On integral schemes over symmetric monoidal categories

TL;DR

This paper extends classical algebraic geometry concepts of integral schemes and function fields to the setting of schemes over symmetric monoidal categories, providing new foundational definitions and characterizations.

Contribution

It introduces notions of Noetherian and integral schemes in symmetric monoidal categories and constructs a corresponding function field for such schemes.

Findings

A definition of Noetherian and integral schemes over symmetric monoidal categories.

Construction of a function field as a commutative monoid object.

Characterization of integrality via reducedness and irreducibility.

Abstract

We propose notions of "Noetherian" and "integral" for schemes over an abelian symmetric monoidal category $(\mathcal C,\otimes,1)$. For Noetherian integral schemes, we construct a "function field" that is a commutative monoid object of $(\mathcal C,\otimes,1)$. Under certain conditions, we show that a Noetherian scheme over $(\mathcal C,\otimes,1)$ is integral if and only if it is reduced and irreducible.