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On the de Rham homology and cohomology of a complete local ring in equicharacteristic zero
Published online by Cambridge University Press: 07 August 2017
Abstract
Let A be a complete local ring with a coefficient field k of characteristic zero, and let Y be its spectrum. The de Rham homology and cohomology of Y have been defined by R. Hartshorne using a choice of surjection R→A where R is a complete regular local k-algebra: the resulting objects are independent of the chosen surjection. We prove that the Hodge–de Rham spectral sequences abutting to the de Rham homology and cohomology of Y, beginning with their E2-terms, are independent of the chosen surjection (up to a degree shift in the homology case) and consist of finite-dimensional k-spaces. These E2-terms therefore provide invariants of A analogous to the Lyubeznik numbers. As part of our proofs we develop a theory of Matlis duality in relation to D-modules that is of independent interest. Some of the highlights of this theory are that if R is a complete regular local ring containing k and D=D(R,k) is the ring of k-linear differential operators on R, then the Matlis dual D(M) of any left D-module M can again be given a structure of left D-module, and if M is a holonomic D-module, then the de Rham cohomology spaces of D(M) are k-dual to those of M.
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- Research Article
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- © The Author 2017
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