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On modules of finite projective dimension

Part of: Commutative algebra: Homological methods Local rings and semilocal rings Theory of modules and ideals

Published online by Cambridge University Press:  11 January 2016

S. P. Dutta
S. P. Dutta*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801, USA, s-dutta@illinois.edu
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Abstract

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We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ring R of mixed characteristic p > 0, where p is a nonzero divisor, if I is an ideal of finite projective dimension over R and p 𝜖 I or p is a nonzero divisor on R/I, then every minimal generator of I is a nonzero divisor. Hence, if P is a prime ideal of finite projective dimension in a local ring R, then every minimal generator of P is a nonzero divisor in R.

MSC classification

Secondary: 13D02: Syzygies, resolutions, complexes 13D22: Homological conjectures (intersection theorems) 13C15: Dimension theory, depth, related rings (catenary, etc.) 13H05: Regular local rings
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 219 , September 2015 , pp. 87 - 111
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Auslander, M. and Buchweitz, R.-O., “The homological theory of maximal Cohen-Macaulay approximations” in Colloque en l’honneur de Pierre Samuel (Orsay, 1987), Mém. Soc. Math. Fr. (N.S.) 38, Soc. Math. Fr., Marseilles, 1989, 5–37. MR 1044344.Google Scholar
[2] Bourbaki, N.,Elements of Mathematics:Commutative Algebra, Addison-Wesley, Reading, MA, 1972. MR 0360549.Google Scholar
[3] Bruns, W. and Herzog, J., Cohen-Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge University Press, Cambridge, 1993. MR 1251956.Google Scholar
[4] Dutta, S. P., On the canonical element conjecture, Trans. Amer. Math. Soc. 299, no. 2 (1987), 803–811. MR 0869233. DOI 10.2307/2000525.Google Scholar
[5] Dutta, S. P., On negativity of higher Euler characteristics, Amer. J. Math. 126 (2004), 1341–1354. MR 2102398.Google Scholar
[6] Dutta, S. P., The monomial conjecture and order ideals, J. Algebra 383 (2013), 232–241. MR 3037977. DOI 10.1016/j.jalgebra.2013.03.006.Google Scholar
[7] Evans, E. G. and Griffith, P., The syzygy problem, Ann. of Math. (2) 114 (1981), 323–333. MR 0632842. DOI 10.2307/1971296.Google Scholar
[8] Evans, E. G. and Griffith, P., Syzygies, London Math. Soc. Lecture Note Ser. 106, Cambridge University Press, Cambridge, 1985. MR 0811636. DOI 10.1017/CBO9781107325661.Google Scholar
[9] Evans, E. G. and Griffith, P.,“Order ideals”inCommutative Algebra(Berkeley,CA,1987), Math. Sci.Res.Inst.Publ. 15, Springer, NewYork, 1989, 213–225.MR1015519. DOI 10.1007/978-1-4612-3660-310.CrossRefGoogle Scholar
[10] Evans, E. G. and Griffith, P., A graded syzygy theorem in mixed characteristic, Math. Res. Lett. 8 (2001), 605–611. MR 1879804. DOI 10.4310/MRL.2001.v8.n5.a2.Google Scholar
[11] Foxby, H.-B., On the μi in a minimal injective resolution, II, Math. Scand. 41 (1977), 19–44. MR 0476801.Google Scholar
[12] Hochster, M., Topics in the Homological Theory of Modules over Commutative Rings, CBMS Reg. Conf. Ser. Math. 24, Amer. Math. Soc., Providence, 1975. MR 0371879.Google Scholar
[13] Hochster, M., Canonical elements in local cohomology modulesand thedirect summandconjecture, J.Algebra 84 (1983), 503–553.MR0723406.DOI10.1016/0021-8693(83)90092-3.Google Scholar
[14] Hochster, M., Big Cohen-Macaulay algebras in dimension three via Heitmann’s theorem, J. Algebra 254 (2002), 395–408. MR 1933876. DOI 10.1016/S0021-8693(02)00086-8.Google Scholar
[15] Peskine, C. and Szpiro, L., Dimension projective finie et cohomologie locale, Publ. Math. Inst. Hautes Études Sci. 42 (1973), 47–119. MR 0374130.Google Scholar
[16] Shamash, J.,The Poincaré series of a local ring, J. Algebra 12 (1969), 453–470. MR 0241411.Google Scholar
[17] Shimomoto, K.,Almost Cohen-Macaulay algebras in mixed characteristic via Fontaine rings, Illinois J. Math. 55 (2011), 107–125. MR 3006682.Google Scholar
[18] Smoke, W., Perfect modules over Cohen-Macaulay local rings, J. Algebra 106 (1987), 367–375. MR 0880963. DOI 10.1016/0021-8693(87)90002-0.Google Scholar