On Proofs and Some Consequences of Serre’s Conjecture on Projective Modules <br/>

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On Proofs and Some Consequences of Serre’s Conjecture on Projective Modules

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 42525 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Jafari, Amir
Abstract:
Jean-Pierre Serre in 1955, in his famous article “Faisceaux Algébriques Cohérents” asked if there is any finitely generated projective module on polynomial rings which is not free. Or equivalently: is there any non-trivial algebraic vector bundle over an affine space? The negative answer to this question is known as Serre’s conjecture and it was an open problem in algebra and affine algebraic geometry until 1976, when it was proved independently by Quillen and Suslin. The challenge of solving this question and its consequences led to a vast amount of research in commutative algebra, homological algebra and algebraic K-theory and inspired generalizations and similar questions. In this thesis we review Serre’s study of algebraic vector bundles and coherent algebraic sheaves that resulted in this question. We also investigate a proof of Serre’s conjecture and at the end we take a look at Hyman Bass’s study of non-finitely generated projective modules
Keywords:
Vector Bundle ; Skew Polynomial Ring ; Algebraic Sheaves ; Hermite Rings ; Projective Module

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