Accessibility and Gorenstein injective envelopes

TL;DR

This paper establishes conditions under which Gorenstein injective cotorsion pairs are complete in Grothendieck categories, with implications for injective envelopes and model structures.

Contribution

It proves the completeness of Gorenstein injective cotorsion pairs under set-theoretic conditions and characterizes their existence via Tate trivial generators.

Findings

Completeness of Gorenstein injective cotorsion pairs is proven under specific generator conditions.

Such cotorsion pairs are equivalent to injective abelian model structures.

Applications include existence of Ding injective envelopes without extra assumptions.

Abstract

Let $\mathcal{G}$ be a Grothendieck category. We prove completeness of the Gorenstein injective cotorsion pair whenever $\mathcal{G}$ admits a set of Tate trivial generators, and show that having such generators is necessary for completeness. In this case it must be a perfect cotorsion pair, cogenerated by a set, and equivalent to an injective abelian model structure on $\mathcal{G}$. Examples include Grothendieck categories (possibly without enough projectives) that admit a generating set consisting of objects of finite projective dimension, such as the category of quasi-coherent sheaves on a quasi-compact and semi-separated scheme. More generally, for a given set $\mathcal{S}$, we characterize the completeness of the Gorenstein $\mathcal{B}$-injective cotorsion pair, where $\mathcal{B} = \mathcal{S}^\perp$, in terms of the existence of a set of $\mathcal{B}$-Tate trivial generators for $\mathcal{G}$. The key ingredient to our proof is the fact that any class of the form $\mathcal{B} :=\mathcal{S}^\perp$ is an accessibly embedded, accessible subcategory of $\mathcal{G}$. The general approach allows for further applications such as the existence of Ding injective envelopes and other relative Gorenstein injective envelopes without imposing additional assumptions on $\mathcal{G}$.