Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T06:55:23.854Z Has data issue: false hasContentIssue false
  • English
  • Français

COFINITENESS OF GENERALIZED LOCAL COHOMOLOGY MODULES

Part of: Commutative algebra: Homological methods

Published online by Cambridge University Press:  29 March 2011

KEIVAN BORNA ,
PARVIZ SAHANDI  and
SIAMAK YASSEMI
KEIVAN BORNA*
Affiliation:
Faculty of Mathematical Sciences and Computer, Tarbiat Moallem University, Tehran, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran (email: borna@ipm.ir)
PARVIZ SAHANDI
Affiliation:
Department of Mathematics, University of Tabriz, Tabriz, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran (email: sahandi@ipm.ir)
SIAMAK YASSEMI
Affiliation:
Department of Mathematics, University of Tehran, Tehran, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran (email: yassemi@ipm.ir)
*
✉For correspondence; e-mail: keivan.borna@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let 𝔞 be an ideal of a Noetherian ring R. Let s be a nonnegative integer and let M and N be two R-modules such that ExtjR(M/𝔞M,Hi𝔞(N)) is finite for all i<s and all j≥0 . We show that HomR (R/𝔞,Hs𝔞(M,N)) is finite provided ExtsR(M/𝔞M,N) is a finite R-module. In addition, for finite R-modules M and N, we prove that if Hi𝔞(M,N) is minimax for all i<s, then HomR (R/𝔞,Hs𝔞(M,N)) is finite. These are two generalizations of the result of Brodmann and Lashgari [‘A finiteness result for associated primes of local cohomology modules’, Proc. Amer. Math. Soc. 128 (2000), 2851–2853] and a recent result due to Chu [‘Cofiniteness and finiteness of generalized local cohomology modules’, Bull. Aust. Math. Soc. 80 (2009), 244–250]. We also introduce a generalization of the concept of cofiniteness and recover some results for it.

Keywords

cofinitenessgeneralized local cohomologylocal cohomologyminimaxness

MSC classification

Secondary: 13D45: Local cohomology
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 3 , June 2011 , pp. 382 - 388
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The research of K. Borna was in part supported by a grant from IPM (No. 89130049). The research of P. Sahandi was in part supported by a grant from IPM (No. 89130051). The research of S. Yassemi was in part supported by a grant from IPM (No. 89130213).

References

[1]Aghapournahr, M. and Melkerson, L., ‘Local cohomology and Serre subcategories’, J. Algebra 320 (2008), 1275–1287.CrossRefGoogle Scholar
[2]Asadollahi, J., Khashyarmanesh, K. and Salarian, Sh., ‘On the finiteness properties of the generalized local cohomology modules’, Comm. Algebra 30 (2002), 859–867.CrossRefGoogle Scholar
[3]Asadollahi, J., Khashyarmanesh, K. and Salarian, Sh., ‘A generalization of the cofiniteness problem in local cohomology modules’, J. Aust. Math. Soc. 75 (2003), 313–324.CrossRefGoogle Scholar
[4]Bahmanpour, K. and Naghipour, R., ‘On the cofiniteness of local cohomology modules’, Proc. Amer. Math. Soc. 136(7) (2008), 2359–2363.CrossRefGoogle Scholar
[5]Borna, K., Sahandi, P. and Yassemi, S., ‘Artinian local cohomology modules’, Canad. Bull. Math. 50(4) (2007), 598–602.Google Scholar
[6]Brodmann, M. P. and Lashgari, A., ‘A finiteness result for associated primes of local cohomology modules’, Proc. Amer. Math. Soc. 128 (2000), 2851–2853.CrossRefGoogle Scholar
[7]Brodmann, M. P. and Sharp, R. Y., Local Cohomology: an Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics, 60 (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
[8]Chu, L., ‘Cofiniteness and finiteness of generalized local cohomology modules’, Bull. Aust. Math. Soc. 80 (2009), 244–250.CrossRefGoogle Scholar
[9]Dibaei, M. T. and Yassemi, S., ‘Associated primes and cofiniteness of local cohomology modules’, Manuscripta Math. 117 (2005), 199–205.CrossRefGoogle Scholar
[10]Grothendieck, A., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Advanced Studies in Pure Mathematics, 2 (North-Holland, Amsterdam, 1968).Google Scholar
[11]Hartshorne, R., ‘Affine duality and cofiniteness’, Invent. Math. 9 (1970), 145–164.CrossRefGoogle Scholar
[12]Herzog, J., ‘Komplexe, Auflösungen und Dualität in der lokalen Algebra’, Preprint, University of Regensburg, 1974.Google Scholar
[13]Khatami, L., Sharif, T. and Yassemi, S., ‘Associated primes of generalized local cohomology modules’, Comm. Algebra 30 (2002), 327–330.Google Scholar
[14]Melkersson, L., ‘Properties of cofinite modules and applications to local cohomology’, Math. Proc. Cambridge Philos. Soc. 125 (1999), 417–423.CrossRefGoogle Scholar
[15]Melkersson, L., ‘Modules cofinite with respect to an ideal’, J. Algebra 285 (2005), 649–668.CrossRefGoogle Scholar
[16]Rotman, J., Introduction to Homological Algebra (Academic Press, New York, 1979).Google Scholar
[17]Suzuki, N., ‘On the generalized local cohomology and its duality’, J. Math. Kyoto Univ. 18 (1978), 71–85.Google Scholar
[18]Yassemi, S., ‘Generalized section functor’, J. Pure Appl. Algebra 95 (1994), 103–119.CrossRefGoogle Scholar
[19]Zöschinger, H., ‘Minimax-Moduln’, J. Algebra 102 (1986), 1–32.CrossRefGoogle Scholar