Koszul and local cohomology, and a question of Dutta

Abstract

For $$(A,\mathfrak {m})$$ a local ring, we study the natural map from the Koszul cohomology module $$H^{\dim A}(\mathfrak {m};\,A)$$ to the local cohomology module $$H^{\dim A}_\mathfrak {m}(A)$$ . 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.