TY - BOOK ID - 6839008 TI - De Rham cohomology of differential modules on algebraic varieties AU - André, Yves AU - Baldassarri, Francesco PY - 2001 VL - 189 SN - 3764363487 3034895224 3034883366 9783764363482 PB - Basel ; Berlin ; Boston : Birkhäuser, DB - UniCat KW - Modules (Algebra) KW - Differential algebra KW - Arithmetical algebraic geometry KW - Géométrie algébrique arithmétique KW - Geometry. KW - Mathematics KW - Euclid's Elements KW - Géométrie algébrique arithmétique. KW - Geometrie algebrique KW - Cohomologie KW - Géométrie algébrique arithmétique. UR - https://www.unicat.be/uniCat?func=search&query=sysid:6839008 AB - 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). ER -