Excellent Rings, Henselian Rings, and the Approximation Property

Rotthaus, Christel

doi:10.1216/rmjm/1181071964

Cited by 19 publications

(11 citation statements)

References 21 publications

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“…Then we claim that A may, under certain circumstances, be quasi-excellent, even though it cannot be excellent. To show this, we first present the following definitions, adapted from[9]:Definition 2.12. A local ring A is quasi-excellent if, for all P ∈ Spec A, the ring A ⊗ A L is regular for every purely inseparable finite field extension L of k(P ) = A P /P A P .…”

mentioning

confidence: 99%

Characterization of completions of noncatenary local domains and noncatenary local UFDs

Avery¹,

Booms²,

Kostolansky³

et al. 2019

Journal of Algebra

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We find necessary and sufficient conditions for a complete local ring to be the completion of a noncatenary local (Noetherian) domain, as well as necessary and sufficient conditions for it to be the completion of a noncatenary local (Noetherian) unique factorization domain. We use our first result to demonstrate a large class of quasi-excellent domains that are not excellent, as well as a large class of catenary domains that are not universally catenary. We use our second result to find a larger class of noncatenary local UFDs than was previously known, and we show that there is no bound on how noncatenary a UFD can be.

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confidence: 99%

Characterization of completions of noncatenary local domains and noncatenary local UFDs

Avery¹,

Booms²,

Kostolansky³

et al. 2019

Journal of Algebra

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“…If information regarding terminology or notation beyond what is provided here is required, either the text of Matsumura [11] or Eisenbud [3] will serve as an adequate reference in most cases. For the properties of excellent rings though, Grothendieck [7] and especially the recent paper by Rotthaus [12] are more thorough sources.…”

Section: Resultsmentioning

confidence: 99%

“…Letting R denote the completion of R with respect to I, it follows that I is generated by a regular sequence if and only if I R is, for R −→ R is faithfully flat. Thus, it suffices to show that the hypotheses of Theorem 5 apply to I R and R. The hypotheses on R guarantee that, in particular, R is a catenary local domain (see [12,Theorem 1.11]), and it is clear that condition (i) of * holds since R/I R = R/I is S 2 and reduced. Second, I R/ I R 2 = I R/I 2 R = I/I 2 ⊗ R is free over R/I R = R/I ⊗ R, so condition (iii) is also satisfied.…”

Section: Proofmentioning

confidence: 99%

A variation on a theme of Vasconcelos

Smith

2000

Journal of Algebra

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A classical result of W. Vasconcelos (1967, J. Algebra 6, 309-316) and D. Ferrand (1967, C. R. Acad. Sci. Paris 264, 427-428) provides necessary and sufficient conditions for an ideal to be generated by a regular sequence. In this paper, we provide generally more accessible conditions guaranteeing this property, as well as some examples indicating that our hypotheses cannot be significantly weakened. The key step in our development is the use of lifting, as presented by M. Auslander, et al. (1993, J. Algebra 156, 273-317).

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“…Define, for any P ∈ Spec A, k(P ) = A P /P A P . As noted in [5], we can consider only the purely inseparable finite field extensions L of k(P ). The following is a consequence of Theorem 31.6 and the definition of formally equidimensional in [4, pp.…”

Section: Preliminariesmentioning

confidence: 99%