Let $R$ be a commutative ring and $\{M\_i\}\_{i\in I}$ be a family of $R$-modules. We know that there are functors $^\widetilde{\ }: R{\rm -Mod}\to {\rm Qco(Spec}R)$ and $\Gamma({\rm Spec}R, -): {\rm Qco}({\rm Spec}R) \to R{\rm -Mod}$ which give us an equivalence of the category of $R$-modules and quasi-coherent $R$-modules.


1) What is the structure of $\widetilde{\prod\_{i\in I}M\_i}$? 


2) Can we describe product objects in ${\rm Qco}({\rm Spec}R)$?



operatiing quasi-coherator functor on a product of modules
Let $P = \prod\_{i \in I}^{\mathsf{Mod}(\mathrm{Spec}(R))} \widetilde{M\_i}$ and $Q = \prod\_{i \in I}^{\mathsf{Qcoh}(\mathrm{Spec}(R))} \widetilde{M\_i}$. The superscript indicates in which category the product is taken.


Then we have $Q = \widetilde{\prod\_{i \in I} M\_i}$, since $M \mapsto \widetilde{M}$ is an equivalence of categories $\mathsf{Mod}(R) \simeq \mathsf{Qcoh}(\mathrm{Spec}(R))$.


By the universal property of $P$, there is a canonical morphism $Q \to P$ in $\mathsf{Mod}(\mathrm{Spec}(R))$. On $D(f)$ it is induced by the canonical homomorphism of $R\_f$-modules $(\prod\_{i \in I} M\_i)\_f \to \prod\_{i \in I} (M\_i)\_f$. It is easy to see that this homomorphism usually fails to be an isomorphism, even for $M\_i=R$. In fact, $P$ is not quasi-coherent.


By the way, if $X$ is an arbitrary scheme, then $\mathsf{Qcoh}(X)$ still has arbitrary products, but they are not so easy to describe. One has to take the [quasi-coherator](http://stacks.math.columbia.edu/tag/077K) of the product taken in $\mathsf{Mod}(X)$.


