The grade conjecture and asymptotic intersection multiplicity
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- by Jesse Beder PDF
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Abstract:
Given a finitely generated module $M$ over a local ring $A$ of characteristic $p$ with $\operatorname {pd} M < \infty$, we study the asymptotic intersection multiplicity $\chi _\infty (M, A/\underline {x})$, where $\underline {x} = (x_1, \ldots , x_r)$ is a system of parameters for $M$. We show that there exists a system of parameters such that $\chi _\infty$ is positive if and only if $\dim \operatorname {Ext}^{d-r}(M, A) = r$, where $d = \dim A$ and $r = \dim M$. We use this to prove several results relating to the grade conjecture, which states that $\operatorname {grade} M + \dim M = \dim A$ for any module $M$ with $\operatorname {pd} M < \infty$.References
- David A. Buchsbaum and David Eisenbud, What makes a complex exact?, J. Algebra 25 (1973), 259–268. MR 314819, DOI 10.1016/0021-8693(73)90044-6
- Sankar P. Dutta, Frobenius and multiplicities, J. Algebra 85 (1983), no. 2, 424–448. MR 725094, DOI 10.1016/0021-8693(83)90106-0
- S. P. Dutta, Ext and Frobenius, J. Algebra 127 (1989), no. 1, 163–177. MR 1029410, DOI 10.1016/0021-8693(89)90281-0
- S. P. Dutta, Ext and Frobenius. II, J. Algebra 186 (1996), no. 3, 724–735. MR 1424590, DOI 10.1006/jabr.1996.0392
- Sankar P. Dutta, M. Hochster, and J. E. McLaughlin, Modules of finite projective dimension with negative intersection multiplicities, Invent. Math. 79 (1985), no. 2, 253–291. MR 778127, DOI 10.1007/BF01388973
- Hans-Bjørn Foxby, Bounded complexes of flat modules, J. Pure Appl. Algebra 15 (1979), no. 2, 149–172. MR 535182, DOI 10.1016/0022-4049(79)90030-6
- O. Gabber, Non-negativity of Serre’s intersection multiplicities, exposé à L’IHES, décembre 1995.
- Henri Gillet and Christophe Soulé, $K$-théorie et nullité des multiplicités d’intersection, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 3, 71–74 (French, with English summary). MR 777736
- Melvin Hochster, The equicharacteristic case of some homological conjectures on local rings, Bull. Amer. Math. Soc. 80 (1974), 683–686. MR 342510, DOI 10.1090/S0002-9904-1974-13548-2
- Ernst Kunz, Characterizations of regular local rings of characteristic $p$, Amer. J. Math. 91 (1969), 772–784. MR 252389, DOI 10.2307/2373351
- Kazuhiko Kurano, On the vanishing and the positivity of intersection multiplicities over local rings with small non-complete intersection loci, Nagoya Math. J. 136 (1994), 133–155. MR 1309384, DOI 10.1017/S0027763000024995
- Stephen Lichtenbaum, On the vanishing of $\textrm {Tor}$ in regular local rings, Illinois J. Math. 10 (1966), 220–226. MR 188249
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 47–119 (French). MR 374130
- Christian Peskine and Lucien Szpiro, Syzygies et multiplicités, C. R. Acad. Sci. Paris Sér. A 278 (1974), 1421–1424 (French). MR 349659
- Paul Roberts, The vanishing of intersection multiplicities of perfect complexes, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 127–130. MR 799793, DOI 10.1090/S0273-0979-1985-15394-7
- Paul Roberts, Le théorème d’intersection, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 7, 177–180 (French, with English summary). MR 880574
- Paul C. Roberts, The homological conjectures, Free resolutions in commutative algebra and algebraic geometry (Sundance, UT, 1990) Res. Notes Math., vol. 2, Jones and Bartlett, Boston, MA, 1992, pp. 121–132. MR 1165322
- Gerhard Seibert, Complexes with homology of finite length and Frobenius functors, J. Algebra 125 (1989), no. 2, 278–287. MR 1018945, DOI 10.1016/0021-8693(89)90164-6
- Jean-Pierre Serre, Local algebra, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. Translated from the French by CheeWhye Chin and revised by the author. MR 1771925, DOI 10.1007/978-3-662-04203-8
Additional Information
- Jesse Beder
- Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801
- Received by editor(s): February 7, 2012
- Received by editor(s) in revised form: January 27, 2013
- Published electronically: August 14, 2014
- Communicated by: Irena Peeva
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 4065-4077
- MSC (2010): Primary 13A35, 13H15
- DOI: https://doi.org/10.1090/S0002-9939-2014-12183-6
- MathSciNet review: 3266978