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On the de Rham homology and cohomology of a complete local ring in equicharacteristic zero

Part of: (Co)homology theory Commutative algebra: Homological methods

Published online by Cambridge University Press:  07 August 2017

Nicholas Switala
Nicholas Switala*
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 SEO (M/C 249), 851 S. Morgan Street, Chicago, IL 60607, USA email nswitala@uic.edu
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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 $R\rightarrow A$ 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 $E_{2}$-terms, are independent of the chosen surjection (up to a degree shift in the homology case) and consist of finite-dimensional $k$-spaces. These $E_{2}$-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 ${\mathcal{D}}$-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 ${\mathcal{D}}={\mathcal{D}}(R,k)$ is the ring of $k$-linear differential operators on $R$, then the Matlis dual $D(M)$ of any left ${\mathcal{D}}$-module $M$ can again be given a structure of left ${\mathcal{D}}$-module, and if $M$ is a holonomic ${\mathcal{D}}$-module, then the de Rham cohomology spaces of $D(M)$ are $k$-dual to those of $M$.

Keywords

local cohomologyalgebraic de Rham cohomologyMatlis dualityD-modules

MSC classification

Primary: 13D45: Local cohomology 14F40: de Rham cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 10 , October 2017 , pp. 2075 - 2146
Copyright
© The Author 2017 

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