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 [math]\displaystyle{ \nabla }[/math] a connection on E. The p-curvature of [math]\displaystyle{ \nabla }[/math] is a map [math]\displaystyle{ \psi: E \to E\otimes \Omega^1_{X/S} }[/math] defined by

[math]\displaystyle{ \psi(e)(D) = \nabla^p_D(e) - \nabla_{D^p}(e) }[/math]

for any derivation D of [math]\displaystyle{ \mathcal{O}_X }[/math] 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 [math]\displaystyle{ \operatorname{Der}_{X/S} \to \operatorname{End}(E) }[/math] 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.