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 $X$.



Base change and relative Ext over noncommutative rings
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 


<http://www.mi.ras.ru/~akuznet/publications/HyperplaneSectionsAndDerivedCategories.pdf>


See Appendix D.


