Total p-differentials on schemes over Z/p2

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.