## Hokkaido Mathematical Journal

### On the annihilators of formal local cohomology modules

Shahram REZAEI

#### 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

https://projecteuclid.org/euclid.hokmj/1550480649

Digital Object Identifier
doi:10.14492/hokmj/1550480649

Mathematical Reviews number (MathSciNet)
MR3914174

Zentralblatt MATH identifier
07055600

#### 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