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For a scheme $X$ defined over the length $2$ $p$-typical Witt vectors$W_2(k)$ of a characteristic $p$ field, we introduce total $p$-differentialswhich interpolate between Frobenius-twisted differentials and Buium's$p$-differentials. They form a sheaf over the reduction $X_0$, and behave as ifthey 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 andKodaira-Spencer classes as in the Katz-Oda formalism. We make connections toFrobenius lifts, Borger-Weiland's biring formalism, and Deligne--Illusieclasses.

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