In mathematics, p-adic cohomology means a
cohomology theory
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
for varieties of characteristic ''p'' whose values are
modules
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computer science and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
...
over a ring of ''p''-adic integers. Examples (in roughly historical order) include:
* Serre's
Witt vector cohomology In mathematics, Witt vector cohomology was an early ''p''-adic cohomology theory for algebraic varieties introduced by . Serre constructed it by defining a sheaf of truncated Witt rings ''W'n'' over a variety ''V'' and then taking the inverse l ...
*
Monsky–Washnitzer cohomology In algebraic geometry, Monsky–Washnitzer cohomology is a ''p''-adic cohomology theory defined for non-singular affine varieties over fields of positive characteristic ''p'' introduced by , who were motivated by the work of . The idea is to lift ...
*
Infinitesimal cohomology In mathematics, infinitesimal cohomology is a cohomology theory for algebraic varieties introduced by . In characteristic 0 it is essentially the same as crystalline cohomology
In mathematics, crystalline cohomology is a Weil cohomology theory for ...
*
Crystalline cohomology
In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ''X'' over a base field ''k''. Its values ''H'n''(''X''/''W'') are modules over the ring ''W'' of Witt vectors over ''k''. It was introduced by and developed by ...
*
Rigid cohomology In mathematics, rigid cohomology is a ''p''-adic cohomology theory introduced by . It extends crystalline cohomology to schemes that need not be proper or smooth, and extends Monsky–Washnitzer cohomology to non-affine varieties. For a scheme '' ...
See also
*
p-adic Hodge theory
In mathematics, ''p''-adic Hodge theory is a theory that provides a way to classify and study ''p''-adic Galois representations of characteristic 0 local fields with residual characteristic ''p'' (such as Q''p''). The theory has its beginnings i ...
*
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu ...
, taking values over a ring of ''l''-adic integers for ''l''≠''p''
Arithmetic geometry
Cohomology theories
{{SIA, mathematics