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Suppose I have a complex variety $X$ and two closed subvarieties $A$ and $B$, with closed immersions $i:A\to X$ and $j:B\to X$. Then we have two $\mathcal O_X$-modules $i_*\mathcal O_A$ and $j_*\mathcal O_B$, and we can take their tensor product $i_*\mathcal O_A\otimes j_*\mathcal O_B$. On the other hand we can consider the intersection $A\cap B$, say with inclusion $\iota:A\cap B\to X$, and we have $\iota_*\mathcal O_{A\cap B}$.

I understand that in general we do not have $i_*\mathcal O_A\otimes j_*\mathcal O_B\cong\iota_*\mathcal O_{A\cap B}$, but there are some cases where this holds, e.g. if $A$ and $B$ are disjoint, then both of these sheaves are trivial.

My question is: what are the general conditions under which $i_*\mathcal O_A\otimes j_*\mathcal O_B\cong\iota_*\mathcal O_{A\cap B}$? Is it a kind of transversality?

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  • $\begingroup$ $A\cap B$ is only a closed subset, you have to specify the scheme structure on it. $\endgroup$ – Cantlog May 22 '14 at 21:15
  • $\begingroup$ I'm thinking of these as varieties, so I have in mind the reduced subscheme structure. $\endgroup$ – bradhd May 23 '14 at 15:11
  • $\begingroup$ Then I don't think there is a general condition insuring the isomorphism. Locally, you are asking when is a sum of radical ideals $I+J$ is radical... A sufficient condition is that the subvarieties are smooth and meet transversally. $\endgroup$ – Cantlog May 23 '14 at 22:19

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