Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Given a smooth variety $X$ and two smooth subvarieties $Y_{1},Y_{2} \subset X$ with smooth intersection $Z=Y_{1} \cap Y_{2}$, I'd like to compute $\mathcal{Ext}^{i}_{X}(\mathcal{O}_{Y_{1}},\mathcal{O}_{Y_{2}})$. Something that I am reading seems to suggest that the answer should be $\Omega^{i}_Z$, or at least something in terms of the normal bundle of $Z$ in $X$. But I don't see this myself.

share|improve this question
4  
See Hartshorne, proof of Theorem III 7.11 - there he computes ${\scr E}xt^r(\mathcal{O}_{Y_1}, \omega_X)$ by applying $Hom(-, \Omega_X)$ to the Koszul complex of $\mathcal{O}_{Y_1}$. The Koszul complex exists only locally, but then he shows that these local calculations glue together. I think you can apply the same argument to $\mathcal{O}_{Y_2}$ in place of $\omega_X$. –  Piotr Achinger Feb 27 '12 at 18:26
add comment

1 Answer

If $c$ is the codimension of $Z$ in $Y_{2}$, then $$\mathcal{E}xt^{i}_{X}(\mathcal{O}_{Y_{1}},\mathcal{O}_{Y_{2}}) = \wedge^{c} N_{Z/Y_{2}}\otimes \wedge^{i -c} \left( N_{Z/X}/N_{Z/Y_{2}}\right).$$ You can see this by Hom-ing one Koszul complex into another. You can find the calculation e.g. in Appendix A.3 of this paper of Katz and Sharpe.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.