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

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

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. A useful property is that the expression is ${\mathcal {O}}_{X}$-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 $\operatorname {Der} _{X/S}\to \operatorname {End} (E)$ 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.