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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.02634 |
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| _version_ | 1866910189191430144 |
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| author | Estrada, Sergio Gillespie, James |
| author_facet | Estrada, Sergio Gillespie, James |
| contents | 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 $\mathcal{G}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02634 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Accessibility and Gorenstein injective envelopes Estrada, Sergio Gillespie, James Category Theory Algebraic Geometry Algebraic Topology Rings and Algebras 18E10, 18C35 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 $\mathcal{G}$. |
| title | Accessibility and Gorenstein injective envelopes |
| topic | Category Theory Algebraic Geometry Algebraic Topology Rings and Algebras 18E10, 18C35 |
| url | https://arxiv.org/abs/2605.02634 |