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 ${\displaystyle \nabla }$ 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

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

for any derivation D of ${\displaystyle {\mathcal {O}}_{X}}$ over S. Here we use that the pth power of a derivation is still a derivation over schemes of characteristic p. A useful property is that the expression is ${\displaystyle {\mathcal {O}}_{X}}$-linear in e, in contrast to the Leibniz rule for connections. Moreover, the expression is p-linear in D.

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.

See also

• Grothendieck–Katz p-curvature conjecture
• Restricted Lie algebra