MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given two smooth projective schemes $X$ and $Y$ over some algebraically closed field $k$, we have $X\times Y$ with the projections $p$ to $X$ and $q$ to $Y$. Furthermore we have a "nice" sheaf of algebras $R$ on $X$, i.e. locally free and of global dimension at most dim($X$), e.g. Azumaya or something similar. Given two $p^{\*}R$-modules $M$ and $N$ on $X\times Y$, which are coherent and torsion free.

Like in the commutative case, i define the i-th relative $\mathcal{E}xt$-sheaf on $Y$ to be: $\mathcal{E}xt^i_{p^{\*}R,q}(M,N):=(R^i(q_{\*}\mathcal{H}om_{p^{\*}R}(M,-))(N)$

Can I expect them to have the same properties as in the commutative case?

For example:

(1) Do we have $\mathcal{E}xt^i_{p^{\*}R,q}(M,N)=0$ for $i>dim(X)$?

(2) Given $y\in Y$ is there a map $\mathcal{E}xt^i_{p^{\*}R,q}(M,N)\otimes k(y) \rightarrow Ext_R^i(M_y,N_y)$

(3) If $Ext_R^i(M_y,N_y)=0$ for all $y\in Y$ does this imply $\mathcal{E}xt^i_{p^{\*}R,q}(M,N)=0$?

(4) Is there a kind of base change theorem for the $\mathcal{E}xt^i_{p^{\*}R,q}(M,N)$?

Or do I need more conditions for $M$ and $N$ to have the desired properties? I'm especially interested in the case, where $M=p^{\*}P$ for some $R$-module $P$ on $X$.

share|cite|improve this question
Can we presume that R is an O_X algebra; ie, an algebra object in the category of quasi-coherent sheaves? Note that this fails for some sheaves of algebras that still have nice localization properties, like differential operators. – Greg Muller Nov 9 '10 at 21:10
Yes, $R$ is supposed to be an $O_X$-algebra. I'm mainly interested in the cases where $R$ is Azumaya, i.e. etale locally just $M_n(O_X)$ or a maximal order of suitable global dimension. – TonyS Nov 9 '10 at 21:39
up vote 1 down vote accepted

Questions of this type are discussed in the paper A.Kuznetsov, Hyperplane sections and derived categories, Izvestiya: Mathematics 70:3 (2006) p. 447-547, which is available at

See Appendix D.

share|cite|improve this answer
Thanks, that looks promising. Maybe there are some questions left after reading the appendix. – TonyS Nov 10 '10 at 18:00

Your Answer


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.