Order ideals of minimal generators

Evans, E. Graham; Griffith, Phillip

doi:10.1090/S0002-9939-1982-0671197-2

Order ideals of minimal generators
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by E. Graham Evans and Phillip Griffith PDF

Proc. Amer. Math. Soc. 86 (1982), 375-378 Request permission

Abstract:

Let $R$ be a local noetherian domain with algebraically closed residue field and let $M$ be a finitely generated module of rank $r$ which is not free. Then there is some minimal generator $x$ of $M$ such that the ideal of images of $x$ under maps of $M$ to $R$ has height at most $r$.

References

Winfried Bruns, “Jede” endliche freie Auflösung ist freie Auflösung eines von drei Elementen erzeugten Ideals, J. Algebra 39 (1976), no. 2, 429–439. MR 399074, DOI 10.1016/0021-8693(76)90047-8
Winfried Bruns, The Eisenbud-Evans generalized principal ideal theorem and determinantal ideals, Proc. Amer. Math. Soc. 83 (1981), no. 1, 19–24. MR 619972, DOI 10.1090/S0002-9939-1981-0619972-3
Winfried Bruns, E. Graham Evans Jr., and Phillip A. Griffith, Syzygies, ideals of height two, and vector bundles, J. Algebra 67 (1980), no. 1, 143–162. MR 595025, DOI 10.1016/0021-8693(80)90313-0
David Eisenbud and E. Graham Evans Jr., A generalized principal ideal theorem, Nagoya Math. J. 62 (1976), 41–53. MR 409440
E. Graham Evans and Phillip Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981), no. 2, 323–333. MR 632842, DOI 10.2307/1971296
Tor H. Gulliksen, Tout idéal premier d’un anneau noethérien est associé à un idéal engendré par trois éléments, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A1206–A1207 (French). MR 271085
G. Horrocks and D. Mumford, A rank $2$ vector bundle on $\textbf {P}^{4}$ with $15,000$ symmetries, Topology 12 (1973), 63–81. MR 382279, DOI 10.1016/0040-9383(73)90022-0