P-adic cohomology

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In mathematics, p-adic cohomology means a cohomology theory for varieties of characteristic p whose values are modules over a ring of p-adic integers. Examples (in roughly historical order) include:

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra. A module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring and a multiplication is defined between elements of the ring and elements of the module.

In mathematics, Witt vector cohmology was an early p-adic cohomology theory for algebraic varieties introduced by Serre (1958). Serre constructed it by defining a sheaf of truncated Witt rings Wn over a variety V and then taking the inverse limit of the sheaf cohomology groups Hi(V, Wn) of these sheaves. Serre observed that though it gives cohomology groups over a field of characteristic 0, it cannot be a Weil cohomology theory because the cohomology groups vanish when i > dim(V). For Abelian varieties Serre (1958b) showed that one could obtain a reasonable first cohomology group by taking the direct sum of the Witt vector cohomology and the Tate module of the Picard variety.

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 Paul Monsky and Gerard Washnitzer (1968) and Monsky (1968), who were motivated by the work of Bernard Dwork (1960). The idea is to lift the variety to characteristic 0, and then take a suitable subalgebra of the algebraic de Rham cohomology of Grothendieck (1966). The construction was simplified by van der Put (1986). Its extension to more general varieties is called rigid cohomology.

In mathematics, infinitesimal cohomology is a cohomology theory for algebraic varieties introduced by Grothendieck (1966). In characteristic 0 it is essentially the same as crystalline cohomology. In nonzero characteristic p Ogus (1975) showed that it is closely related to etale cohomology with mod p coefficients, a theory known to have undesirable properties.

See also

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. The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the notion of Hodge–Tate representation. Hodge–Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence the name p-adic Hodge theory. Further developments were inspired by properties of p-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts of the field.

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 conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.

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