Popescu's theorem explained
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
External links
Notes and References
- Dorin. Popescu. General Néron desingularization. Nagoya Mathematical Journal. 100 . 1985. 97–126. 0818160. 10.1017/S0027763000000246. free.
- Dorin. Popescu. General Néron desingularization and approximation. Nagoya Mathematical Journal. 104 . 1986. 85–115. 0868439. 10.1017/S0027763000022698. free.
- Conrad . Brian . Brian Conrad . de Jong . Aise Johan . Aise Johan de Jong . 10.1016/S0021-8693(02)00144-8 . 2 . . 1935511 . 489–515 . Approximation of versal deformations . 255 . 2002., Theorem 1.3.
- Ogoma. Tetsushi. General Néron desingularization based on the idea of Popescu. Journal of Algebra. 167. 1994. 1. 57–84. 10.1006/jabr.1994.1175. 1282816. free.
- Book: Swan, Richard G.. Richard Swan
. Richard Swan. Néron–Popescu desingularization. Algebra and geometry (Taipei, 1995). 135–192. Lect. Algebra Geom.. 2. International Press. Cambridge, MA. 1998. 1697953.
- 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. 10.1090/s0894-0347-99-00294-5. 1647069. free.
- Book: Cisinski. Denis-Charles. Déglise. Frédéric. Triangulated Categories of Mixed Motives. Springer Monographs in Mathematics. 2019. 10.1007/978-3-030-33242-6. 0912.2110. 978-3-030-33241-9.