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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).

Modules (Algebra) --- Differential algebra --- Arithmetical algebraic geometry --- Géométrie algébrique arithmétique --- Geometry. --- Mathematics --- Euclid's Elements --- Géométrie algébrique arithmétique. --- Geometrie algebrique --- Cohomologie

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Algebraic geometry --- Geometry, Algebraic --- Commutative algebra --- 512.7 --- Geometry --- Algebra --- Algebraic geometry. Commutative rings and algebras --- Commutative algebra. --- Geometry, Algebraic. --- 512.7 Algebraic geometry. Commutative rings and algebras --- Géométrie algébrique

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Algebraic geometry --- Ordered algebraic structures --- Geometry, Algebraic --- Noncommutative algebras --- Representations of algebras --- Géométrie algébrique --- Algèbres non commutatives --- Représentations d'algèbres --- Géométrie algébrique --- Algèbres non commutatives --- Représentations d'algèbres

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Logic --- Algebraic geometry --- Categories (Mathematics) --- Geometry, Algebraic --- Logic, Symbolic and mathematical --- 510.6 --- #WWIS:ALTO --- Mathematical logic --- Toposes. --- Congresses. --- 510.6 Mathematical logic --- Toposes --- Catégories (Mathématiques) --- Géométrie algébrique --- Logique symbolique et mathématique --- Congresses --- Congrès --- Catégories (mathématiques) --- Logique mathématique --- Intuitionnisme --- Categories (Mathematics) - Congresses --- Geometry, Algebraic - Congresses --- Logic, Symbolic and mathematical - Congresses --- Geometrie algebrique

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The purpose of this unique book is to establish purely algebraic foundations for the development of certain parts of topology. Some topologists seek to understand geometric properties of solutions to finite systems of equations or inequalities and configurations which in some sense actually occur in the real world. Others study spaces constructed more abstractly using infinite limit processes. Their goal is to determine just how similar or different these abstract spaces are from those which are finitely described. However, as topology is usually taught, even the first, more concrete type of problem is approached using the language and methods of the second type. Professor Brumfiel's thesis is that this is unnecessary and, in fact, misleading philosophically. He develops a type of algebra, partially ordered rings, in which it makes sense to talk about solutions of equations and inequalities and to compare geometrically the resulting spaces. The importance of this approach is primarily that it clarifies the sort of geometrical questions one wants to ask and answer about those spaces which might have physical significance.

Categories (Mathematics) --- Commutative rings. --- Rings (Algebra) --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Commutative rings --- 512.7 --- 512.7 Algebraic geometry. Commutative rings and algebras --- Algebraic geometry. Commutative rings and algebras --- Categories (Mathematics). --- Ordered algebraic structures --- Algebraic geometry --- Anneaux commutatifs --- Catégories (Mathématiques) --- Anneaux (algèbre) --- Géométrie algébrique --- Structures algébriques ordonnées