  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.
