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. $p$ -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 $p$ .

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.

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.

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p-curvature

Definition

See also

References