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 \( {\displaystyle \psi :E\to E\otimes \Omega _{X/S}^{1}} \) defined by

\( {\displaystyle \psi (e)(D)=\nabla _{D}^{p}(e)-\nabla _{D^{p}}(e)} \)

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 \( {\displaystyle \operatorname {Der} _{X/S}\to \operatorname {End} (E)} \) 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.