A criterion for completeness

Abstract:

Let $(R,\mathfrak {m})$ denote a local ring with $E = E_R(k)$ the injective hull of $k = R/\mathfrak {m}$, its residue field. Let $M$ denote a finitely generated $R$-module. By Jensen’s result it follows that $\mathrm {Ext}^1_R(F,M) = 0$ for any flat $R$-module $F$ if and only if $M$ is complete. Let $\underline {x} = x_1,\ldots ,x_r$ be a system of elements of $R$ such that $\mathrm {Rad}\underline {x}R = \mathfrak {m}$. In the main result it is shown that the vanishing of $\mathrm {Ext}_R^1(F,M), F = \bigoplus _{i = 1}^r R_{x_i},$ implies that $M$ is complete. It is known from work of Enochs and Jenda that $\mathrm {Hom}_R(E_R(R/\mathfrak {p}), E) \simeq \widehat {R_{\mathfrak {p}}^{\mu _{\mathfrak {p}}}}$ for a certain finite or infinite number $\mu _{\mathfrak {p}}$. We discuss which $\mu _{\mathfrak {p}}$ might occur for certain primes with $\dim R/\mathfrak {p} = 1$.