Local monomialization of transcendental extensions

Cutkosky, Steven Dale

Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1517-1586.

Soient $R\subset S$ deux anneaux locaux réguliers, essentiellement de type fini sur un corps $k$ de caractéristique zéro. Si $V$ est un anneau de valuation du corps des fractions $K$ de $S$ dominant $S$, nous montrons qu’il existe des suites de transformés monoidaux (éclatements d’idéaux premiers réguliers) $R\to R_1$ et $S\to S_1$ le long de $V$ tels que $R_1\to S_1$ est une application monomiale. Il s’ensuit qu’un morphisme de variétés non singulières peut-être rendu monomial le long d’une valuation après éclatement de sous-variétés non singulières.

Suppose that $R\subset S$ are regular local rings which are essentially of finite type over a field $k$ of characteristic zero. If $V$ is a valuation ring of the quotient field $K$ of $S$ which dominates $S$, then we show that there are sequences of monoidal transforms (blow ups of regular primes) $R\to R_1$ and $S\to S_1$ along $V$ such that $R_1\to S_1$ is a monomial mapping. It follows that a morphism of nonsingular varieties can be made to be a monomial mapping along a valuation, after blow ups of nonsingular subvarieties.

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