In
algebraic geometry, -curvature is an invariant of a
connection on a
coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with ref ...
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 In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if
*(i) it is locally of finite presentation
*(ii) it is flat, and
*(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular.
(iii) mea ...
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 ''p''th 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 algebra In mathematics, a restricted Lie algebra is a Lie algebra together with an additional "''p'' operation."
Definition
Let ''L'' be a Lie algebra over a field ''k'' of characteristic ''p>0''. A ''p'' operation on ''L'' is a map X \mapsto X^ satisfyi ...
s, 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 In mathematics, a restricted Lie algebra is a Lie algebra together with an additional "''p'' operation."
Definition
Let ''L'' be a Lie algebra over a field ''k'' of characteristic ''p>0''. A ''p'' operation on ''L'' is a map X \mapsto X^ satisfyi ...
References
* Katz, N., "Nilpotent connections and the monodromy theorem", ''IHES Publ. Math.'' 39 (1970) 175–232.
* Ogus, A., "Higgs cohomology, {{mvar, p-curvature, and the Cartier isomorphism", ''Compositio Mathematica'', ''140.1'' (Jan 2004): 145–164.
*
*
Connection (mathematics)
Algebraic geometry