| de Rham Cohomology of Differential Modules on Algebraic Varieties 2001 Edition Contributor(s): André, Yves (Author), Baldassarri, Francesco (Author) |
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ISBN: 3764363487 ISBN-13: 9783764363482 Publisher: Birkhauser OUR PRICE: $104.49 Product Type: Hardcover - Other Formats Published: December 2000 Annotation: The book offers a systematic treatment of the theory of differential modules on algebraic varieties over a field of characteristic 0. Its final purpose is to give a proof of a conjecture of Baldassarri comparing the algebraic and p-adic analytic De Rham cohomologies of such a module. Along the way, the authors present a purely algebraic treatment of the theory of regularity and irregularity in several variables, give original elementary proofs of the main results on De Rham cohomology of differential modules, and then develop a new approach to the classical algebraic/analytic comparison theorems (concerning regular modules) which unifies the complex and p-adic situations and avoids resolution of singularities. The main results should be of interest to arithmetic-algebraic geometers. The methods should be of interest to specialists of D-modules. On the other hand, the greater part of the book can be used as an introduction to the subject and should be accessible to non-specialists and graduate students with a background in algebraic geometry. |
| Additional Information |
| BISAC Categories: - Mathematics | Topology - General - Medical - Mathematics | Geometry - General |
| Dewey: 514.23 |
| LCCN: 00044433 |
| Series: Progress in Mathematics |
| Physical Information: 0.56" H x 6.14" W x 9.21" (1.10 lbs) 214 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: This is a study of algebraic differential modules in several variables, and of some of their relations with analytic differential modules. Let us explain its source. The idea of computing the cohomology of a manifold, in particular its Betti numbers, by means of differential forms goes back to E. Cartan and G. De Rham. In the case of a smooth complex algebraic variety X, there are three variants: i) using the De Rham complex of algebraic differential forms on X, ii) using the De Rham complex of holomorphic differential forms on the analytic an manifold X underlying X, iii) using the De Rham complex of Coo complex differential forms on the differ- entiable manifold Xdlf underlying Xan. These variants tum out to be equivalent. Namely, one has canonical isomorphisms of hypercohomology: While the second isomorphism is a simple sheaf-theoretic consequence of the Poincare lemma, which identifies both vector spaces with the complex cohomology H (XtoP, C) of the topological space underlying X, the first isomorphism is a deeper result of A. Grothendieck, which shows in particular that the Betti numbers can be computed algebraically. This result has been generalized by P. Deligne to the case of nonconstant coeffi- cients: for any algebraic vector bundle .M on X endowed with an integrable regular connection, one has canonical isomorphisms The notion of regular connection is a higher dimensional generalization of the classical notion of fuchsian differential equations (only regular singularities). |