Top formal local cohomology module

Abstract

Let I be an ideal of a local commutative noetherian ring ($R, {{\mathfrak {m}}}$) and M a finitely generated R-module. We study some properties of the top formal local cohomology module ${\mathcal {F}}^l_I(M)=\mathop {\varprojlim }H^l_{{{\mathfrak {m}}}}(M{/}I^tM)$ with $l=\mathrm {dim}M{/}I M$. In particular, we show that, in the case $M \ne IM$, ${\mathcal {F}}^l_I(M)$ is artinian if and only if $l>\mathrm {dim}{\overline{M}}/I{\overline{M}}$ where ${\overline{M}}=M/ H^0_I(M)$. As a consequence, we have $\mathrm {dim}{\overline{M}}/I{\overline{M}}=\sup \{i \in {\mathbb {Z}}\mid {\mathcal {F}}^i_I(M) \text { is not artinian}\},$ provided that ${\overline{M}} \ne I{\overline{M}}$.