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p-curvature

From Wikipedia, the free encyclopedia

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

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Suppose X/S is a smooth morphism of schemes of finite characteristic p > 0, E a vector bundle on X, and a connection on E. The p-curvature of is a map defined by

for any derivation D of 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 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

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References

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  • 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.