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To Serre's dismay, this problem quickly became known as Serre's conjecture. (Serre wrote, "I objected as often as I could .") The statement does not immediately follow from the proofs given in the topological or holomorphic case. These cases only guarantee that there is a continuous or holomorphic
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Serre made some progress towards a solution in 1957 when he proved that every finitely generated projective module over a polynomial ring over a field was
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with a finitely generated free module, it became free. The problem remained open until 1976, when
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Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Exposé 23
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later gave a simpler and much shorter proof of the theorem, which can be found in
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561:-modules projectifs de type fini qui ne soient pas libres." Serre,
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Geometrically, finitely generated projective modules over the ring
529:-bundles on affine space are all trivial, this is not true for
688:[Projective modules over polynomial rings are free],
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independently proved the result. Quillen was awarded the
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in 1978 in part for his proof of the Serre conjecture.
30:"Serre's problem" redirects here. For other uses, see
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289:{\displaystyle M\to {\widetilde {M}}}
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296:(Hartshorne II.5, page 110).
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691:Doklady Akademii Nauk SSSR
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682:Suslin, Andrei A.
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428:{\displaystyle k}
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345:{\displaystyle A}
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448:direct sum
161:Background
684:(1976),
574:Lam, p. 1
565:, p. 243.
407:…
281:~
272:→
192:…
766:(2006),
728:(2002),
308:and the
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730:Algebra
704:0469905
675:0427303
642:0177011
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484:regular
472:Algebra
356:over a
156:History
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