Benutzer:Frogfol/Spielwiese/Quillen–Suslin theorem

Der Satz von Quillen–Suslin, auch bekannt als Serre Vermutung, ist eine Theorem der in Kommutativen Algebra über die Beziehung zwischen [freien und projektiven Moduln über Polynomringen. Er sagt aus, dass jeder endlich erzeugte projektiver Modul über einem Polynomring frei ist.

Geometrisch entsprechen endlich erzeugte projektive Moduln Vektorraumbündeln über affinen Räumen. Freie Moduln entsprechen trivialen Vektorraumbündeln. Der affine reelle oder komplexe Raum ist zusammenziehbar, daher hat er keine nicht-trivialen Vektorraumbündel. A simple argument using the exponential exact sequence and the d-bar Poincaré lemma shows that it also admits no non-trivial holomorphic vector bundles.

Jean-Pierre Serre, in his 1955 paper "Faisceaux algébriques cohérents", remarked that the equivalent question was not known for algebraic vector bundles: "It is not known if there exist projective A-modules of finite type which are not free." Here A is a polynomial ring over a field, that is, A = k[x₁, ..., x_n].

To Serre's dismay, this problem quickly became known as Serre's conjecture. (Serre wrote, "I objected as often as I could [to the name].") The statement is not immediately obvious from the topological and holomorphic cases, because these cases only guarantee that there is a continuous or holomorphic trivialization, not an algebraic trivialization. Instead, the problem turns out to be extremely difficult. 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 stably free, meaning that after forming its direct sum with a finitely generated free module, it became free. The problem remained open until 1976, when Daniel Quillen and Andrei Suslin independently proved that the answer was affirmative. Quillen was awarded the Fields Medal in 1978 in part for his proof of the Serre conjecture. Leonid Vaseršteĭn later gave a simpler and much shorter proof of the theorem which can be found in Serge Lang's Algebra.

Einzelnachweise

{{Sortierung: Satz von Quillen-Suslin

Kategorie: Kommutative Algebra Kategorie: Algebraische Geometrie