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 ψ : E → E ⊗ Ω X / S 1 {\displaystyle \psi :E\to E\otimes \Omega _{X/S}^{1}} defined by
ψ ( e ) ( D ) = ∇ D p ( e ) − ∇ D p ( e ) {\displaystyle \psi (e)(D)=\nabla _{D}^{p}(e)-\nabla _{D^{p}}(e)}
for any derivation D of O X {\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 O X {\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 Der X / S → End ( E ) {\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
References
Katz, N., "Nilpotent connections and the monodromy theorem", IHES Publ. Math. 39 (1970) 175–232. Ogus, A., "Higgs cohomology, p-curvature, and the Cartier isomorphism", Compositio Mathematica, 140.1 (Jan 2004): 145–164.
