Valuations dominating regular local rings and proximity relations
Let \(R\) be a (commutative) regular noetherian local ring of dimension \(n\geq 2\) and let \(V\) be a valuation ring of the quotient field of \(R\) such that \(V\) dominates \(R\). There is a sequence of regular noetherian local rings (of dimension \(n\)) associated with \((R,V)\), \((R_i): R_0\subset R_1\subset\dots\subset R_i\subset\dots\subset V\), such that \(R_i\) is the quadratic transform of \(R_{i-1}\) along \(V\), for \(i\geq 1\). The main purpose of this paper is to study the proximity relations for valuations centered in such regular local rings. One question is when \(V=\bigcup_{i=0}^\infty R_i\). If \(v\) is the valuation associated with \(V\) that is zero-dimensional, a related question is: what information about \(v\) is recoverable from the sequence \((R_i)\) and vice-versa. The case \(n=2\) being fairly settled, the authors introduce the proximity matrix and the multiplicity sequence (thus extending the classical definitions for the case \(n=2\)), associated with the sequence \((R_i)\). The authors describe precise relations between these invariants and look into their properties.
scientific article; zbMATH DE number 5129947