Serre decided to try taking coefficients in the Witt vectors as an early attempt at a Weil cohomology theory. Ultimately, it wasn’t successful for this purpose, but has been generalized in several ways for other purposes with great success.
Let be a scheme over a perfect field of positive characteristic . Let and be the functors of Witt vectors and truncated Witt vectors respectively. The functorial nature allows us to define a sheaf of Witt vectors and just by taking Witt vectors of the rings of sections of .
Note that as a sheaf of sets is just . The ring structure is just the addition and multiplication of the Witt vectors. The operations on the Witt vectors sheafify as well. When we have the exact sequence . If we take , then we get the sequence
The sheaf of Witt vectors is an abelian sheaf, so we just define cohomology as the standard sheaf cohomology (on the Zariski site of ). Let , then since are -modules annihilated by , we get that are also -modules annihilated by .
In fact, all of our old operators , , and still act on . They are easily seen to satisfy the formulas , , and for . If is projective then is a finite -module.
What is usually referred to with Witt cohomology is which is defined to be . Note that even if is projective, this limit does not have to be a finite type -module.