When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof In the following we have $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ a quasi-coherent sheaf on $X$ and I am referring to proposition 5.8 page 115 in Hartshorne.
To prove that the pushforward of a quasi-coherent sheaf is quasi-coherent, there are the following assumptions:
either $X$ is noetherian, or $f$ is quasi-compact and separated.
Both these assumptions lead to some finiteness of affine covers on $X$ where we can write $\mathcal{F}$ as $\widetilde{M}$.
My question is: Why do we need these assumptions? In the proof, after reducing to $Y$ affine, the argument leads to the exact sequence
\begin{equation}
0 \rightarrow f_* \mathcal{F} \rightarrow \bigoplus_i f_* \left(\mathcal{F}|_{U_i}\right) \rightarrow \bigoplus_{i,j,k} f_* \left(\mathcal{F}|_{U_{i,j,k}}\right),
\end{equation}
where the $U_i$'s are affines in $X$ and the $U_{i,j,k}$'s are affines covering the intersection $U_i \cap U_j$. The assumption (at least it's what I understood) are needed to make this family of indexes finite. But wouldn't the proof work even in the infinite case?
Of course I have to be mistaken somewhere, but I cannot see where and why!

The category of quasi-coherent sheaves is not abelian in general, infinite
direct products of quasi-coherent sheaves are not quasi-coherent in general. That's why the fact that the family of indexes is finite is crucial.
Remark for general schemes : see "Modules vs. quasi-coherent modules" of Thomason & Trobaugh in "Higher Algebraic K-theory of Schemes and Derived Categories, The Grothendieck Festschrift, Vol. III", Thomason wrote that it was as this time a priori unknown if the category of quasi-coherent sheaves over a general scheme has all limits. Since the existence of arbitrary products would imply the existence of limits...
What is the current situation on this ? Answer : as of last version of the Stacks Project, it is still unknown.
