Download e-book for iPad: De Rham Cohomology of Differential Modules on Algebraic by Yves André

By Yves André

ISBN-10: 3034883366

ISBN-13: 9783034883368

ISBN-10: 3034895224

ISBN-13: 9783034895224

This is a research of algebraic differential modules in different variables, and of a few in their kinfolk with analytic differential modules. allow us to clarify its resource. the belief of computing the cohomology of a manifold, particularly its Betti numbers, through differential varieties is going again to E. Cartan and G. De Rham. with regards to a tender advanced algebraic kind X, there are 3 versions: i) utilizing the De Rham complicated of algebraic differential varieties on X, ii) utilizing the De Rham complicated of holomorphic differential kinds at the analytic an manifold X underlying X, iii) utilizing the De Rham complicated of Coo complicated differential types at the vary­ entiable manifold Xdlf underlying Xan. those versions tum out to be similar. specifically, one has canonical isomorphisms of hypercohomology: whereas the second one isomorphism is a straightforward sheaf-theoretic end result of the Poincare lemma, which identifies either vector areas with the advanced cohomology H (XtoP, C) of the topological area underlying X, the 1st isomorphism is a deeper results of A. Grothendieck, which exhibits particularly that the Betti numbers should be computed algebraically. This outcome has been generalized via P. Deligne to the case of nonconstant coeffi­ cients: for any algebraic vector package .M on X endowed with an integrable commonplace connection, one has canonical isomorphisms The concept of standard connection is the next dimensional generalization of the classical inspiration of fuchsian differential equations (only typical singularities).

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Extra resources for De Rham Cohomology of Differential Modules on Algebraic Varieties

Example text

Yn). Soit M un K(x, y)-module differentiel integrable libre de rang s. Consicterons Y comme des parametres et M comme un K(y)(x)-module differentiel. Alors Theoreme: Soit a independants de y. E K. Les exposants de ce module differentiel en a sont Demonstration. Prenons a = O. Soil F = (::) une b",e de M. Les motrires de derivation sont o -F=HF ox ou H, G i E o -F=Gi F °Yi ' Mat (s, K (x, y)). L' hypothese que M est integrable se traduit par o 0 - H +HG' = -G· +G ·H 0Yi I ox I I pour I :s i :s n.

O (ei is the valuation of the image in Oc,Q of a local equation of Zi in Ox"p). 37 Regularity in several variables Now V induces a composite map The map Ui is nothing but the specialization of Reszj V at P, and v 0 U is nothing but ResQh*V. Since the Reszj V mutually commute, the endomorphisms v 0 Ui = eiUi of the finite dimensional K = K{Q)-vector space (h*£)Q ® K{Q) also commute and the eigenvalues of Li v 0 Ui are of the form Li eiAi, for eigenvalues Ai of Ui. We conclude that EXPQ{h*V) C (L ei Expzj (V)) + z.

1 The exponents (at v) of a regular F I K -differential module £ are the exponents of the induced FIC-differential module (in one variable x) £IC. 3). 2 Expv(£) C K. 2). 4 Exponents of a regular connection along a divisor Let (X, Z) be a model of a divisorially valued function field (F, v). 1». 1 The exponents of V along Z are the exponents of Expz(V) = EXPv(£). £ at v. 2 Let (X, Z)beamodeland(E, V), (EI, V]), (E2, V2)becoherent sheaves with integrable regular connections on X \ Z. Regularity in several variables 35 i) If £ sits in a horizontal exact sequence 0 ----+ £) ----+ £ ----+ £2 ----+ 0, then Expz(V) = Expz(V]) U EXpZ(V2).

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