# Computation of $\mathcal{Tor}_i^{O_X}(O_{Z},O_{Z})$ for non-complete intersection $Z$ in $X$

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, $$\mathcal{Tor}_1^{O_{\mathbb P^2}}(O_{\mathbb P^1},O_{\mathbb P^1}) \cong O_{\mathbb P^1}(-1)$$.

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

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).