Let $i : U \to X$ be a quasi-compact open immersion of schemes. Under which conditions is the natural map


$i\_\* M \otimes i\_\* N \to i\_\* (M \otimes N)$


for all $M,N \in \text{Qcoh}(U)$ an isomorphism? We may assume that $X=\text{Spec}(A)$ is affine. If $U$ is affine, then we may reduce to the case $M=N=\mathcal{O}\_U$ (using presentations) and use that $i^{\#}$ is a flat epimiorphism to get an affirmative answer. If there are counterexamples for general $U$, what conditions are sufficient?



When does the direct image functor commute with tensor products?
Here's a counterexample: over a field $k$, let $X=\mathbb{A}^4=\mathbb{A}^2\times \mathbb{A}^2$, and $U$ the complement of the origin. Put $Y=\mathbb{A}^2\times\{0\}$, $Z=\{0\}\times\mathbb{A}^2$, $Z'=Z\cap U$, $Y'=Y\cap U$. 


Take $M=\mathcal{O}\_{Y'}$ and $N=\mathcal{O}\_{Z'}$. Then $M\otimes N$ is zero, while $ i\_\* M=\mathcal{O}\_{Y}$ and $i\_\* N=\mathcal{O}\_{Z}$, so $i\_\* M\otimes i\_\* N$ is the structure sheaf of the origin.


