On Noetherian schemes over (C,⊗,1) and the category of quasi-coherent sheaves

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Let (C,⊗,1) be an abelian symmetric monoidal category satisfying certain conditions and let X be a scheme over (C,⊗,1) in the sense of Toën and Vaquié. 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 (C,⊗,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 (C,⊗,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.