Formal prime ideals of infinite value and their algebraic resolution

Cutkosky, Steven Dale; ElHitti, Samar

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 3-4, pp. 635-649.

Soit $R$ un domaine local essentiellement de type fini sur un corps de caractéristique $0$, et $\nu$ une valuation du corps de fractions de $R$ qui domine $R$. Le rang d’une telle valuation augmente souvent en prolongeant la valuation a une valuation dominante $\widehat{R}$ qui est la completion de $R$. Dans le cas où le rang de $\nu$ est égal a $1$, Cutkosky et Ghezzi manipule ce phénomène en résolvant l’ideal premier de valeur infinie. Dans le cas ou le rang est plus grand que 1, ils donnent un example qui montre qu’il n’y a aucun ideal naturel dans $\widehat{R}$ qui mene a cette obstruction. Nous généralisons leurs résultats de resolution des ideaux premiers de valeurs infinies aux valuations de rang arbitraire.

Suppose that $R$ is a local domain essentially of finite type over a field of characteristic $0$, and $\nu$ a valuation of the quotient field of $R$ which dominates $R$. The rank of such a valuation often increases upon extending the valuation to a valuation dominating $\widehat{R}$, the completion of $R$. When the rank of $\nu$ is $1$, Cutkosky and Ghezzi handle this phenomenon by resolving the prime ideal of infinite value, but give an example showing that when the rank is greater than $1$, there is no natural ideal in $\widehat{R}$ that leads to this obstruction. We extend their result on the resolution of prime ideals of infinite value to valuations of arbitrary rank.

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