Extension of a valuation.
(Extension d’une valuation.)

*(French)*Zbl 1121.13006The goal of this paper is to determine the extensions of a valuation \(\nu\) on \(K\), to a field \(L\supset K\), where \(L\) is either the quotient field of the polynomial ring \(K[x]\) or \(L=K[x]/(G)\), where \(G\) is an irreducible polynomial. This problem was yet considered by Saunders Mac Lane, assuming that \(\nu\) is a discrete valuation. The main point in the paper under review is that Vaquié considers any valuation, in this case appear some families of extended valuations \(S=(\mu_\alpha )_{\alpha \in A}\), indexed by a totally ordered set \(A\) which is not necessarily a countable set, and proves that any extended valuation can be obtained as a “limit” of an admissible family of some extended valuations lying on “simple” admissible families, which are non necessarily countable sets.

The notion of admissible family is a generalization of the notion of approximate roots of an irreducible polynomial \(f\) in two variables. In the case of an analytically irreducible curve there exists only one extension, the admissible family is finite and corresponds to the approximative roots of \(f\). He also proves that finding an admissible family for a valuation \(\mu \) of \(K[x]\) allows to find a system of generators of the graded algebra \(\text{ gr}_\mu K[x]\), which plays an important role in the local uniformization problem.

The notion of admissible family is a generalization of the notion of approximate roots of an irreducible polynomial \(f\) in two variables. In the case of an analytically irreducible curve there exists only one extension, the admissible family is finite and corresponds to the approximative roots of \(f\). He also proves that finding an admissible family for a valuation \(\mu \) of \(K[x]\) allows to find a system of generators of the graded algebra \(\text{ gr}_\mu K[x]\), which plays an important role in the local uniformization problem.

Reviewer: Marcel Morales (Saint-Martin-d’Heres)

##### MSC:

13A18 | Valuations and their generalizations for commutative rings |

12J10 | Valued fields |

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |

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\textit{M. Vaquié}, Trans. Am. Math. Soc. 359, No. 7, 3439--3481 (2007; Zbl 1121.13006)

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