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This monograph tells the story of a philosophy of J-P. Serre and his vision of relating that philosophy to problems in affine algebraic geometry. It gives a lucid presentation of the Quillen-Suslin theorem settling Serre's conjecture. The central topic of the book is the question of whether a curve in $n$-space is as a set an intersection of $(n-1)$ hypersurfaces, depicted by the central theorems of Ferrand, Szpiro, Cowsik-Nori, Mohan Kumar, Boratýnski. The book gives a comprehensive introduction to basic commutative algebra, together with the related methods from homological algebra, which will enable students who know only the fundamentals of algebra to enjoy the power of using these tools. At the same time, it also serves as a valuable reference for the research specialist and as potential course material, because the authors present, for the first time in book form, an approach here that is an intermix of classical algebraic K-theory and complete intersection techniques, making connections with the famous results of Forster-Swan and Eisenbud-Evans. A study of projective modules and their connections with topological vector bundles in a form due to Vaserstein is included. Important subsidiary results appear in the copious exercises. Even this advanced material, presented comprehensively, keeps in mind the young student as potential reader besides the specialists of the subject.

Projective modules (Algebra) --- Ideals (Algebra) --- Commutative rings. --- Generators. --- Rings (Algebra) --- Generators of ideals (Algebra) --- Modules (Algebra)

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Ordered algebraic structures --- Lie algebras --- Projective modules (Algebra) --- Modules (Algebra) --- Algebras, Lie --- Algebra, Abstract --- Algebras, Linear --- Lie groups --- Lie, Algèbres de --- Modules projectifs (algèbre) --- Lie, Algèbres de.

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Group theory --- Ordered algebraic structures --- Discrete mathematics --- Associative rings. --- Group theory. --- Projective modules (Algebra) --- Trees (Graph theory) --- Projective modules (Algebra). --- Trees (Graph theory). --- Associative rings --- Anneaux associatifs --- Théorie des groupes --- Arbres (Théorie des graphes) --- Théorie des groupes --- Arbres (Théorie des graphes) --- Modules projectifs (algèbre) --- Groupes, Théorie des --- Groupes, Théorie des --- Algebres et anneaux associatifs --- Ideaux et modules --- Modules projectifs (algèbre)

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Extending modules are generalizations of injective modules and, dually, lifting modules generalize projective supplemented modules. There is a certain asymmetry in this duality. While the theory of extending modules is well documented in monographs and text books, the purpose of our monograph is to provide a thorough study of supplements and projectivity conditions needed to investigate classes of modules related to lifting modules. The text begins with an introduction to small submodules, the radical, variations on projectivity, and hollow dimension. The subsequent chapters consider preradicals and torsion theories (in particular related to small modules), decompositions of modules (including the exchange property and local semi-T-nilpotency), supplements in modules (with specific emphasis on semilocal endomorphism rings), finishing with a long chapter on lifting modules, leading up their use in the theory of perfect rings, Harada rings, and quasi-Frobenius rings. Most of the material in the monograph appears in book form for the first time. The main text is augmented by a plentiful supply of exercises together with comments on further related material and on how the theory has evolved.

Modules (Algebra) --- Moduli theory. --- Projective modules (Algebra) --- Ideals (Algebra) --- Modules (Algèbre) --- Variétés topologiques à 4 dimensions --- Idéaux (Algèbre) --- Ideals (Algebra). --- Modules (Algebra). --- Projective modules (Algebra). --- Moduli theory --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Algebraic ideals --- Theory of moduli --- Finite number systems --- Modular systems (Algebra) --- Mathematics. --- Algebra. --- Mathematical analysis --- Math --- Science --- Algebraic fields --- Rings (Algebra) --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Finite groups

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Ordered algebraic structures --- Projective modules (Algebra) --- Intersection theory. --- Intersection theory (Mathematics) --- Mathematical Theory --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Intersecties [Theorie van ] --- Intersection theory --- Intersections [Théorie des ] --- Modules [Projective ] (Algebra) --- Theorie van intersecties --- Théorie des intersections --- Commutative algebra. --- Commutative rings. --- Commutative Rings and Algebras. --- Rings (Algebra)