Vanishing and localization of $(d,\frak{b})$-ideal transforms

Document Type : Research Paper

Author

Department of Mathematics, Payame Noor University, P.O. Box 19395-4697, Tehran, Iran

Abstract

Let $R$ be a commutative Noetherian ring, $\!M$ an $\!R$-module and $d$ a non-negative integer. Let $\Sigma$ denote the set of ideals $\frak{I}$ of $R$ such that $\mathrm{dim}(R/\frak{I})\!\leq\!d$. For an ideal $\frak{b}$ of $R$, we define the $(d,\i\frak{b})$-transform $D_{d,\i\i\frak{b}}(M)$ and study its properties. Then a criterion for $D_{d,\frak{b}}(R)\!=\!\bigcap_{\frak{p}\notin W(d,\frak{b})}R_{\frak{p}}$ will be given, where $W(d,\frak{b})$ contains all ideals $\frak{a}$ of $R$ such that $\frak{I}\subseteq \frak{a}+\frak{b}$ for some $\frak{I}\in \Sigma$. For each $i\geq 0$, let $D^i_{d,\frak{b}}(-)$ denote the $i$-th right derived functor of $D_{d,\frak{b}}(M)$. We study the localization of the module $D^i_{d,\frak{b}}(M)$ and prove that  $D^i_{d,\frak{b}}(M)_\frak{p}\cong D^i_{d-\textrm{dim}(R/(\frak{p}+\frak{b})),\frak{b}_\frak{p}}(M_\frak{p})$ for all $\frak{p}\in\mathrm{Spec}(R)$ and all $i\geq 0$. Finally, we establish vanishing theorems for $D^i_{d,\frak{b}}(M)$.

Keywords

Main Subjects

References

[1] Banica, C., & Soia, M. (1976). Singular sets of a module on local cohomology. Boll. Un. Mat. Ital. B, 16, 923-934.
[2] Brodmann, M. P., & Sharp, R. Y. (1998). Local Cohomology- An Algebraic Introduction with Geometric Applications. Cambridge University Press. https://doi.org/10.1017/CBO9780511629204
[3] Matsumura, H. (1986). Commutative ring theory. Cambridge University Press. https://doi.org/10.1017/CBO9781139171762
[4] Rotman, J. (1979). Introduction to homological algebra. Academic Press. https://doi.org/10.1007/978-0-387-68324-9
[5] R. Takahashi, R., & Yoshino, Y., & Yoshizawa, T. (2009). Local cohomology based on a nonclosed support de ned by a pair of ideals. J. Pure Appl. Algebra, 213(4), 582-600. https://doi.org/10.1016/j.jpaa.2008.09.008
[6] Zamani, N., & Bijan-zadeh, M. H., & Sayedsadeghi, M. (2016). cohomology with supports of dimension  d. Journal of Algebra and Its Applications, 15(3), 1650042(1)-1650051(10). https://doi.org/10.1142/S0219498816500420
[7] Zamani, N., & Bijan-zadeh, M. H., & Sayedsadeghi, M. (2014). d-Transform Functor and Some Finiteness and Isomorphism Results. Vie. J. Math., 42, 179-186. https://doi.org/10.1007/s10013-013-0042-2

Files

History

  • Receive Date: 14 November 2024
  • Revise Date: 11 April 2025
  • Accept Date: 02 May 2025

Share

How to cite

Statistics