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!
 A: The answer by "user10000100_u", marked correct, is false.
Pavel Coupek's remark is true. Perhaps to clarify where the confusion lies:
"-The category of quasi-coherent sheaves is not abelian in general, infinite direct products of quasi-coherent sheaves are not quasi-coherent in general."
The category of quasi-coherent sheaves has all limits, so these direct products exist and ARE quasi-coherent. However, one can also take the direct product as O_X-module sheaves, and this product yields a different object; in particular usually an O_X-module sheaf which is not quasi-coherent.
A: 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.  
A: 1) There is  no reason to believe that $ f_* \mathcal{F}$ is coherent whenever $X$ is affine : Hartshorne certainly does not claim that, contrary to your typo [now corrected !].  
2) The reason that some hypotheses must be made on $f$ is that the result may be false without them!
There exists an example due to Altman-Hoobler-Kleiman, page 36 where indeed for some  morphism  $f:X \rightarrow Y$ and some quasi-coherent sheaf $\mathcal F$ on $X$ the image sheaf $ f_* \mathcal{F}$ is not quasi-coherent.
Such examples are  non trivial: for example  Dieudonné-Grothendieck claim to give one  (in the new edition of EGA, page 314) but their description is incorrect.
Edit: A simple counterexample
Here is an example of a non quasi-coherent image of a  quasi-coherent sheaf,  simpler than the one in Altman-Hoobler-Kleiman :    
a) Let $X_i\; (i\in \mathbb N)$ and $S$ be copies of $\operatorname {Spec}(\mathbb  Z)$ and let $X=\coprod X_i$ be the disjoint union of the $X_i$'s equipped with its natural morphism $f=\coprod_i id_i:X\to S$, the one restricting for every $i$ to the identity $id_i:X_i=\operatorname {Spec}(\mathbb  Z)\to S=\operatorname {Spec}(\mathbb  Z)$.
The counterexample will simply be $\mathcal F= \mathcal O_X$: I will now show that $f_* \mathcal O_X$ is a non quasi-coherent sheaf on $S$.  
b) Since for a quasi coherent sheaf $\mathcal G$ on $S$ the canonical morphism $$m_U:\mathcal G(S) \otimes _{\mathcal O_S(S)}\mathcal O_X(U)\to \mathcal G(U)$$ must be bijective  for all affine $U\subset S$, it suffices to show that this property is violated for  $\mathcal G =f_*(\mathcal O_X)     $ and $U= D(2)=\operatorname {Spec}(\mathbb  Z)\setminus \{(2)\}\subset S=\operatorname {Spec}(\mathbb  Z) $.  
c) In our situation we get the canonical morphism $m_U:(\prod_{i\in \mathbb N} \mathbb  Z) \otimes _{\mathbb Z} \mathbb Z[{\frac 12}]\to \prod_{i\in \mathbb N}\mathbb Z[{\frac 12}]$.
This morphism is not surjective because the image  $m_U(t) $ of an element  $t\in (\prod_{i\in \mathbb N} \mathbb  Z) \otimes _{\mathbb Z} \mathbb Z[{\frac 12}]$  is  a sequence  of rational numbers  $(\frac {a_i}{2^M})_i\in\prod_{i\in \mathbb N}\mathbb Z[{\frac 12}]$ with some fixed denominator $2^M$.
Whereas in $\prod_{i\in \mathbb N}\mathbb Z[{\frac 12}]$ there exists sequences  $(\frac {b_i}{2^{s_i}})_i $ of rational numbers with $b_i$ odd and  denominators $2^{s_i}$ tending to infinity. 
Conclusion:  $f_* \mathcal O_X$ is a non quasi-coherent sheaf on $S=\operatorname {Spec}(\mathbb  Z)$ .
A: 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.
