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.
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L.M. was supported by NSF Grant DMS 1901672, NSF FRG Grant DMS 1952366, and by a fellowship from the Sloan Foundation, A.K.S. by NSF Grant DMS 1801285, and U.W. by the Simons Foundation Collaboration Grant for Mathematicians 580839. A.K.S. thanks Purdue University and his coauthors for their hospitality. The authors are grateful to Srikanth B. Iyengar and to the referee for several helpful comments.
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Ma, L., Singh, A.K. & Walther, U. Koszul and local cohomology, and a question of Dutta. Math. Z. 298, 697–711 (2021). https://doi.org/10.1007/s00209-020-02619-0
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DOI: https://doi.org/10.1007/s00209-020-02619-0