Total $p$-differentials on schemes over $Z/p^2$

TL;DR

This paper introduces total p-differentials on schemes over W_2(k), bridging Frobenius-twisted differentials and p-differentials, enabling new geometric structures and connections in characteristic p geometry.

Contribution

It defines total p-differentials on schemes over W_2(k), extending differential concepts and constructing analogues of Gauss-Manin connections and Kodaira-Spencer classes.

Findings

Total p-differentials interpolate between Frobenius-twisted and Buium's p-differentials.

Construction of sheaves of differentials over schemes in characteristic p.

Development of analogues of classical geometric connections in positive characteristic.

Abstract

For a scheme $X$ defined over the length $2$ $p$-typical Witt vectors $W_2(k)$ of a characteristic $p$ field, we introduce total $p$-differentials which interpolate between Frobenius-twisted differentials and Buium's $p$-differentials. They form a sheaf over the reduction $X_0$, and behave as if they were the sheaf of differentials of $X$ over a deeper base below $W_2(k)$. This allows us to construct the analogues of Gauss-Manin connections and Kodaira-Spencer classes as in the Katz-Oda formalism. We make connections to Frobenius lifts, Borger-Weiland's biring formalism, and Deligne--Illusie classes.