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Analytically unramified ring — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Analytically_unramified_ring
License (as archived): CC-BY-SA-4.0
Archived: 2026-11-09T00:52:37.746Z
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Analytically unramified ring — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Analytically_unramified_ring (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_2cd07e99ff37 (Metraipolis archive dc_2cd07e99ff37, sha256 ee1ef081ab1be3a57c9a5bc95fbc999c59357b896af3ceae05c8f4cef0ab4c60)

BibTeX

@misc{dc2cd07e99ff37,
  title = {Analytically unramified ring — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Analytically_unramified_ring},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_2cd07e99ff37; sha256 ee1ef081ab1be3a57c9a5bc95fbc999c59357b896af3ceae05c8f4cef0ab4c60},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Analytically_unramified_ring. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In algebra, an analytically unramified ring is a local ring whose completion is reduced (has no nonzero nilpotent). The following rings are analytically unramified: pseudo-geometric reduced ring. excellent reduced ring. Chevalley (1945) showed that every local ring of an algebraic variety is analytically unramified. Schmidt (1936) gave an example of an analytically ramified reduced local ring. Krull showed that every 1-dimensional normal Noetherian local ring is analytically unramified; more precisely he showed that a 1-dimensional normal Noetherian local domain is analytically unramified if and only if its integral closure is a finite module. This prompted Zariski (1948) to ask whether a local Noetherian domain such that its integral closure is a finite module is always analytically unramified. However Nagata (1955) gave an example of a 2-dimensional normal analytically ramified Noetherian local ring. Nagata also showed that a slightly stronger version of Zariski's question is correct: if the normalization of every finite extension of a given Noetherian local ring R is a finite module, then R is analytically unramified. There are two classical theorems of David Rees (1961) that characterize analytically unramified rings. The first says that a Noetherian local ring (R, m) is analytically unramified if and only if there are a m-primary ideal J and a sequence nj→∞ such that Jj‾⊂Jnj, where the bar means the integral closure of an ideal. The second says that a Noetherian local domain is analytically unramified if and only if, for every finitely-generated R-algebra S lying between R and the field of fractions K of R, the integral closure of S in K is a finitely generated module over S. The second follows from the first.

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Fonte: Wikipedia in inglese, voce «Analytically unramified ring» — https://en.wikipedia.org/wiki/Analytically_unramified_ring
Wikidata Q17097738 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q17097738
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Analytically_unramified_ring?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1299133418, consultata il 2026-11-09T00:49:04.041Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: ac8710147ebfbab68c43c0a4800c35bba07f58d6222d0747d2bb2acc63032906