P-curvature

In algebraic geometry, p-curvature is an invariant of a connection on a coherent sheaf for schemes of characteristic p > 0. It is a construction similar to a usual curvature, but only exists in finite characteristic.

Suppose X/S is a smooth morphism of schemes of finite characteristic p > 0, E a vector bundle on X, and ${\displaystyle \nabla }$ a connection on E. The p-curvature of ${\displaystyle \nabla }$ is a map ${\displaystyle \psi :E\to E\otimes \Omega _{X/S}^{1}}$ defined by

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