Finiteness dimensions and cofiniteness of local cohomology modules

Abstract

Let $R$ be a commutative Noetherian ring with nonzero identity, $\mathfrak{𝔞}$ an ideal of $R$, and $n$ a nonnegative integer. For an arbitrary $R$-module $X$ which is not necessarily finite, we prove the following results:

(i) $f_{\mathfrak{𝔞}}^n\left(X\right)=inf${$i\in {\mathbb{N}}_0:H_{\mathfrak{𝔞}}^i\left(X\right)$ is not an ${FD}_{<n}$ $R$-module} if ${Ext}_R^i\left(R∕\mathfrak{𝔞},X\right)$ is an ${FD}_{<n}$ $R$-module for all $i$.

(ii) $f_{\mathfrak{𝔞}}^1\left(X\right)=inf${$i\in {\mathbb{N}}_0:H_{\mathfrak{𝔞}}^i\left(X\right)$ is not a minimax $R$-module} if ${Ext}_R^i\left(R∕\mathfrak{𝔞},X\right)$ is finite for all $i$.

(iii) $f_{\mathfrak{𝔞}}^2\left(X\right)=inf${$i\in {\mathbb{N}}_0:H_{\mathfrak{𝔞}}^i\left(X\right)$ is not a weakly Laskerian $R$-module} if $R$ is semilocal and ${Ext}_R^i\left(R∕\mathfrak{𝔞},X\right)$ is finite for all $i$.

(iv) $H_{\mathfrak{𝔞}}^i\left(X\right)$ is $\mathfrak{𝔞}$-cofinite for all $i<f_{\mathfrak{𝔞}}^2\left(X\right)$ and ${Ass}_R\left(H_{\mathfrak{𝔞}}^{f_{\mathfrak{𝔞}}^2\left(X\right)}\left(X\right)\right)$ is finite if ${Ext}_R^i\left(R∕\mathfrak{𝔞},X\right)$ is finite for all $i\le f_{\mathfrak{𝔞}}^2\left(X\right)$.

Here, $f_{\mathfrak{𝔞}}^n\left(X\right)=inf${$f_{\mathfrak{𝔞}{R}_{\mathfrak{𝔭}}}\left(X_{\mathfrak{𝔭}}\right):\mathfrak{𝔭}\in Spec\left(R\right)$ and ${dim}_R\left(R∕\mathfrak{𝔭}\right)\ge n$} is the $n$-th finiteness dimension of $X$ with respect to $\mathfrak{𝔞}$ and $f_{\mathfrak{𝔞}}\left(X\right)=inf${$i\in {\mathbb{N}}_0:H_{\mathfrak{𝔞}}^i\left(X\right)$ is not a finite $R$-module} is the finiteness dimension of $X$ with respect to $\mathfrak{𝔞}$.