Popescu's theorem: Difference between revisions
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For example, if ''A'' is a local [[G-ring]] (e.g., local [[excellent ring]]) and ''B'' its completion, then the map ''A'' →''B'' is regular by definition and the theorem applies. |
For example, if ''A'' is a local [[G-ring]] (e.g., local [[excellent ring]]) and ''B'' its completion, then the map ''A'' →''B'' is regular by definition and the theorem applies. |
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The usual proof of the [[Artin approximation theorem]] relies crucially on Popescu's theorem. Popescu's result was proved by an alternate method, and somewhat strengthened, by Spivakovsky.<ref>{{Cite journal|last=Spivakovsky|first=Mark|date=1999|title=A new proof of D. Popescu's theorem on smoothing of ring homomorphisms|url=https://www.ams.org/journals/jams/1999-12-02/S0894-0347-99-00294-5/|journal=[[Journal of the American Mathematical Society]]|language=en-US|volume=12|issue=2|pages=381–444|doi=10.1090/s0894-0347-99-00294-5|issn=0894-0347|via=}}</ref> |
The usual proof of the [[Artin approximation theorem]] relies crucially on Popescu's theorem. Popescu's result was proved by an alternate method, and somewhat strengthened, by Spivakovsky.<ref>{{Cite journal|last=Spivakovsky|first=Mark|date=1999|title=A new proof of D. Popescu's theorem on smoothing of ring homomorphisms|url=https://www.ams.org/journals/jams/1999-12-02/S0894-0347-99-00294-5/|journal=[[Journal of the American Mathematical Society]]|language=en-US|volume=12|issue=2|pages=381–444|doi=10.1090/s0894-0347-99-00294-5|issn=0894-0347|via=}}</ref><ref>{{Cite journal|last=Cisinski|first=Denis-Charles|last2=Déglise|first2=Frédéric|date=2009-12-10|title=Triangulated categories of mixed motives|url=http://arxiv.org/abs/0912.2110|journal=arXiv:0912.2110 [math]}}</ref> |
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== References == |
== References == |
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Revision as of 17:01, 19 February 2018
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In abstract algebra, Popescu's theorem, introduced by Dorin Popescu, states:[1]
- Let A be a noetherian ring and B a noetherian algebra over it. Then, the structure map A →B is a regular morphism if and only if B is a direct limit of smooth A-algebras.
For example, if A is a local G-ring (e.g., local excellent ring) and B its completion, then the map A →B is regular by definition and the theorem applies.
The usual proof of the Artin approximation theorem relies crucially on Popescu's theorem. Popescu's result was proved by an alternate method, and somewhat strengthened, by Spivakovsky.[2][3]
References
- ^ Conrad & De Jong, Theorem 1.3.
- ^ Spivakovsky, Mark (1999). "A new proof of D. Popescu's theorem on smoothing of ring homomorphisms". Journal of the American Mathematical Society. 12 (2): 381–444. doi:10.1090/s0894-0347-99-00294-5. ISSN 0894-0347.
- ^ Cisinski, Denis-Charles; Déglise, Frédéric (2009-12-10). "Triangulated categories of mixed motives". arXiv:0912.2110 [math].
- Conrad, Brian; De Jong, A. J. "Approximation of versal deformations" (PDF). Retrieved 1 December 2014.
External links
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