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 AB 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., a local excellent ring) and B its completion, then the map AB 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[edit]


  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. 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 (December 10, 2009). "Triangulated categories of mixed motives". arXiv:0912.2110.

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