Cohomology of the Weil group of a \(p\)-adic field
Let \(K\) be a finite extension of the \(p\)-adic field \({\mathbb Q}_p\) and let \(\overline{K}\) be an algebraic closure of \(K\). The Weil group \(W\) of \(K\) is the subgroup of \(G=\mathrm {Gal}(\overline{K}/K)\) consisting of those automorphisms which induce an integer power of Frobenius on residue fields. The topology of \(W\) is determined by the requirement that the inertia subgroup \(I\) of \(G\) (with the Krull topology) is an open subgroup of \(W\). This paper proves several theorems about Weil groups cohomology. For instance, it is shown that if \(M\) is a \(W\)-module which is finitely generated as an abelian group then \(H^3(W,M)=0\). In the case where \(M\) is a discrete group a comparison theorem is proved relating the Weil cohomology groups \(H^n(W,M)\) to the Galois cohomology groups \(H^n(G,M)\). In the case where \(A\) is a connected commutative algebraic group a similar theorem is proved relating \(H^n(W,A(\overline{L}))\) to \(H^n(G,A(\overline{K}))\), where \(L\) is the completion of the maximal unramified extension of \(K\). In both settings, the comparison theorem allows one to determine the Galois cohomology groups in terms of the Weil cohomology groups.
scientific article; zbMATH DE number 6138269
scientific article