Popescu's theorem
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In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu,[1][2] states:[3]
- Let A be a Noetherian ring and B a Noetherian algebra over it. Then, the structure map A → B is a regular homomorphism if and only if B is a direct limit of smooth A-algebras.
For example, if A is a local G-ring (e.g., a local excellent ring) and B its completion, then the map A → B is regular by definition and the theorem applies.
Another proof of Popescu's theorem was given by Tetsushi Ogoma,[4] while an exposition of the result was provided by Richard Swan.[5]
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 Mark Spivakovsky.[6][7]
See also
References
- ↑ Popescu, Dorin (1985). "General Néron desingularization". Nagoya Mathematical Journal 100: 97–126. doi:10.1017/S0027763000000246.
- ↑ Popescu, Dorin (1986). "General Néron desingularization and approximation". Nagoya Mathematical Journal 104: 85–115. doi:10.1017/S0027763000022698.
- ↑ Conrad, Brian; de Jong, Aise Johan (2002). "Approximation of versal deformations". Journal of Algebra 255 (2): 489–515. doi:10.1016/S0021-8693(02)00144-8. https://math.stanford.edu/~conrad/papers/approx.pdf., Theorem 1.3.
- ↑ Ogoma, Tetsushi (1994). "General Néron desingularization based on the idea of Popescu". Journal of Algebra 167 (1): 57–84. doi:10.1006/jabr.1994.1175.
- ↑ Swan, Richard G. (1998). "Néron–Popescu desingularization". Algebra and geometry (Taipei, 1995). Lect. Algebra Geom.. 2. Cambridge, MA: International Press. pp. 135–192.
- ↑ 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. https://www.ams.org/journals/jams/1999-12-02/S0894-0347-99-00294-5/.
- ↑ Cisinski, Denis-Charles; Déglise, Frédéric (2019). Triangulated Categories of Mixed Motives. Springer Monographs in Mathematics. doi:10.1007/978-3-030-33242-6. ISBN 978-3-030-33241-9.
External links
- "Presenting [math]\displaystyle{ \mathbb{Q} }[/math][[t as an explicit colimit of smooth [math]\displaystyle{ \mathbb{Q} }[/math]-algebras: an explicit example for the Popescu's theorem". MathOverflow. https://mathoverflow.net/q/155235.
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