Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On endomorphism rings and dimensions of local cohomology modules


Author: Peter Schenzel
Journal: Proc. Amer. Math. Soc. 137 (2009), 1315-1322
MSC (2000): Primary 13D45; Secondary 13H10, 14M10
DOI: https://doi.org/10.1090/S0002-9939-08-09676-7
Published electronically: November 12, 2008
MathSciNet review: 2465654
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (R,\mathfrak{m})$ denote an $ n$-dimensional complete local Gorenstein ring. For an ideal $ I$ of $ R$ let $ H^i_I(R), i \in \mathbb{Z},$ denote the local cohomology modules of $ R$ with respect to $ I.$ If $ H^i_I(R) = 0$ for all $ i \not= c = \operatorname{height} I,$ then the endomorphism ring of $ H^c_I(R)$ is isomorphic to $ R$. Here we prove that this is true if and only if $ H^i_I(R) = 0$ for $ i = n, n-1$, provided $ c \geq 2$ and $ R/I$ has an isolated singularity, resp. if $ I$ is set-theoretically a complete intersection in codimension at most one. Moreover, there is a vanishing result of $ H^i_I(R)$ for all $ i > m, m$ a given integer, and an estimate of the dimension of $ H^i_I(R).$


References [Enhancements On Off] (What's this?)

  • [1] H. BASS: On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28. MR 0153708 (27:3669)
  • [2] A. GROTHENDIECK: `Local cohomology', Notes by R. Hartshorne, Lect. Notes in Math. 41, Springer-Verlag, 1967. MR 0224620 (37:219)
  • [3] R. HARTSHORNE: Affine duality and cofiniteness, Invent. Math. 9 (1969/1970), 145-164. MR 0257096 (41:1750)
  • [4] M. HELLUS, P. SCHENZEL: On cohomologically complete intersections, J. Algebra 320 (2008), 3733-3748.
  • [5] M. HELLUS, J. STÜCKRAD: On endomorphism rings of local cohomology modules, Proc. Amer. Math. Soc. 136 (2008), 2333-2341. MR 2390499
  • [6] C. HUNEKE, G. LYUBEZNIK: On the vanishing of local cohomology modules, Invent. Math. 102 (1990), 73-93. MR 1069240 (91i:13020)
  • [7] K. KAWASAKI: On the highest Lyubeznik number, Math. Proc. Cambr. Phil. Soc. 132 (2002), 409-417. MR 1891679 (2003b:13026)
  • [8] G. LYUBEZNIK: Finiteness properties of local cohomology modules (an application of $ D$-modules to commutative algebra), Invent. Math. 113 (1993), 41-55. MR 1223223 (94e:13032)
  • [9] P. SCHENZEL: On birational Macaulayfications and Cohen-Macaulay canonical modules, J. Algebra 275 (2004), 751-770. MR 2052635 (2005i:13017)
  • [10] C. WEIBEL: `An Introduction to Homological Algebra', Cambridge Stud. in Advanced Math. 38, Cambridge Univ. Press, 1994. MR 1269324 (95f:18001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13D45, 13H10, 14M10

Retrieve articles in all journals with MSC (2000): 13D45, 13H10, 14M10


Additional Information

Peter Schenzel
Affiliation: Institut für Informatik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany
Email: peter.schenzel@informatik.uni-halle.de

DOI: https://doi.org/10.1090/S0002-9939-08-09676-7
Keywords: Local cohomology, vanishing, cohomological dimension
Received by editor(s): April 21, 2008
Received by editor(s) in revised form: June 17, 2008
Published electronically: November 12, 2008
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society