Witt vector cohomology

In mathematics, Witt vector cohomology 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 W_n over a variety V and then taking the inverse limit of the sheaf cohomology groups Hⁱ(V, W_n) 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.

References

• Serre, J.P. (1958), "Sur la topologie des variétés algébriques en caractéristique p", 1958 Symposium internacional de topología algebraica, Mexico City: Universidad Nacional Autónoma de México and UNESCO, pp. 24–53, MR 0098097
• Serre, Jean-Pierre (1958b), "Quelques propriétés des variétés abéliennes en caractéristique p", Amer. J. Math., 80: 715–739, doi:10.2307/2372780, MR 0098100