A note on cofinite modules over Noetherian complete local rings

Abstract

Let $(R,{\frak m})$ be a commutative Noetherian complete local ring and let $I$ be a proper ideal of $R$ such that the category of all $I$-cofinite $R$-modules is an Abelian subcategory of the category of all $R$-modules. Then for each $I$-cofinite $R$-module $M$ it is shown that $\dim R/(I+{\rm Ann}_R M)=\dim M$. As a consequence of this result it is shown that if $J$ is an ideal of $R$ such that for each ${\frak p}\in {\rm mAss}_R R$, $\dim R/(J+{\frak p})\leq 1$ or ${\rm cd}(J,R/{\frak p})\leq 1$, then $\dim R/(J+{\rm Ann}_R M)=\dim M$, for each $J$-cofinite $R$-module $M$.