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# Auslander–Buchsbaum theorem

In commutative algebra, the Auslander–Buchsbaum theorem states that regular local rings are unique factorization domains.

The theorem was first proved by Maurice Auslander and David Buchsbaum (1959). They showed that regular local rings of dimension 3 are unique factorization domains, and Masayoshi Nagata (1958) had previously shown that this implies that all regular local rings are unique factorization domains.

References

Auslander, Maurice; Buchsbaum, D. A. (1959), "Unique factorization in regular local rings", Proceedings of the National Academy of Sciences of the United States of America 45: 733–734, doi:10.1073/pnas.45.5.733, ISSN 0027-8424, JSTOR 90213, MR 0103906, PMC 222624, PMID 16590434

Nagata, Masayoshi (1958), "A general theory of algebraic geometry over Dedekind domains. II. Separably generated extensions and regular local rings", American Journal of Mathematics 80: 382–420, ISSN 0002-9327, JSTOR 2372791, MR 0094344

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