Noetherian schemes over abelian symmetric monoidal categories

Abstract

In this paper, we develop basic results of algebraic geometry over abelian symmetric monoidal categories. Let [Formula: see text] be a commutative monoid object in an abelian symmetric monoidal category [Formula: see text] satisfying certain conditions and let [Formula: see text]. If the subobjects of [Formula: see text] satisfy a certain compactness property, we say that [Formula: see text] is Noetherian. We study the localization of [Formula: see text] with respect to any [Formula: see text] and define the quotient [Formula: see text] of [Formula: see text] with respect to any ideal [Formula: see text]. We use this to develop appropriate analogues of the basic notions from usual algebraic geometry (such as Noetherian schemes, irreducible, integral and reduced schemes, function field, the local ring at the generic point of a closed subscheme, etc.) for schemes over [Formula: see text]. Our notion of a scheme over a symmetric monoidal category [Formula: see text] is that of Toën and Vaquié.