P curvature - Alchetron, The Free Social 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

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. p-curvature of ∇ is a map ψ : E → E ⊗ Ω X / S 1 defined by

ψ ( e ) ( D ) = ∇ D p ( e ) − ∇ 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 p.

By the definition p-curvature measures the failure of the map Der X / S → 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

P-curvature Wikipedia

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