Gorenstein projective objects in functor categories

TL;DR

This paper constructs and analyzes a Frobenius subcategory of Gorenstein projective objects within functor categories, providing criteria for when it coincides with the entire Gorenstein projective subcategory, aiding explicit computations.

Contribution

It introduces a Frobenius exact subcategory of Gorenstein projective objects in functor categories and establishes conditions for its equality with the full Gorenstein projective subcategory.

Findings

Constructed a Frobenius exact subcategory of Gorenstein projectives.

Provided criteria for the subcategory to equal the entire Gorenstein projectives.

Demonstrated explicit computations of Gorenstein projectives in examples.

Abstract

Let $k$ be a commutative ring, let $\mathcal{C}$ be a small, $k$-linear, Hom-finite, locally bounded category, and let $\mathcal{B}$ be a $k$-linear abelian category. We construct a Frobenius exact subcategory $\mathcal{GP}(\mathcal{G}_P\operatorname{proj}(\mathcal{B}^{\mathcal{C}}))$ of the functor category $\mathcal{B}^{\mathcal{C}}$, and we show that it is a subcategory of the Gorenstein projective objects $\mathcal{GP}(\mathcal{B}^{\mathcal{C}})$ in $\mathcal{B}^{\mathcal{C}}$. Furthermore, we obtain criteria for when $\mathcal{GP}(\mathcal{G}_P\operatorname{proj}(\mathcal{B}^{\mathcal{C}}))=\mathcal{GP}(\mathcal{B}^{\mathcal{C}})$. We show in examples that this can be used to compute $\mathcal{GP}(\mathcal{B}^{\mathcal{C}})$ explicitly.