Cohomological dimension of certain algebraic varieties
HTML articles powered by AMS MathViewer
- by K. Divaani-Aazar, R. Naghipour and M. Tousi
- Proc. Amer. Math. Soc. 130 (2002), 3537-3544
- DOI: https://doi.org/10.1090/S0002-9939-02-06500-0
- Published electronically: May 14, 2002
- PDF | Request permission
Abstract:
Let $\mathfrak a$ be an ideal of a commutative Noetherian ring $R$. For finitely generated $R$-modules $M$ and $N$ with $\operatorname {Supp} N\subseteq \operatorname {Supp} M$, it is shown that $\mathrm {cd}(\mathfrak {a},N)\leq \mathrm {cd}(\mathfrak {a},M)$. Let $N$ be a finitely generated module over a local ring $(R,\mathfrak m)$ such that $\operatorname {Min}_{\hat {R}}\hat {N}=\operatorname {Assh}_{\hat {R}}\hat {N}$. Using the above result and the notion of connectedness dimension, it is proved that $\mathrm {cd}(\mathfrak {a},N)\geq \dim N-c(N/\mathfrak {a} N)-1.$ Here $c(N)$ denotes the connectedness dimension of the topological space $\operatorname {Supp} N$. Finally, as a consequence of this inequality, two previously known generalizations of Faltings’ connectedness theorem are improved.References
- Yôichi Aoyama, Some basic results on canonical modules, J. Math. Kyoto Univ. 23 (1983), no. 1, 85–94. MR 692731, DOI 10.1215/kjm/1250521612
- Y\B{o}ichi Aoyama and Shiro Goto, On the endomorphism ring of the canonical module, J. Math. Kyoto Univ. 25 (1985), no. 1, 21–30. MR 777243, DOI 10.1215/kjm/1250521156
- M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics, vol. 60, Cambridge University Press, Cambridge, 1998. MR 1613627, DOI 10.1017/CBO9780511629204
- Donatella Delfino, A vanishing theorem for local cohomology modules, J. Pure Appl. Algebra 115 (1997), no. 2, 107–111. MR 1431156, DOI 10.1016/S0022-4049(96)00006-0
- K. Divaani-Aazar, P. Schenzel, Ideal topologies, local cohomology and connectedness, Math. Proc. Camb. Phil. Soc 131 (2001), 211–226.
- Gerd Faltings, Some theorems about formal functions, Publ. Res. Inst. Math. Sci. 16 (1980), no. 3, 721–737. MR 602466, DOI 10.2977/prims/1195186927
- Gerd Faltings, Über lokale Kohomologiegruppen hoher Ordnung, J. Reine Angew. Math. 313 (1980), 43–51 (German). MR 552461, DOI 10.1515/crll.1980.313.43
- Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux $(SGA$ $2)$, Advanced Studies in Pure Mathematics, Vol. 2, North-Holland Publishing Co., Amsterdam; Masson & Cie, Editeur, Paris, 1968 (French). Augmenté d’un exposé par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962. MR 0476737
- Robin Hartshorne, Cohomological dimension of algebraic varieties, Ann. of Math. (2) 88 (1968), 403–450. MR 232780, DOI 10.2307/1970720
- Melvin Hochster and Craig Huneke, Indecomposable canonical modules and connectedness, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 197–208. MR 1266184, DOI 10.1090/conm/159/01509
- C. Huneke and G. Lyubeznik, On the vanishing of local cohomology modules, Invent. Math. 102 (1990), no. 1, 73–93. MR 1069240, DOI 10.1007/BF01233420
- Wolmer V. Vasconcelos, Divisor theory in module categories, North-Holland Mathematics Studies, No. 14, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1974. MR 0498530
Bibliographic Information
- K. Divaani-Aazar
- Affiliation: Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran – and – Department of Mathematics, Az-Zahra University, Tehran, Iran
- Email: kdivaani@ipm.ir
- R. Naghipour
- Affiliation: Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran – and – Department of Mathematics, University of Tabriz, Tabriz, Iran
- Email: naghipour@tabrizu.ac.ir
- M. Tousi
- Affiliation: Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran – and – Department of Mathematics, Shahid Beheshti University, Tehran, Iran
- Email: mtousi@vax.ipm.ac.ir
- Received by editor(s): October 17, 2000
- Received by editor(s) in revised form: August 3, 2001
- Published electronically: May 14, 2002
- Additional Notes: This research was supported in part by a grant from IPM
- Communicated by: Wolmer V. Vasconcelos
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 3537-3544
- MSC (2000): Primary 13D45, 14B15
- DOI: https://doi.org/10.1090/S0002-9939-02-06500-0
- MathSciNet review: 1918830
Dedicated: Dedicated to Professor Hossein Zakeri