De Rham \(\mathcal E\)-factors
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 \(CC({\mathcal M})\). Therefore the homotopy point \(\mathrm{gr}_S([{\mathcal M}])\) may be viewed as a micro-local \(K\)-theory animation of \(CC({\mathcal M})\). In the case of curves Deligne gave a construction of the geometric \(\varepsilon\)-factorization in the de Rham setting. \textit{A Beilinson} [in: Motives and algebraic cycles. A celebration in honour of Spencer J. Bloch. Providence, RI: American Mathematical Society (AMS); Toronto: The Fields Institute for Research in Mathematical Sciences. Fields Institute Communications 56, 15--82 (2009; Zbl 1186.14019)] showed that the de Rham factors form an \(\varepsilon\)-factorization theory. The construction given in this article is a generalization of the previous theory to the higher-dimensional case.
scientific article; zbMATH DE number 6125038