Hokkaido Mathematical Journal

On the annihilators of formal local cohomology modules

Shahram REZAEI

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $\frak{a}$ denote an ideal in a commutative Noetherian local ring $(R,\frak{m})$ and $M$ a non-zero finitely generated $R$-module of dimension $d$. Let $d:=\dim(M/\frak{a} M)$. In this paper we calculate the annihilator of the top formal local cohomology module $\mathfrak{F}_{\frak{a}}^d (M)$. In fact, we prove that ${\rm Ann}_R(\mathfrak{F}_{\frak{a}}^d (M))={\rm Ann}_R(M/U_R(\frak{a}, M))$, where $$ U_R(\frak{a}, M):=\cup\lbrace N: N\leqslant M \text{ and } \dim(N/\frak{a}N) \lt \dim(M/\frak{a}M) \rbrace. $$ We give a description of $U_R(\frak{a}, M)$ and we will show that $$ {\rm Ann}_R (\mathfrak{F}_{\frak{a}}^d(M)) = {\rm Ann}_R (M/\cap_{\frak{p}_j \in {\rm Assh}_R M \cap {\rm V}(\frak{a})} N_j), $$ where $0=\bigcap_{j=1}^{n} N_{j}$ denotes a reduced primary decomposition of the zero submodule $0$ in $M$ and $N_j$ is a $\frak{p}_j$-primary submodule of $M$, for all $j=1,\dots, n$. Also, we determine the radical of the annihilator of $\mathfrak{F}_{\frak{a}}^d (M)$. We will prove that $$ \sqrt{{\rm Ann}_R(\mathfrak{F}_{\frak{a}}^d (M))} = {\rm Ann}_R(M/G_R(\frak{a}, M)), $$ where $G_R(\frak{a}, M)$ denotes the largest submodule of $M$ such that ${\rm Assh}_R(M)\cap {\rm V}(\frak{a}) \subseteq {\rm Ass}_R(M/G_R(\frak{a}, M))$ and ${\rm Assh}_R(M)$ denotes the set $\{\frak{p} \in {\rm Ass} M:\dim R/\frak{p} = \dim M\}.$

Article information

Source
Hokkaido Math. J., Volume 48, Number 1 (2019), 195-206.

Dates
First available in Project Euclid: 18 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1550480649

Digital Object Identifier
doi:10.14492/hokmj/1550480649

Mathematical Reviews number (MathSciNet)
MR3914174

Zentralblatt MATH identifier
07055600

Subjects
Primary: 13D45: Local cohomology [See also 14B15] 13E05: Noetherian rings and modules

Keywords
attached primes local cohomology annihilator

Citation

REZAEI, Shahram. On the annihilators of formal local cohomology modules. Hokkaido Math. J. 48 (2019), no. 1, 195--206. doi:10.14492/hokmj/1550480649. https://projecteuclid.org/euclid.hokmj/1550480649


Export citation