De Rham \(\mathcal E\)-factors
type: Item
instance of: scholarly article
title: De Rham \(\mathcal E\)-factors
zbMATH Open document ID: 1275.14019
author: Deepam Patel
published in: Inventiones Mathematicae
publication date: 2013-01-14T00:00:00Z
review text: Let \(X\) be a smooth projective variety over a field \(k\) of characteristic 0, \({\mathcal M}\) a \({\mathcal D}_X\)-module. The determinant of the de Rham cohomology yields a homotopy point \([R\Gamma_{\mathrm{dR}}(X, sM)]\) of the \(K\)-theory spectrum \(K(k)\). This article contains a micro-local description of the homotopy point \([R\Gamma_{\mathrm{dR}}(X,{\mathcal M})]\). Let \(S= SS({\mathcal M})\) denote the singular support of \({\mathcal M}\), contained in the cotangent bundle \(T^*(X)\). Given a subvariety \(Y\subset X\) and a 1-form \(\nu\) on \(X-Y\), taking values in the complement of \(S\), one constructs a natural homotopy point \({\mathcal E}_{\nu,Y}({\mathcal M})\) of \(K(k)\). The homotopy point \({\mathcal E}_{\nu,Y}({\mathcal M})\) has local nature and is determined by restrictions of \({\mathcal M}\) and \(\nu\) on any open neighborhood of \(Y\). There is also a natural identification of the homotopy points \({\mathcal E}_{\nu,Y}({\mathcal M})\) and \([R\Gamma_{\mathrm{dR}}(X, sM)]\). Passing to determinants gives a graded line \(\varepsilon_{\nu,Y}({\mathcal M})= \det({\mathcal E}_{\nu,Y}({\mathcal M}))\). If \(Y\) is a disjoint union of finetely many \(Y_i\) then \[ \det(R\Gamma_{\mathrm{dR}}(X,{\mathcal M}))= \bigoplus_i \varepsilon_{\nu, Y_i}({\mathcal M}). \] The construction of \({\mathcal E}\)-factors follow from a micro-local \(K\)-theoretic animation of the characteristic cycle \(CC({\mathcal M})\) of \({\mathcal M}\). Using the results of Quillen on the morphism of spectra \(K({\mathcal D}_X)K(T^*_X)\). The author constructs a lifting of this morphism to \(K\)-theory with support in \(S\). By doing so one gets a natural homotopy morphism \(\mathrm{gr}_S: K_S({\mathcal D}_X)\to K_S(T^*_X)\). The induced morphism \(\mathrm{gr}: K_{0,S}({\mathcal D}_X)\to K_{0,S}(T^*_X)\) is given by sending \({\mathcal M}\) to the associated graded with respect to some good filtration. The image in the Chow groups is precisely \(CC({\mathcal M})\). The…
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 19D99
zbMATH DE Number: 6125038
reviewed by: Claudio Pedrini
MaRDI profile type: Publication
full work available at URL: s00222-012-0381-8
OpenAlex ID: W1974916171
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DOI: 10.1007/S00222-012-0381-8
description: scientific article; zbMATH DE number 6125038