Let $X$ be a scheme with structure sheaf $\mathcal O_X$ and let $\mathcal B$ be a quasi-coherent $\mathcal O_X$-algebra. Let $\mathcal F$ be any sheaf of $\mathcal B$-modules.
In EGA I, § 9, Prop. 9.6.1 it is asserted that then the following holds:
Then $\mathcal F$ is quasi-coherent over $\mathcal B$ if and only if $\mathcal F$ is quasi-coherent over $\mathcal O_ X$.
The proof of the implication $\Rightarrow$ is easy, but the other direction is hard to digest for me: EGA argues as follows: we may work locally on $X$, hence assume $X=Spec(A)$, whence $\mathcal B$ corresponds to a $A$-algebra $B$ and $\mathcal F$ corresponds to a $B$-module $M$ (namely, because $\mathcal B$ and $\mathcal F$ are quasi-coherent over $\mathcal O_X$).
Then it reads without explanation that $M$ is isomorphic to the cokernel of a map
$(*) \quad B^{(I)} \rightarrow B^{(J)}$,
and hence by taking the tilde-construction we see that $\mathcal F$ is quasi-coherent over $\mathcal B$.
Apriori, one should only be able to write $M$ as a cokernel of a map $(*)$ with $B$ replaced by $A$, but tensoring with $B$ doesn't seem to give $M$ as cokernel, but $M\otimes_A B$.
My question is why one may write $M$ as a cokernel of a map $(*)$, as claimed in EGA.
Or any other proof of the desired implication
"$\mathcal F$ quasi-coherent over $\mathcal O_X \Rightarrow \mathcal F$ quasi-coherent over $\mathcal B$"
would be very appreciated.
Basically Zhen already answered the question, but let me add this as an answer so that it won't bump up again (and to see if this really answered the question):
By assumption $M$ is a $B$-module, hence any presentation with generators and relations gives an exact sequence $B^{(I)} \to B^{(J)} \to M \to 0$ of $B$-modules. The rest of the EGA proof is quite self-explanatory; if not you can ask further questions.