The equivariant Serre problem for abelian groups.(English)Zbl 0884.14007

Let $$G$$ be a reductive algebraic group over $$\mathbb{C}$$, $$G^{0}$$ the connected component of the identity, $$B$$ a finite dimensional rational $$G$$-module. The main result of this paper is the following:
If $$G$$ is abelian then any $$G$$-vector bundle on $$B$$ is trivial.
This is an equivariant version of a celebrated theorem of Quillen and Suslin on a conjecture of Serre. It was shown by F. Knop, Invent. Math. 105, No. 1, 217-220 (1991; Zbl 0739.20019) that, if $$G^{0}$$ is not abelian, then there exists a $$G$$-module $$B$$ supporting non-trivial $$G$$-bundles. To prove the main result, the authors relate it to the quotient problem: Is any vector bundle over the quotient $$B // G$$ trivial? A positive answer for this last problem in case $$G$$ abelian follows from a theorem by J. Gubeladze [Math. USSR, Sb. 63, No. 1, 165-180 (1989); translation from Mat. Sb., Nov. Ser. 135(177), No. 2, 169-185 (1988; Zbl 0668.13011)] and also by R. G. Swan [in: Azumaya algebras, actions, and modules, Proc. Conf. Hon. Azumaya’s 70th birthday, Bloomington 1990, Contemp. Math. 124, 215-250 (1992; Zbl 0742.13005)]. A further ingredient of the proof is a key result by M. Masuda and T. Petrie [J. Am. Math. Soc. 8, No. 3, 687-714 (1995; Zbl 0862.14009)].

MSC:

 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14L30 Group actions on varieties or schemes (quotients)
Full Text: