在不改变值群的情况下将局部域的赋值扩展到完备局部域
Extending valuations of local domains to complete local domains without changing the value group
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中文总结 AI 辅助
研究在不改变值群的情况下，将局部域的赋值扩展到完备局部域的问题，通过给出泰西耶猜想的证明，为正特征下局部一致化的两种方法提供重要步骤。
中文摘要 AI 辅助
设\((R,m,k)\)是一个优秀的局部诺特域，其分式域为\(K\)。设\(\nu:K^*\twoheadrightarrow\Gamma\)是一个以\(R\)为中心的赋值，\(R_\nu\)是\(K\)对应的赋值环，支配\(R\)。用\(\widehat R\)表示\(R\)的\(m\)进完备化。在赋值理论应用于交换代数和奇点研究中，常需将\(R\)替换为其\(m\)进完备化\(\widehat R\)，并将\(\nu\)替换为到\(\frac{\widehat R}P\)的合适扩展\(\widehat\nu_-\)，其中\(P\)是合适选取的素理想且\(P\cap R=(0)\)。之前文章给出了所有此类扩展\(\widehat\nu_-\)的系统描述并定义了紧密扩展概念。若\(\widehat\nu_-\)是紧密扩展，则其分次代数与\(\nu\)的分次代数双有理（反之未知且可能不成立），特别地，\(\widehat\nu_-\)的值群是\(\Gamma\)。本文给出了泰西耶猜想的证明，该结果的一个预期应用是正特征下局部一致化两种近期方法中的重要一步。
英文摘要
Let $(R,m,k)$ be an excellent local noetherian domain with field of fractions $K$. Let $$ ν:K^*\twoheadrightarrowΓ$$ be a valuation centered at $R$ and let $R_ν$ be the corresponding valuation ring of $K$, dominating $R$. Denote by $\widehat R$ the $m$-adic completion of $R$. In the applications of valuation theory to commutative algebra and the study of singularities, one is often induced to replace $R$ by its $m$-adic completion $\widehat R$ and $ν$ by a suitable extension $\widehatν_-$ to $\frac{\widehat R}P$ for a suitably chosen prime ideal $P$, such that $$ P\cap R=(0). $$ In a previous article we gave a systematic description of all such extensions $\widehatν_-$ and defined the notion of tight extensions that are of particular interest for applications (see Herrera, Olalla, Spivakovsky and Teissier, Extending a valuation centered in a local domain to its formal completion, Proc. London Math. Soc. (3) 105 (2012) 571--621). If $\widehatν_-$ is a tight extension then its graded algebra is birational to that of $ν$ (the converse is not known and might not be true). In particular, the value group of $\widehatν_-$ is $Γ$. The existence of tight extensions was conjectured by the last author (see Teissier, Valuations, deformations, and toric geometry, Fields Institute Communications, 33, 2003, 361-459). In the present paper we give a proof of Teissier's conjecture. An intended application of this result is an important step in two recent approaches to local uniformization in positive characteristic.
