On the graded algebra relative to a valuation
Let \(v\) be a valuation of a field \(F\) which is the quotient field of a noetherian local domain \(R\) centered at \(R\) having residue field \(K\). It is shown that the value semigroup \(S\) of \(v\) is a CF (= combinatorial finite) semigroup. It is also shown how the semigroup theoretical properties of \(S\) allow to describe a minimal system of homogeneous generators of the \(S\)-graded \(K\)-algebra \(\text{gr}_vR\) relative to \(v\) and to obtain an \(S\)-graded minimal resolution of \(\text{gr}_vR\) as \(A[v]\)-module, \(A[v]\) being a certain associated polynomial ring. The paper generalizes some results proved earlier for semigroup \(K\)-algebras [see, e.g., \textit{A. Campillo} and \textit{C. Marijuan} in: Sémin. Théor. Nombres Bordx., Sér. II 3, No. 2, 249-260 (1991; Zbl 0818.20078)].
scientific article; zbMATH DE number 1019707
scientific article