1
$\begingroup$

If we get a morphism $f : X \to Y$ of schemes over $k$, how should I define the direct image functor $$ f_* : Crys(X) \to Crys(Y)? $$ By a crystal I mean a quasi-coherent sheaf $M$ on $X$ with a collection of isomorphisms (a stratification) between the pullbacks $x^*M \simeq y^*M$ whenever $x$ and $y$ are morphisms from $Spec(R)$ for a commuting ring $R$ which give the same morphisms from $Spec(R/Nil(R))$. Also there are other conditions when $f : Spec(R) \to Spec(S)$ but I omit these.

So my question is how to define the supposed direct image functor, and any help with this would be very appreciated.

$\endgroup$

1 Answer 1

1
$\begingroup$

If $f$ is quasi-finite, you can follow chapter IV of Berthelot's thesis (Springer Lecture notes 407). Otherwise, you will need to enter the derived world, much like in the case of D-modules. Gaitsgory-Rozenblyum defines a crystalline pushforward by taking a derived pushforward of ind-coherent sheaves along the corresponding morphism of de Rham stacks.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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