Local Algebra

Let (R, m, k) be a local noetherian domain with field of fractions K and Rν a valuation ring, dominating R (not necessarily birationally). Let ν|K:K* ↠Γ be the restriction of ν to K by definition, ν|K is centred at R. Let R^denote 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 R^ and ν by a suitable extension ν^− to R^/P for a suitably chosen prime ideal P, such that P∩R=(0). The purpose of this paper is to give, assuming that R is excellent, a systematic description of all such extensions ν^− and to identify certain classes of extensions which are of particular interest for applications.

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“…Proof. By Lemma 4 of Appendix 4 of [17], the ideals Pβ are cofinal among the ideals P β for β ∈ Δ . Proof.…”

Section: Stable Rings and Primality Of Their Implicit Idealsmentioning

confidence: 99%

Extending a valuation centred in a local domain to the formal completion

Govantes

Acosta

Spivakovsky

et al. 2012

Proceedings of the London Mathematical Society

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“…If in addition the extension of v from K to L is unique, then the lemma of Ostrowski (see [22], Chapter VI, §12, Corollary to Theorem 25) says that…”

Section: Tame Extensions and Defect Extensionsmentioning

confidence: 99%

On maximal immediate extensions of valued fields

Blaszczok

Kuhlmann

2016

Mathematische Nachrichten

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We study the maximal immediate extensions of valued fields whose residue fields are perfect and whose value groups are divisible by the residue characteristic if it is positive. In the case where there is such an extension which has finite transcendence degree we derive strong properties of the field and the extension and show that the maximal immediate extension is unique up to isomorphism, although these fields need not be Kaplansky fields. If the maximal immediate extension is an algebraic extension, we show that it is equal to the perfect hull and the completion of the field.

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“…Proof. 7T e q \ q 2 , so R q = X[[TI]] for some subfield K of R q containing F (see [7], Chapter VIII, Theorem 27 and its Corollary). As neF[v], K [[n]] is a subring of LEMMA…”

Section: Suppose Vek and K Is Finite Over F(v) Let R Be The Integralmentioning

confidence: 99%

Congruence Topologies in Commutative Rings

Segal

1979

Bulletin of the London Mathematical Society

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Local Algebra

Cited by 107 publications

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Extending a valuation centred in a local domain to the formal completion

Extending a valuation centred in a local domain to the formal completion

On maximal immediate extensions of valued fields

Congruence Topologies in Commutative Rings

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