The central sheaf of a Grothendieck category

TL;DR

This paper studies the center of a Grothendieck category, showing it forms a sheaf of rings over stable localizing subcategories, with results extending to locally noetherian categories and arbitrary coverings.

Contribution

It introduces a sheaf of centers on the lattice of stable localizing subcategories of a Grothendieck category, expanding understanding of their structure and sheaf properties.

Findings

The center forms a sheaf of rings on stable localizing subcategories.

Sheaf condition holds for finite coverings in general.

Sheaf condition extends to arbitrary coverings when the category is locally noetherian.

Abstract

The center $Z(\mathcal{A})$ of an abelian category $\mathcal{A}$ is the endomorphism ring of the identity functor on that category. A localizing subcategory of a Grothendieck category $\mathcal{C}$ is said to be stable if it is stable under essential extensions. The set $\mathbf{L}^{st}(\mathcal{C})$ of stable localizing subcategories of $\mathcal{C}$ is partially ordered under reverse inclusion. We show $\mathcal{L} \mapsto Z(\mathcal{C}/\mathcal{L})$ defines a sheaf of commutative rings on $\mathbf{L}^{st}(\mathcal{C})$ with respect to finite coverings. When $\mathcal{C}$ is assumed to be locally noetherian, we also show that the sheaf condition holds for arbitrary coverings.