Flat model structures and Gorenstein objects in functor categories

TL;DR

This paper develops a flat model structure for functor categories from a small preadditive category to module categories, linking Gorenstein objects in these categories to their evaluations, and extends existing results in the field.

Contribution

It constructs a new flat model structure on functor categories and characterizes Gorenstein objects via evaluations, improving prior results by Dell'Ambrogio, Stevenson, and Šťovíček.

Findings

Established a flat model structure on functor categories.

Characterized Gorenstein objects through pointwise evaluation.

Extended previous results on Gorenstein properties in functor categories.

Abstract

We construct a flat model structure on the category $_{\mathcal{Q},R}{\mathsf{Mod}}$ of additive functors from a small preadditive category $\mathcal{Q}$ satisfying certain conditions to the module category $_{R}{\mathsf{Mod}}$ over an associative ring $R$, whose homotopy category is the $\mathcal{Q}$-shaped derived category introduced by Holm and Jorgensen. Moreover, we prove that for an arbitrary associative ring $R$, an object in $_{\mathcal{Q},R}{\mathsf{Mod}}$ is Gorenstein projective (resp., Gorenstein injective, Gorenstein flat, projective coresolving Gorenstein flat) if and only if so is its value on each object of $\mathcal{Q}$, and hence improve a result by Dell'Ambrogio, Stevenson and \v{S}\v{t}ov\'{\i}\v{c}ek.