact, to Grothendieck categories. 1(3) if the exact sequence ( * ) is pure and the R-module L is pure-injective, then the R-module M is pure-injective ( Št'ovíček 2014 [44]); (4) in particular, if the R-module M is fp-injective and the R-module L is injective, then the R-module M is injective; (5) if the R-module L is cotorsion, then the R-module M is cotorsion (Bazzoni, Cortés-Izurdiaga, and Estrada 2017 [4]); (6) if the ring R is right coherent and the right R-module L is fp-projective, then the R-module M is fp-projective ( Šaroch and Št'ovíček 2018 [37]); (7) over any ring R, if the R-module L is fp-projective, then the R-module M is weakly fp-projective (Bazzoni, Hrbek, and the present author 2022 [5]). Periodicity phenomena are linked to behavior of the modules of cocycles in acyclic complexes. This means that the assertions (1-7) can be restated as follows:(1 c ) in any acyclic complex of projective modules with flat modules of cocycles, the modules of cocycles are actually projective (so the complex is contactible); (2 c ) in any pure acyclic complex of pure-projective modules, the modules of cocycles are pure-projective (so the complex is contractible); (3 c ) in any pure acyclic complex of pure-injective modules, the modules of cocycles are pure-injective (so the complex is contractible); (4 c ) in any acyclic complex of injective modules with fp-injective modules of cocycles, the modules of cocycles are actually injective (so the complex is contractible); (5 c ) in any acyclic complex of cotorsion modules, the modules of cocycles are cotorsion; (6 c ) in any acyclic complex of fp-projective right modules over a right coherent ring, the modules of cocycles are fp-projective; (7 c ) in any acyclic complex of fp-projective modules (over any ring), the modules of cocycles are weakly fp-projective. We refer to the introduction to the preprint [5] for a more detailed discussion of the periodicity theorems (1-7) and (1 c -7 c ). 0.1. The aim of this paper is to obtain a common generalization of ( 1) and (6-7), and also a common generalization of ( 4) and ( 5), in the context of a chosen class of modules or objects in a Grothendieck category. Let us start with presenting the most symmetric and nicely looking formulation of a special case of our main results, and then proceed to further g...

Derived, coderived, and contraderived categories of locally presentable abelian categories

“…It remains to consider the induced short exact sequence of direct limits Proof. In view of [36,Proposition 6.1], it suffices to show that any closed morphism into L ˛from a countably presented CDG-module T ˛factorizes through a countably generated projective CDG-module in Z 0 (B ˛-Mod). Indeed, by the ℵ 1 -direct limit assumption on L ˛, the closed morphism T ˛−→ L ˛factorizes through a gradedprojective CDG-module P ˛over B ˛.…”