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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?

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2 Answers 2

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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.

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  • $\begingroup$ You might want to change most of your $X$s to $Z$s. $\endgroup$
    – S. Carnahan
    Dec 12, 2010 at 17:33
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This is an addition to the Laurent's answer. First, it should be said that if one derives all the functors, one will get an isomorphim --- $Ri_*M \otimes^L Ri_*N \cong Ri_* (M \otimes^L N)$. Indeed, it is a simple corollary of the projection formula: $$ Ri_*M \otimes^L Ri_*N \cong Ri_*(M \otimes^L i^*Ri_*N) \cong Ri_*(M \otimes^L N) $$ (the second isomorphism is by the flat base change). What goes wrong with the underived version is that $M$ and $N$ have higher direct images which then have $Tor$'s all of which eventually get canceled. In the particular example of Laurent one has $$ R^1i_*O_{Y'} = y_1^{-1}y_2^{-1}k[y_1^{-1},y_2^{-1}], \quad R^1i_*O_{Z'} = z_1^{-1}z_2^{-1}k[z_1^{-1},z_2^{-1}], $$ where are $y_1,y_2$ are coordinates on $Y$ and $z_1,z_2$ are coordinates on $Z$. In addition to $O_Y\otimes O_Z = k$ we have $$ Tor_2(O_Y,R^1i_*O_{Z'}) = Tor_2(R^1i_*O_{Y'},O_Z) = k, $$ $$ Tor_4(R^1i_*O_{Y'},R^1i_*O_{Z'}) = k, $$ and it is easy to see that all this cancels in the spectral sequence calculating $Ri_*O_{Y'} \otimes^L Ri_*O_{Z'}$.

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