P-curvature
In algebraic geometry, -curvature is an invariant of a connection on a coherent sheaf for schemes of characteristic. 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, E a vector bundle on X, and a connection on E. The -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.
By the definition -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.
