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

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.