Auslander–Buchsbaum theorem

Not to be confused with Auslander–Buchsbaum formula.

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

The theorem was first proved by MauriceAuslander and David Buchsbaum  ( 1959 ). They showed that regular local rings of dimension 3 are unique factorization domains, and MasayoshiNagata  ( 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 (5): 733–734, Bibcode:1959PNAS...45..733A, 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 (2): 382–420, doi:10.2307/2372791, ISSN   0002-9327, JSTOR   2372791, MR   0094344