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Let $Z \hookrightarrow X$ be a closed subvariety of a smooth projective variety. How do we compute $\mathcal{Tor}_i^{O_X}(O_{Z},O_{Z})$ $(i>0)$ as coherent sheaves on $Z$ where $Z$ is not of complete intersection inside $X$? Are there some good examples?

The completely intersection case is somehow easier using Koszul complex. For instance, $\mathcal{Tor}_1^{O_{\mathbb P^2}}(O_{\mathbb P^1},O_{\mathbb P^1}) \cong O_{\mathbb P^1}(-1)$.

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    $\begingroup$ An obvious remark: it is always true that $\mathcal{T}or_1^{\mathcal{O}_X}(\mathcal{O}_Z,\mathcal{O}_Z)\cong \mathcal{I}_Z/\mathcal{I}_Z^2$. $\endgroup$
    – abx
    Commented Nov 22, 2019 at 16:21
  • $\begingroup$ If $Z$ is a LOCALLY complete intersection, the answer is $\wedge^i(\mathcal{I}_Z/\mathcal{I}_Z^2)$. $\endgroup$
    – Sasha
    Commented Nov 22, 2019 at 18:23

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If $Z \hookrightarrow X$ is a locally complete intersection (for instance if $Z$ is also smooth), then we still have:

$$ Tor^k_{\mathcal{O}_X}(\mathcal{O}_Z,\mathcal{O}_Z) = \bigwedge^k \mathcal{I}_Z/\mathcal{I}_Z^2.$$

In the case $Z$ arbitrary, I guess things are pretty unpredictable as soon as $k \geq 2$ (as abx notes, the $Tor^1$ is nonetheless always equal to the conormal sheaf).

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