p-curvature

In algebraic geometry, Template:Mvar-curvature is an invariant of a connection on a coherent sheaf for schemes of characteristic Template:Math. 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 Template:Math, E a vector bundle on X, and $\mathrm{\nabla }$ a connection on E. The Template:Mvar-curvature of $\mathrm{\nabla }$ is a map $\psi :E\to E\otimes {\mathrm{\Omega }}_{X/S}^1$ defined by

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

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

By the definition Template:Mvar-curvature measures the failure of the map ${\mathrm{Der}}_{X/S}\to \mathrm{End}(E)$ 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, Template:Mvar-curvature, and the Cartier isomorphism", Compositio Mathematica, 140.1 (Jan 2004): 145–164.