Tokyo Journal of Mathematics

The Derived Category Analogue of the Hartshorne-Lichtenbaum Vanishing Theorem

Kamran DIVAANI-AAZAR and Marziyeh HATAMKHANI

Full-text: Open access

Abstract

Let $\mathfrak{a}$ be an ideal of a local ring $(R,\mathfrak{m})$ and $X$ a $d$-dimensional homologically bounded complex of $R$-modules whose all homology modules are finitely generated. We show that $H^d_{\mathfrak{a}}(X)=0$ if and only if $\dim \widehat{R}/\mathfrak{a} \widehat{R}+\mathfrak{p}>0$ for all prime ideals $\mathfrak{p}$ of $\hat{R}$ such that $\dim \hat{R}/\mathfrak{p}-\inf (X\otimes_R\hat{R})_{\mathfrak{p}}=d$.

Article information

Source
Tokyo J. Math., Volume 36, Number 1 (2013), 195-205.

Dates
First available in Project Euclid: 22 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1374497519

Digital Object Identifier
doi:10.3836/tjm/1374497519

Mathematical Reviews number (MathSciNet)
MR3112383

Zentralblatt MATH identifier
1277.13013

Subjects
Primary: 13D45: Local cohomology [See also 14B15]
Secondary: 13D02: Syzygies, resolutions, complexes 14B15: Local cohomology [See also 13D45, 32C36]

Citation

HATAMKHANI, Marziyeh; DIVAANI-AAZAR, Kamran. The Derived Category Analogue of the Hartshorne-Lichtenbaum Vanishing Theorem. Tokyo J. Math. 36 (2013), no. 1, 195--205. doi:10.3836/tjm/1374497519. https://projecteuclid.org/euclid.tjm/1374497519


Export citation

References

  • M. Brodmann and C. Huneke, A quick proof of the Hartshorne-Lichtenbaum vanishing theorem, Algebraic geometry and its applications, (West Lafayette, IN, 1990), 305–308, Springer, New York, 1994.
  • M. Brodmann and R. Y. Sharp, Local cohomology$:$ An algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge, 1998.
  • F. W. Call and R. Y. Sharp, A short proof of the local Lichtenbaum-Hartshorne theorem on the vanishing of local cohomology, Bull. London Math. Soc. 18(3), (1986), 261–264.
  • L. W. Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, 1747, Springer-Verlag, Berlin, 2000.
  • K. Divaani-Aazar and A. Hajikarimi, Generalized local cohomology modules and homological Gorenstein dimensions, Comm. Algebra 39(6), (2011), 2051–2067.
  • K. Divaani-Aazar, R. Naghipour and M. Tousi, The Lichtenbaum-Hartshorne theorem for generalized local cohomology and connectedness, Comm. Algebra 30(8), (2002), 3687–3702.
  • H-B. Foxby, Hyperhomological algebra & commutative rings, in preparation.
  • H-B. Foxby, A homological theory of complexes of modules, Preprint Series no. 19 a & 19 b, Department of Mathematics, University of Copenhagen, 1981.
  • H-B. Foxby, Bounded complexes of flat modules, J. Pure Appl. Algebra 15(2), (1979), 149–172.
  • R. Hartshorne, Residues and duality, Lecture Notes in Mathematics 20, Springer-Verlag, Berlin-New York, 1966.
  • J. Herzog, Komplex Auflösungen und Dualität in der lokalen Algebra, Habilitationsschrift, Universität Regensburg, (1974).
  • J. Lipman, Lectures on local cohomology and duality, Local cohomology and its applications (Guanajuato, 1999), Lecture Notes in Pure and Appl. Math. 226, Dekker, New York, (2002), 39–89.
  • P. Schenzel, Explicit computations around the Lichtenbaum-Hartshorne vanishing theorem, Manuscripta Math. 78(1), (1993), 57–68.
  • R. Takahashi, Y. Yoshino and T. Yoshizawa, Local cohomology based on a nonclosed support defined by a pair of ideals, J. Pure Appl. Algebra 213(4), (2009), 582–600.
  • S. Yassemi, Generalized section functors, J. Pure Appl. Algebra 95(1), (1994), 103–119.
  • Y. Yoshino and T. Yoshizawa, Abstract local cohomology functors, Math. J. Okayama Univ. 53 (2011), 129–154.
  • N. Zamani, On graded generalized local cohomology, Arch. Math. (Basel) 86(4), (2006), 321–330.