On integral schemes over symmetric monoidal categories

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.

One result added from previous version. This paper will not be published in its present form. The results and methods herein will be subsumed into two further papers