Popescu's theorem

In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu,{{cite journal|first=Dorin|last= Popescu|title= General Néron desingularization|journal=Nagoya Mathematical Journal|volume= 100 |year=1985|pages= 97–126|mr=0818160|doi=10.1017/S0027763000000246|doi-access=free}}{{cite journal|first=Dorin|last= Popescu|title= General Néron desingularization and approximation|journal=Nagoya Mathematical Journal|volume= 104 |year=1986|pages= 85–115|mr=0868439|doi=10.1017/S0027763000022698|doi-access=free}}

states:{{cite journal | last1 = Conrad | first1 = Brian | author1-link = Brian Conrad | last2 = de Jong | first2 = Aise Johan | author2-link = Aise Johan de Jong | doi = 10.1016/S0021-8693(02)00144-8 | issue = 2 | journal = Journal of Algebra | mr = 1935511 | pages = 489–515 | title = Approximation of versal deformations | url = https://math.stanford.edu/~conrad/papers/approx.pdf | volume = 255 | year = 2002}}, Theorem 1.3.

: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,{{cite journal|last=Ogoma|first= Tetsushi|title=General Néron desingularization based on the idea of Popescu|

journal=Journal of Algebra|volume= 167|year=1994|issue= 1| pages=57–84|doi=10.1006/jabr.1994.1175|mr=1282816|doi-access=free}} while an exposition of the result was provided by Richard Swan.{{cite book|last=Swan|first= Richard G.|

authorlink=Richard Swan| contribution=Néron–Popescu desingularization| title= Algebra and geometry (Taipei, 1995)|pages= 135–192|series= Lect. Algebra Geom.|volume= 2|publisher= International Press| location=Cambridge, MA|year= 1998|mr=1697953}}

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.{{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|volume=12|issue=2|pages=381–444|doi=10.1090/s0894-0347-99-00294-5|mr=1647069|doi-access=free}}{{Cite book|last1=Cisinski|first1=Denis-Charles|last2=Déglise|first2=Frédéric|title=Triangulated Categories of Mixed Motives|chapter=|series=Springer Monographs in Mathematics|year=2019|doi=10.1007/978-3-030-33242-6|arxiv=0912.2110|isbn=978-3-030-33241-9}}