One may define the higher direct image $R^if_*(E)$ of a quasi coherent sheaf $E$ on $X$, where $f:X \rightarrow Y$ is a morphism of schemes. How does this functor behave with respect to tensor products?
https://en.wikipedia.org/wiki/Direct_image_functor
Note: The direct image functor is right adjoint to the inverse image functor, which means that for any continuous $f : X → Y$ and sheaves $F , G$ on $X, Y$, there is a natural isomorphism:
$$Hom_{Sh(X)}(f^{−1}G , F) = Hom_{Sh(Y)}( G , f_∗F ).$$
Question: Let $X$ be a normal complex variety and $U=X \setminus X_{sing}$ its regular part. Let $\iota: U\to X$ be the canonical open immersion. let $F$ and $G$ be two coherent sheaves on $U$. Is there an isomorphism
$$\iota_*(F\otimes G)=\iota_*F \otimes \iota_*G?$$
Answer: If $A \rightarrow S^{-1}A$ is the localization at $S \subseteq A$, and if $M$ is an $S^{-1}A$-module it follows the canonical map $i:M \rightarrow S^{-1}M$ is an isomorphism: If $\frac{m}{s}\in S^{-1}M$ it follows the element $m':=\frac{1}{s}m \in M$ is a well defined element. It follows there is an equality in $S^{-1}M: \frac{m'}{1}=\frac{m}{s}.$
Hence the canonical map $M \cong S^{-1}M$ is an isomorphism. If $S:=\{f^n\}$ it follows for any two $A_f$-modules $E,F$ there is an isomorphism
$$E\otimes_{A_f} F \cong (E\otimes_A A_f) \otimes_{A_f} (A_f \otimes_A F)\cong E\otimes_A F.$$
Hence if $j: Spec(A_f):=U \rightarrow X:=Spec(A)$ is the inclusion, it follows there is an isomorphism
$$j_*(\tilde{E}\otimes_{\mathcal{O}_U} \tilde{F}) \cong j_*(\tilde{E})\otimes_{\mathcal{O}_X}j_*(\tilde{F}).$$
In general if $M,N$ are $S^{-1}A$-modules, there is a canonical isomorphism
$$M\otimes_A N \cong M\otimes_{S^{-1}A} N.$$
of $S^{-1}A$-modules.