Three facts of valuation theory
The author gives a sketch of results and their proofs that are to appear in the Obergurgl conference proceedings, 1997. For a field \(k\), a local integral \(k\)-algebra \((R,m)\) with a field of fractions \(K\), this is a study of \(k\)-valuations \(v\) of \(K\) with center in \(R\) (their valuation ring \((R_v, m_v)\) contains \(R\) and \(m_v\cap k=(0)\)). The results concern the following: (1) The Spivakovsky graded algebra is isomorphic to a certain quotient of a polynomial ring in possibly countably many indeterminates. Thus it is a deformation of a (non-normal) toric variety, possibly of infinite embedding dimension. (2) The graded \(k\)\(_v\)-algebra, \(\text{gr}_vR_v\), is a filtering direct limit of termic maps between polynomial subalgebras in rank \(r(\Phi)\) variables (\(\Phi\) is the totally ordered group of the valuation). (3) Let \(R\) be an analytically irreducible Noetherian local ring, \(v_1\) the valuation of height one with which \(v\) is composed, \(p=m_{v_1}\cap R\), the center of \(v_1\) on \(R\); then the completion \(\widetilde R^v\) with respect to the topology defined by the ideal filtration \(P_i\), \(i\in\Phi_+\), is isomorphic (as topological rings) to a quotient of the \(p\)-adic completion \(\widetilde R^p\) of \(R\). Moreover, it is Noetherian and in particular, if \(R\) is excellent, so is \(\widetilde R^v\).
scientific article; zbMATH DE number 1268559