Masuda, Mikiya; Petrie, Ted Stably trivial equivariant algebraic vector bundles. (English) Zbl 0862.14009 J. Am. Math. Soc. 8, No. 3, 687-714 (1995). In this article, the authors describe a method for constructing non-trivial equivariant algebraic vector bundles over representation spaces for reductive groups (the equivariant Serre problem). The idea is to define invariants, which distinguish bundles, which are explicitly given as subbundles of trivial vector bundles. H. Kraft and G. Schwarz, using another method, also produced many such non-trivial bundles. They considered exclusively the case where the base is a representation with a one-dimensional quotient. The authors do not restrict to this case. They use their method to find the first non-trivial \(G\)-vector bundles over a \(G\)-module, for \(G\) a finite group. For example, they show that the non-trivial \({\mathcal O} (2, \mathbb{C})\)-vector bundles originally given by Schwarz remain non-trivial for large enough dihedral groups. For a certain case, one can choose \(G\) to be the dihedral group of order 14.The equivariant Serre problem is strongly related to the linearity problem, which asks if there are algebraic actions of a reductive group on affine space which are not linearisable. An action is called linearisable if it is conjugate to a linear action. Schwarz’s first examples of non-trivial \(G\)-vector bundles over \(G\)-modules provided the first examples of non-linearisable actions on affine space. In this article, the authors give a new result, which allows them to use their non-trivial \(G\)-vector bundles for finite groups to obtain non-linearisable actions of finite groups on affine space. They prove, for example, that there exist non-linearisable actions of \(D_{10}\), the dihedral group of order 20, on \(\mathbb{C}^4\). Also, there exist infinite families of inequivalent actions of \(D_{18}\) on \(\mathbb{C}^4\). Reviewer: L.Moser-Jauslin (Dijon) Cited in 3 ReviewsCited in 6 Documents MSC: 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14L30 Group actions on varieties or schemes (quotients) 20G20 Linear algebraic groups over the reals, the complexes, the quaternions Keywords:reductive group actions; non-linearizable actions; equivariant algebraic vector bundles; equivariant Serre problem; linearity problem PDFBibTeX XMLCite \textit{M. Masuda} and \textit{T. Petrie}, J. Am. Math. Soc. 8, No. 3, 687--714 (1995; Zbl 0862.14009) Full Text: DOI References: [1] H. Bass and W. Haboush, Linearizing certain reductive group actions, Trans. Amer. Math. Soc. 292 (1985), no. 2, 463 – 482. · Zbl 0602.14047 [2] H. Bass and W. Haboush, Some equivariant \?-theory of affine algebraic group actions, Comm. Algebra 15 (1987), no. 1-2, 181 – 217. · Zbl 0612.14047 [3] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001 [4] Friedrich Knop, Nichtlinearisierbare Operationen halbeinfacher Gruppen auf affinen Räumen, Invent. Math. 105 (1991), no. 1, 217 – 220 (German). · Zbl 0739.20019 [5] Hanspeter Kraft, \?-vector bundles and the linearization problem, Group actions and invariant theory (Montreal, PQ, 1988) CMS Conf. Proc., vol. 10, Amer. Math. Soc., Providence, RI, 1989, pp. 111 – 123. [6] Hanspeter Kraft and Gerald W. Schwarz, Reductive group actions with one-dimensional quotient, Inst. Hautes Études Sci. Publ. Math. 76 (1992), 1 – 97. · Zbl 0783.14026 [7] Mikiya Masuda and Ted Petrie, Equivariant algebraic vector bundles over representations of reductive groups: theory, Proc. Nat. Acad. Sci. U.S.A. 88 (1991), no. 20, 9061 – 9064. , https://doi.org/10.1073/pnas.88.20.9061 Mikiya Masuda, Lucy Moser-Jauslin, and Ted Petrie, Equivariant algebraic vector bundles over representations of reductive groups: applications, Proc. Nat. Acad. Sci. U.S.A. 88 (1991), no. 20, 9065 – 9066. · Zbl 0753.14005 [8] Mikiya Masuda and Ted Petrie, Algebraic families of \?(2)-actions on affine space \?\(^{4}\), Algebraic groups and their generalizations: classical methods (University Park, PA, 1991) Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 347 – 354. · Zbl 0854.22016 [9] Mikiya Masuda and Ted Petrie, Equivariant algebraic vector bundles over representations of reductive groups: theory, Proc. Nat. Acad. Sci. U.S.A. 88 (1991), no. 20, 9061 – 9064. , https://doi.org/10.1073/pnas.88.20.9061 Mikiya Masuda, Lucy Moser-Jauslin, and Ted Petrie, Equivariant algebraic vector bundles over representations of reductive groups: applications, Proc. Nat. Acad. Sci. U.S.A. 88 (1991), no. 20, 9065 – 9066. · Zbl 0753.14005 [10] -, Equivariant algebraic vector bundles for connected reductive groups, preprint. [11] -, Equivariant algebraic vector bundles over cones with smooth one dimensional quotient, preprint. · Zbl 0928.14013 [12] -, The equivariant Serre problem for abelian groups, preprint. · Zbl 0884.14007 [13] Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. · Zbl 0441.13001 [14] Daniel Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167 – 171. · Zbl 0337.13011 [15] Gerald W. Schwarz, Exotic algebraic group actions, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 2, 89 – 94 (English, with French summary). · Zbl 0688.14040 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.