Taiwanese Journal of Mathematics

FINITENESS RESULT FOR GENERALIZED LOCAL COHOMOLOGY MODULES

Abolfazl Tehranian

Full-text: Open access

Abstract

Let $R$ be a Noetherian ring, let $M$ and $N$ be finitely generated modules and let $\mathfrak{a}$ and $\mathfrak{b}$ be ideals of $R$. Let $s$ be an integer such that $\mathfrak{b}_{\mathfrak{p}} \subseteq \sqrt{\mbox{Ann} \mbox{H}^i_{\mathfrak{a}_{\mathfrak{p}}}(M_{\mathfrak{p}} ,N_{\mathfrak{p}})}$ for all $i \le s$ and all prime ideal $\mathfrak{p}$ of $R$. Then we show the following statements hold:

(1) If $\mbox{H}^i_{\mathfrak{b}}(N) = 0$ for all $i \lt s$, then $\mbox{H}_{\mathfrak{a}}^{i}(M,N)$ is finitely generated for all $i \leq s$.

(2) $\mathfrak{b} \subseteq \sqrt{\mbox{Ann} \mbox{H}_{\mathfrak{a}}^{2}(M,N)}$.

These statements generalize the corresponding results which are shown in [6] and [1] for standard local cohomology module.

Article information

Source
Taiwanese J. Math., Volume 14, Number 2 (2010), 447-451.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405800

Digital Object Identifier
doi:10.11650/twjm/1500405800

Mathematical Reviews number (MathSciNet)
MR2655780

Zentralblatt MATH identifier
1198.13017

Subjects
Primary: 13D45: Local cohomology [See also 14B15] 13E99: None of the above, but in this section

Keywords
local cohomology modules finite over a local homomorphism Artinian module secondary representation

Citation

Tehranian, Abolfazl. FINITENESS RESULT FOR GENERALIZED LOCAL COHOMOLOGY MODULES. Taiwanese J. Math. 14 (2010), no. 2, 447--451. doi:10.11650/twjm/1500405800. https://projecteuclid.org/euclid.twjm/1500405800


Export citation

References

  • M. Brodmann, Ch. Rotthaus and R. Y. Sharp, On annihilators and associated primes of local cohomology modules, J. Pure Applied Algebra, 153 (2000), 197-227
  • M. Brodmann and R. Sharp, Local cohomology-an algebraic introduction with geometric applications, Cambridge studies in advanced Mathematics No. 60, Cambridge University Press, 1998.
  • G. Faltings, Über die Annulatoren lokaler Kohomologiegruppen, Arch. Math., 30 (1978), 473-476.
  • G. Faltings, Der Endlichkeitssatz der lokalen kohomologie, Math. Ann., 255 (1981), 45-56.
  • J. Herzog, Komplexe, Auflösungen und dualität in der lokalen Algebra, Preprint 1974, Univ. Regensburg.
  • K. N. Raghavan, Local-global principle for annihilation of local cohomology, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), 329-331, Contemp. Math., 159, Amer. Math. Soc., Providence, RI, 1994.
  • N. Suzuki, On the generalized local cohomology and its duality, J. Math. Kyoto Univ., 18 (1978), 71-85.
  • S. Yassemi, Generalized section functor, J. Pure Applied Algebra, 95 (1994), 103-119.
  • S. Yassemi, L. Khatami and T. Sharif, Associated primes of generalized local cohomology modules, Comm. Algebra, 30 (2002), 327-330.