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.

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

for any derivation D of $$\mathcal{O}_X$$ over S. Here we use that the pth power of a derivation is still a derivation over schemes of characteristic $p$.

By the definition $p$-curvature measures the failure of the map $$\operatorname{Der}_{X/S} \to \operatorname{End}(E)$$ to be a homomorphism of restricted Lie algebras, just like the usual curvature in differential geometry measures how far this map is from being a homomorphism of Lie algebras.