Popescu’s theorem
In algebra, Popescu’s theorem, introduced by D. Popescu, states:[1]
- Let A be a noetherian ring and B a noetherian algebra over it. Then, the structure map A →B 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., local excellent ring) and B its completion, then the map A →B is regular by definition and the theorem applies.
The usual proof of the Artin approximation theorem relies crucially on Popescu's theorem.
References
- ↑ Conrad & De Jong, Theorem 1.3.
- Conrad, Brian; De Jong, A. J. "Approximation of versal deformations" (PDF). Retrieved 1 December 2014.
External links
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