TITLE: When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof
QUESTION [27 upvotes]: 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!

REPLY [4 votes]: To expand on Pavel's comment, the category of quasicoherent sheaves is always abelian (in fact, by Gabber's result, it is always Grothendieck).  Moreover, the pullback functor $f^\ast:\operatorname{QCoh}(Y)\to \operatorname{QCoh}(X)$ for a map of schemes $f:X\to Y$ always admits a right adjoint $f_\sharp:\operatorname{QCoh}(X)\to \operatorname{QCoh}(Y).$ 
This follows from the following observations: 

The category of quasicoherent sheaves on a scheme is closed under colimits in the category of $\mathcal{O}_X$-modules. 
Gabber's result: Gabber's result implies that $\operatorname{QCoh}(X)$ satisfies the hypotheses of the special adjoint functor theorem, namely that $\operatorname{QCoh}(X)$ is locally presentable for any scheme $X$.  
The special adjoint functor theorem:  Since $\operatorname{QCoh}(Y)$ is closed under colimits in $\mathcal{O}_Y$-modules, the pullback functor $f^\ast$ preserves colimits of $\mathcal{O}_Y$-modules and sends quasicoherent sheaves to quasicoherent sheaves, and both  $\operatorname{QCoh}(X)$ and $\operatorname{QCoh}(Y)$ are locally presentable, we see that $f^\ast$ admits a right adjoint by the special adjoint functor theorem.

It is not true in general, however, that this right adjoint functor $f_\sharp$ agrees with the restriction of $f_\ast$ to $\operatorname{QCoh}(X)$.  In general, it can be constructed using the 'associated Quasicoherent sheaf' functor.