On the tensor product of linear sites and Grothendieck categories

TL;DR

This paper introduces a tensor product for linear sites and Grothendieck categories, linking it to locally presentable categories and describing its application to non-commutative and projective schemes.

Contribution

It defines a tensor product for linear sites and Grothendieck categories, connecting it to existing tensor products and schemes, and characterizes its properties in various contexts.

Findings

Tensor product of linear sites and Grothendieck categories is defined.

The tensor product aligns with the tensor product of locally presentable linear categories.

For projective schemes, the tensor product corresponds to the usual product scheme.

Abstract

We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally presentable linear categories, and that the tensor product of locally coherent Grothendieck categories is locally coherent if and only if the Deligne tensor product of their abelian categories of finitely presented objects exists. We describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for projective schemes our tensor product corresponds to the usual product scheme.