Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am reading this Questions about Serre duality, and there is one part in the answer that I'd like to know how it works. But after many tries I didn't get anywhere. So here is the problem.

Let $X$ and $B$ be algebraic varieties over an algebraically closed field, $\pi_1$ and $\pi_2$ be the projections from $X\times B$ onto $X$ and $B$, respectively. Then it was claimed that $R^q\pi_{2,*} \pi_1^* \Omega_X^p \cong H^q(X, \Omega^p_X)\otimes \mathcal{O}_B$.

I am guessing it works for any (quasi)coherent sheaf on $X$.

Basically, I have two tools available, either Proposition III8.1 of Hartsshorne or going through the definition of the derived functors.

Thank you.

share|improve this question
add comment

1 Answer

up vote 3 down vote accepted

Using flat base-change (Prop. III.9.3 of Hartshorne), one sees that $$R^1\pi_{2 *} \pi_1^*\Omega_X^p = \pi_1^* H^q(X,\Omega^p_X) = H^q(X,\Omega^p_X)\otimes \mathcal O_B.$$

share|improve this answer
    
How come I didn't think about that? Thanks, Matt. I learned a lot from you on Math.SE. –  Jiangwei Xue May 19 '11 at 17:23
    
@Jiangwei: Dear Jiangwei, You're welcome. Regards, –  Matt E May 20 '11 at 3:51
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.