Recall that for any category $\mathcal C$, there is a unique finest topology, the canonical topology on $\mathcal C$ for which all representable functors are sheaves. I am interested in the example $\mathcal C=\mathsf{Sch}_S$, the category of schemes over some base scheme $S$. In particular, if $S=\operatorname{Spec}(k)$ and $H\subset G$ are algebraic groups over $k$, one defines $G/H$ by taking the quotient as fppf sheaves and then proving the quotient sheaf is representable by a $k$-scheme.

Question 1: If $G$ is an algebraic group defined over $k$, $H\subset G$ a closed $k$-subgroup, is the quotient $(G/H)_\mathrm{can}$ (quotient in the canonical topology) representable by a $k$-scheme? If so, is it the same $k$-scheme as $G/H$ (computed with the fppf topology)?

Question 2: Let $\tau$ be a subcanonical topology on $\mathsf{Sch}_k$ (main examples I'm interested in: etale and fppf topologies). Let $\mathscr F$ be a (representable) $\tau$-sheaf of abelian groups. Does $\operatorname{H}^i_\tau(k,\mathscr F)$ agree with $\operatorname{H}^i_\mathrm{can}(k,\mathscr F)$? If not in general, does it under some natural restrictions on $\mathscr F$ (for example, constructible)?

Question 3: (This one is much more ambitious, and only should only be addressed if the answer to the first two question is "yes".) Let $S$ be an integral noetherian scheme (even Dedekind if that helps). Do quotients (of algebraic groups) and sheaf cohomology (of nice classes of representable sheaves) computed in the canonical topology agree with quotients and cohomology computed with respect to the fppf (resp. etale) topology?

Any partial answers, or even pointers to references, are greatly appreciated.


Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.