Auslander–Buchsbaum theorem: Difference between revisions
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In commutative algebra, the '''Auslander–Buchsbaum theorem''' states that [[regular local ring]]s are [[unique factorization domain]]s. |
{{distinguish|Auslander–Buchsbaum formula}} |
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In [[commutative algebra]], the '''Auslander–Buchsbaum theorem''' states that [[regular local ring]]s are [[unique factorization domain]]s. |
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The theorem was first proved by {{ |
The theorem was first proved by {{harvs|txt|first=Maurice|last=Auslander|authorlink=Maurice Auslander|first2=David|last2=Buchsbaum|author2-link=David Buchsbaum|year=1959}}. They showed that regular [[local ring]]s of dimension 3 are unique factorization domains, and {{harvs|txt|first1=Masayoshi | author1-link=Masayoshi Nagata |last=Nagata|year=1958}} had previously shown that this implies that all regular local rings are unique factorization domains. |
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==References== |
==References== |
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*{{Citation | last1=Auslander | first1=Maurice | last2=Buchsbaum | first2=D. A. | title=Unique factorization in regular local rings | jstor=90213 | mr=0103906 | year=1959 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=45 | issue=5 | pages=733–734 | doi=10.1073/pnas.45.5.733 | pmid=16590434 | pmc=222624| bibcode=1959PNAS...45..733A | doi-access=free }} |
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*{{Citation | last1=Nagata | first1=Masayoshi | author1-link=Masayoshi Nagata | title=A general theory of algebraic geometry over Dedekind domains. II. Separably generated extensions and regular local rings | jstor=2372791 | mr=0094344 | year=1958 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=80 | issue=2 | pages=382–420 | doi=10.2307/2372791}} |
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{{DEFAULTSORT:Auslander-Buchsbaum theorem}} |
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[[Category:Theorems in ring theory]] |
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{{commutative-algebra-stub}} |
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Latest revision as of 22:38, 12 August 2023
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
[edit]- 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
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