On the surjectivity criterion for Buchsbaum modules

Abstract:

Let $R$ be a Cohen-Macaulay local ring with maximal ideal $m$ and suppose that $\dim R \geq 2$. Then $R$ is regular if (and only if) for any Buchsbaum $R$-module $M$ and for any integer $i,i \ne {\dim _R}M$, the canonical map ${\text {Ext}}_R^i\left ( {R/m,M} \right ) \to H_m^i\left ( M \right ): = \lim _{\substack {\to \\n}} \mathrm {Ext}_R^i \left (R/m^n, M \right )$ is surjective. The hypothesis that $R$ is Cohen-Macaulay is not superfluous. Two examples are given.