Koszul and local cohomology, and a question of Dutta

Abstract

For a local ring $(A,\mathfrak{m})$ of dimension $n$, we study the natural map from the Koszul cohomology module $H^n(\mathfrak{m};A)$ to the local cohomology module $H_{\mathfrak{m}}^n(A)$. We prove that the injectivity of this map characterizes the Cohen-Macaulay property of the ring $A$. We also answer a question of Dutta by constructing normal rings $A$ for which this map is zero.