Der kanonische Modul eines Invariantenrings. (The canonical module of a ring of invariants). (German) Zbl 0716.20021

Summary: We calculate the canonical module of a ring of invariants R of a reductive group acting on an affine variety. From that we derive a criterion for the Gorenstein property to hold for R. In case R is the ring of invariants of a graded ring we obtain lower and upper bounds for the degree of the generating function of R.


20G05 Representation theory for linear algebraic groups
14L30 Group actions on varieties or schemes (quotients)
14L24 Geometric invariant theory
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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[1] Boutot, J.-F, Singularités rationelles et quotients par LES groupes réductifs, Invent. math., 88, 65-68, (1987) · Zbl 0619.14029
[2] Hartshorne, R, Stable reflexive sheaves, Math. ann., 254, 121-176, (1980) · Zbl 0431.14004
[3] ()
[4] Hochster, M; Roberts, J, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Adv. in math., 13, 115-175, (1974) · Zbl 0289.14010
[5] Kempf, G, The hochster-Roberts theorem in invariant theory, Michigan math. J., 26, 19-32, (1979) · Zbl 0409.13004
[6] Knop, F, Über die glattheit von quotientenabbildungen, Manuscripta math., 56, 419-427, (1986) · Zbl 0585.14033
[7] Knop, F; Littelmann, P, Der Grad erzeugender funktionen von invariantenringen, Math. Z., 196, 211-231, (1987) · Zbl 0635.20017
[8] Luna, D, Slices étales, Bull. soc. math. France, 33, 81-105, (1973) · Zbl 0286.14014
[9] Luna, D; Richardson, R.W, A generalization of the Chevalley restriction theorem, Duke math. J., 46, 487-496, (1976) · Zbl 0444.14010
[10] Popov, V.L, The constructive theory of invariants, Math. USSR-izv., 19, 359-376, (1982) · Zbl 0501.14006
[11] Raynaud, M; Gruson, L, Critères de platitude et de projectivité. techniques de “platification” d’un module, Invent. math., 13, 1-89, (1971) · Zbl 0227.14010
[12] Stanley, R, Hilbert functions of graded algebras, Adv. in math., 28, 57-83, (1978) · Zbl 0384.13012
[13] Stanley, R, Combinatorics and commutative algebras, ()
[14] Watanabe, K, Certain invariant subrings are Gorenstein, I. Osaka J. math., 11, 1-8, (1974) · Zbl 0281.13007
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