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In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu, states:

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, while an exposition of the result was provided by Richard Swan.

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.

See also

• Ring with the approximation property

References

1. Popescu, Dorin (1985). "General Néron desingularization". Nagoya Mathematical Journal. 100: 97–126. doi:10.1017/S0027763000000246. MR 0818160.
2. Popescu, Dorin (1986). "General Néron desingularization and approximation". Nagoya Mathematical Journal. 104: 85–115. doi:10.1017/S0027763000022698. MR 0868439.
3. Conrad, Brian; de Jong, Aise Johan (2002). "Approximation of versal deformations" (PDF). Journal of Algebra. 255 (2): 489–515. doi:10.1016/S0021-8693(02)00144-8. MR 1935511., Theorem 1.3.
4. 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. MR 1282816.
5. Swan, Richard G. (1998). "Néron–Popescu desingularization". Algebra and geometry (Taipei, 1995). Lect. Algebra Geom. Vol. 2. Cambridge, MA: International Press. pp. 135–192. MR 1697953.
6. 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. MR 1647069.
7. Cisinski, Denis-Charles; Déglise, Frédéric (2019). Triangulated Categories of Mixed Motives. Springer Monographs in Mathematics. arXiv:0912.2110. doi:10.1007/978-3-030-33242-6. ISBN 978-3-030-33241-9.

External links

• "Presenting ${\displaystyle \mathbb {Q} }$[[t as an explicit colimit of smooth ${\displaystyle \mathbb {Q} }$-algebras: an explicit example for the Popescu's theorem". MathOverflow.

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