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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)].

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