On Noetherian schemes over $(\mathcal C,\otimes,1)$ and the category of quasi-coherent sheaves

TL;DR

This paper generalizes the theory of quasi-coherent sheaves on schemes over symmetric monoidal categories, showing they can be built from finitely generated submodules and characterizing scheme points via functors, extending recent algebraic results.

Contribution

It introduces a framework for schemes over monoidal categories, demonstrates that quasi-coherent sheaves are colimits of finitely generated modules, and characterizes scheme points through functor categories over field objects.

Findings

Quasi-coherent sheaves are colimits of finitely generated submodules on certain schemes.

Points of schemes over field objects can be recovered from functors between sheaf categories.

Generalizes recent results of Brandenburg and Chirvasitu to a broader categorical setting.

Abstract

Let $(\mathcal C,\otimes,1)$ be an abelian symmetric monoidal category satisfying certain conditions and let $X$ be a scheme over $(\mathcal C,\otimes,1)$ in the sense of To\"en and Vaqui\'{e}. In this paper we show that when $X$ is quasi-compact and semi-separated, any quasi-coherent sheaf on $X$ may be expressed as a directed colimit of its finitely generated quasi-coherent submodules. Thereafter, we introduce a notion of "field objects" in $(\mathcal C,\otimes,1)$ that satisfy several properties similar to those of fields in usual commutative algebra. Finally we show that the points of a Noetherian, quasi-compact and semi-separated scheme $X$ over such a field object $K$ in $(\mathcal C,\otimes,1)$ can be recovered from certain kinds of functors between categories of quasi-coherent sheaves. The latter is a partial generalization of some recent results of Brandenburg and Chirvasitu.