Grosshans, Frank D. Contractions of the actions of reductive algebraic groups in arbitrary characteristic. (English) Zbl 0778.20018 Invent. Math. 107, No. 1, 127-133 (1992). Let \(k\) be an algebraically closed field, and \(A\) a commutative \(k\)- algebra. Let \(G\) be a reductive algebraic group which acts rationally on \(A\), and \(B=TU\) a Borel subgroup of \(G\), and \(B^ -=TU^ -\) the Borel subgroup of \(G\) opposite to \(B\). Denote by \(A^ U\) the set of fixed points of \(A\) under the action of \(U\). The author of this paper relates properties of \(A\) to those of \(A^ U\). The author also constructs a \(G\)- action on \(R=(A^ U \otimes_ k k[G/U^ -])^ T\), and shows that there is a graded algebra \(\text{gr }A\) and an injective \(G\)-equivariant algebra homomorphism \(\Phi:\text{gr }A\to R\). Moreover, the action of \(G\) on \(\text{Spec}(A)\) is a flat deformation of the action of \(G\) on \(\text{Spec(gr }A)\), and \(\Phi\) is an isomorphism if and only if \(A\) has a good filtration. Reviewer: Li Fuan (Beijing) Cited in 1 ReviewCited in 16 Documents MSC: 20G15 Linear algebraic groups over arbitrary fields 14L30 Group actions on varieties or schemes (quotients) 13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 20G05 Representation theory for linear algebraic groups Keywords:commutative \(k\)-algebra; reductive algebraic group; Borel subgroup; \(G\)- action; graded algebra; \(G\)-equivariant algebra homomorphism; flat deformation; good filtration PDF BibTeX XML Cite \textit{F. D. Grosshans}, Invent. Math. 107, No. 1, 127--133 (1992; Zbl 0778.20018) Full Text: DOI EuDML References: [1] Donkin, S.: Invariants of unipotent radicals. Math. Z.198, 117-125 (1988) · Zbl 0627.14013 · doi:10.1007/BF01183043 [2] Donkin, S.: Rational representations of algebraic groups: Tensor products and filtrations. (Lect. Notes Math., vol. 1140) Berlin Heidelberg New York: Springer 1985 · Zbl 0586.20017 [3] Grosshans, F.: Finitely generated rings of invariants having rational singularities. (Can. Math. Soc. Conf. Proc., vol. 10, pp. 53-60) Providence: Am. Math Soc. 1989 · Zbl 0716.14004 [4] Grosshans, F.: The invariants of unipotent radicals of parabolic subgroups. Invent. Math.73, 1-9 (1983) · Zbl 0511.14006 · doi:10.1007/BF01393822 [5] Jantzen, J.C.: Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne. J. Reine Angew. Math.317, 157-199 (1980) · Zbl 0451.20040 · doi:10.1515/crll.1980.317.157 [6] Matsumura, H.: Commutative ring theory. (Camb. Stud. Adv. Math., vol. 8) Cambridge: Cambridge University Press 1986 · Zbl 0603.13001 [7] Nagata, M.: Lectures on the fourteenth problem of Hilbert. Bombay: Tata Institute 1965 · Zbl 0182.54101 [8] Newstead, P.E.: Introduction to moduli problems and orbit spaces. Bombay: Tata Institute 1978 · Zbl 0411.14003 [9] Popov, V.L.: Contraction of the actions of reductive algebraic groups. Math. USSR Sb.58 (2), 311-335 (1987) · Zbl 0627.14033 · doi:10.1070/SM1987v058n02ABEH003106 [10] Zariski, O. and Samuel, P.: Commutative algebra, vol. II. Princeton: Van Nostrand 1960 · Zbl 0121.27801 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.