Journal of the
Korean Mathematical Society JKMS

The aim of the paper is to obtain information about the flat covers and minimal flat resolutions of Artinian modules over a Noetherian ring. Let $R$ be a commutative Noetherian ring and let $A$ be an Artinian $R$-module. We prove that the flat cover of $A$ is of the form $\prod_{p\in {\rm Att}_R(A)} T_p$, where $T_p$ is the completion of a free $R_p$-module. Also, we construct a minimal flat resolution for $R/xR$-module $0:_Ax$ from a given minimal flat resolution of $A$, when $x$ is a non-unit and non-zero divisor of $R$ such that $A=xA$. This result leads to a description of the structure of a minimal flat resolution for ${\rm H}^n_{\underline m}(R)$, $n$th local cohomology module of $R$ with respect to the ideal $\underline m$, over a local Cohen-Macaulay ring $(R, \underline m)$ of dimension $n$.

Keywords: Artinian module, flat cover, minimal flat resolution

MSC numbers: 13C11, 13E10