On the \(\pi\)-adic theory -- Galois cohomology
Let \(\pi\) be an irreducible element in the ring \(F[t]\) of polynomials over a finite field \(F\), let \(A\) be the ring of integers of the \(\pi\)- adic completion of the field \(F(t)\), and let \(q\) be the number of elements of the residue field of \(A\). Let \(K\) be a complete discrete valuation field which is an extension of \(F(t)\) and for which the residue field is perfect, let \(\overline{K}\) be the separable algebraic closure of \(K\) to which the valuation of \(K\) is extended, let \(M\) be the maximal ideal of the ring of integers of \(\overline{K}\), and let \(G\) be the Galois group of \(\overline{K}\) over \(K\). The ring \(A\) is isomorphic to the ring of formal power series in \(\pi\) over a field of \(q\) elements, and \(M\) becomes a module over \(A\) by setting \([\pi]m=\pi m+m^ q\) for \(m\) in \(M\). The Lubin-Tate module over \(A\) is the torsion submodule of \(M\); and the action of \(G\) on this submodule is given by \(\sigma(m)=[\chi(\sigma)]m\), where \(\chi\) is a homomorphism of \(G\) into the multiplicative group of units of \(R\). The group \(G\) acts continuously on the completion \(C\) of \(\overline{K}\); and for each integer \(r\), denote by \(C(r)\) the \(G\)-module obtained by setting \(\sigma\cdot y=\chi(\sigma)^ r \sigma(y)\) for \(y\) in \(C\). The author constructs an element \(c\) of \(C\) such that \(\sigma(c)=\chi(c)c\), and multiplication by \(c^ r\) is an isomorphism of \(C(r)\) onto \(C\). Then for Galois cohomology, \(H^ 0(G,C(r))=c^{-r}H^ 0 (G,C)\) and \(H^ k(G,C(r))=H^ k(G,C)\) for \(k\geq 1\). \textit{J. Ax} [J. Algebra 15, 417-428 (1971; Zbl 0216.047)] proved that \(H^ 0(G,C)\) is the completion of the purely inseparable closure of \(K\), and the author proves that \(H^ 1(G,C)=0\).
\(\pi\)-adic theory in positive characteristic
scientific article; zbMATH DE number 221218