Let $X$ be a scheme. Consider the global section functor $\Gamma : \mathrm{Qcoh}(X) \to \mathrm{Ab}$. It is well-known that $\Gamma$ preserves filtered colimits if $X$ is assumed to be quasi-compact and quasi-separated. In other words, $\mathcal{O}_X$ is a compact object. Even more, it can be shown (using the results of EGA I, 6.9) that compact objects are precisely the quasi-coherent modules which are locally of finite presentation. If we define a locally finitely presentable tensor category to be a tensor category $C$, whose underlying category is locally finitely presentable and such that a) $1_C$ is compact, b) the compact objects are closed under $\otimes_C$, then $\mathrm{Qcoh}(X)$ turns out to be such a category.
Does the converse also hold? This would be a categorical interpretation of this typical finiteness condition in algebraic geometry.
Question. When $\Gamma$ preserves filtered colimits, does it follow that $X$ is quasi-compact and quasi-separated? If not, what if we even assume that $\mathrm{Qcoh}(X)$ is a locally finitely presentable tensor category?
There are other natural conditions which might imply that $X$ is qc+qs, for example that even all cohomology functors $H^i : \mathrm{Qcoh}(X) \to \mathrm{Ab}$ preserve filtered colimits. Besides there are relative versions of this question: When $f : X \to Y$ is a morphism of schemes, then $f$ is quasi-compact and quasi-separated iff $f_\*$ or $f^\*$ has which property?
EDIT: Let us spell out what it means that $\Gamma$ preserves direct sums: Let $M_i$ be (infinitely many) quasi-coherent modules on $X$, $s_i \in \Gamma(M_i)$ global sections, and $X = \cup_j X_j$ an arbitrary open covering, then the following holds: If for every $j$ we have $s_i |_{X_j} = 0$ for almost all $i$, then $s_i = 0$ for almost all $i$. So the bounds for the $i$ with $s_i \neq 0$ on each $X_j$ are bounded. Why should this happen for a scheme which is not quasi-compact? But in order to prove this, we would have to construct appropriate quasi-coherent modules, and this can only be done my transfinite methods if $X$ is general (see for example here).
Best Answer
Let me show that if $\Gamma$ preserves filtered colimits, then $X$ is quasicompact. (At the moment I don't know about 'quasiseparated'; but, as Martin points out, I only use the injectivity of $\varinjlim\circ \Gamma \to \Gamma\circ \varinjlim$ for filtered inductive systems).
Assume $X$ is not quasicompact. Then there is a filtered decreasing family $(Y_i)$ of nonempty closed subschemes of $X$, with empty intersection. The structure sheaves $\mathcal{O}_{Y_i}$ form an inductive system with colimit zero, but the unit section of any $Y_{i_0}$ is an element of $\varinjlim_i \Gamma(\mathcal{O}_{Y_i})$ which is nonzero because each $Y_i$ is nonempty.