On the Milnor \(K\)-groups of complete discrete valuation fields
Suppose that \(K\) is a discrete valuation field of characteristic \(0\), with residue field \(F\). Assume that \(p = \text{char}(k)\) is odd, and that \(e=v_{K}(p)\) is prime to \(p\). For \(i > ep/(p-1)\) let \(n\) be the maximal integer such that \(i-ne \geq e/(p-1)\), and let \(s=v_{p}(i-ne)\) where \(v_{p}\) is the \(p\)-adic order. Then the main theorem of this paper (theorem 1.1) asserts that the \(i\)th associated graded piece \(\text{gr}^{i}K_{q}^{M}(K)\) of the Milnor \(K\)-group \(K_{q}^{M}(K)\) can be calculated in terms of the \(q\)th exterior power \(\Omega^{q}_{F}\) of the module \(\Omega^{1}_{F/{\mathbb Z}}\) of absolute differentials of \(F\) by the isomorphism \[ \text{gr}^{i}K_{q}^{M}(K) \cong \Omega_{F}^{q-1}/B^{q-1}_{s+n}. \] Here, \(\{ B_{i}^{q-1} \}\) is an ascending sequence of subgroups of \(\Omega_{F}^{q-1}\) which is inductively defined by the Cartier operator. The proof of the main theorem occupies most of the paper.
scientific article; zbMATH DE number 1446307