Koszul and local cohomology, and a question of Dutta
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Mathematische Zeitschrift
Koszul and local cohomology, and a question of Dutta Linquan Ma1 · Anurag K. Singh2 · Uli Walther1 Received: 24 February 2020 / Accepted: 23 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract For (A, m) a local ring, we study the natural map from the Koszul cohomology module dim A (A). We prove that the injectivity of H dim A (m; A) to the local cohomology module Hm 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.
1 Introduction For a commutative Noetherian local ring (A, m), we study the natural map from the Koszul dim A (A), and use cohomology module H dim A (m; A) to the local cohomology module Hm this to answer a question raised by Dutta [6], Question 1.1 below. The motivation for Dutta’s question stems from Hochster’s monomial conjecture [10, page 33] that occupies a central place in local algebra; this is the conjecture that if z := z 1 , . . . , z n form a system of parameters for a local ring A, then for each t ∈ N one has / (z 1t+1 , . . . , z nt+1 )A. (z 1 · · · z n )t ∈ An equivalent formulation of the conjecture is that the natural map ϕ zn : H n (z; A) −→ HznA (A),
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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Anurag K. Singh [email protected] Linquan Ma [email protected] Uli Walther [email protected]
1
Department of Mathematics, Purdue University, 150 N University St, West Lafayette, IN 47907, USA
2
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA
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L. Ma et al.
as discussed in §2, is nonzero. The monomial conjecture was proved for rings containing a field by Hochster, in the same paper where it was first formulated. The case of rings of dimension at most two is straightforward; for decades, the conjecture remained unresolved for mixed characteristic rings of dimension greater than or equal to three, as did its equivalent formulations, the direct summand conjecture, the canonical element conjecture, and the improved new intersection conjecture. In [9] Heitmann proved these equivalent conjectures for mixed characteristic rings of dimension three; more recently, André [1] settled the mixed characteristic case in full generality, with Bhatt [2] establishing a derived variant. Related homological conjectures including Auslander’s zerodivisor conjecture and Bass’s conjecture had been settled earlier by Roberts [16]. For the setup of Dutta’s question, let A be a complete local ring. Using the Cohen structure theorem, A can be written as the homomorphic image of a complete regular local ring; this surjection may be facto
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