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COHOMOLOGY OF WITT VECTORS

Let L/K be a Galois extension of p-adic fields with the Galois group G. Let OL( resp. OK) denotes the ring of integers of L (resp. K). Let W (OL) denote the ring of Witt vectors of OL. We know that W (OL) = lim ←−Wn(OL), where Wn(OL) denotes the ring of Witt vectors of length n. It is a Galois module. Hesselholt conjectured that the pro-abelian group {H(G,Wn(OL))}n∈N is zero. He observed that it is enough to prove the conjecture in the case L/K is totally wildly ramified and in particular for degree-p extensions. Now, for degree p wildly ramified extensions let s = s(L/K) be the ramification break in the ramification filtration of G. This is the largest integer such that G acts trivially on OL/P L , where PL is the maximal ideal of OL. He proved the conjecture for the class of extensions where s(L/K) ≥ eK/(p−1). We proved that the conjecture holds for all Galois extensions. Similarly one can expect the same vanishing result for the Galois extensions of local fields in equi-characteristic case i.e. when K/Fq((t)); q = p and characteristic of the residue fields k is also p. We proved that this is the case. This result can be viewed as analogue of Hilbert’s theorem-90. In this talk I will give a sketch of the proof.