Noetherian schemes over abelian symmetric monoidal categories

In this paper, we develop basic results of algebraic geometry over abelian symmetric monoidal categories. Let A be a commutative monoid object in an abelian symmetric monoidal category (C, circle times, 1) satisfying certain conditions and let epsilon(A) = Hom(A-Mod)(A, A). If the subobjects of A satisfy a certain compactness property, we say that A is Noetherian. We study the localization of A with respect to any s epsilon epsilon(A) and define the quotient A/I of A with respect to any ideal I subset of epsilon(A). 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 (C, circle times, 1). Our notion of a scheme over a symmetric monoidal category (C, circle times, 1) is that of Toen and Vaqui