mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Witt vector cohomology was an early ''p''-adic cohomology theory for

algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

introduced by . Serre constructed it by defining a

sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics) In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open s ...

of truncated Witt rings ''W''_''n'' over a variety ''V'' and then taking the

inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ca ...

of the

sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions (holes) to solving a geometric problem glob ...

groups ''H''^''i''(''V'', ''W_n'') of these sheaves. Serre observed that though it gives cohomology groups over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

of characteristic 0, it cannot be a Weil cohomology theory because the cohomology groups vanish when ''i'' > dim(''V''). For

Abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular f ...

, 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.

References

* *{{citation, mr=0098100 , last=Serre, first= Jean-Pierre , title=Quelques propriétés des variétés abéliennes en caractéristique p , journal=Amer. J. Math. , volume=80, year= 1958b , pages=715–739, doi=10.2307/2372780 Algebraic geometry Cohomology theories