Über die Glattheit von Quotientenabbildungen. (On the smoothness of quotient maps)
type: Item
instance of: scholarly article
title: Über die Glattheit von Quotientenabbildungen. (On the smoothness of quotient maps)
zbMATH Open document ID: 0585.14033
DOI: 10.1007/BF01168503
author: Friedrich Knop
published in: Manuscripta Mathematica
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155179
review text: Let G be a semisimple algebraic group acting on a factorial Gorenstein algebra S defined over a field of characteristic 0. Let \(X:=Spec S^ G\) and \(\pi: X\to Y\) be the quotient map. The main results are: 1. Let x be a smooth point of X whose orbit has maximal dimension and such that \(\pi\) (x) is a smooth point of Y. Then \(\pi\) is smooth at x. 2. Let S be positively graded and let \(\chi_ S(t)\) be its generating function which is the power series expansion of a rational function. Then deg \(\chi\) \({}_ S\leq \deg \chi_ SG\). This answers a conjecture of \textit{V. L. Popov} [Math. USSSR, Izv. 20, 333-354 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, 347-370 (1982; Zbl 0547.20034)].
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14B25
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 20G35
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 3937335
zbMATH Keywords: smoothness of quotient maps
zbMATH Keywords: semisimple algebraic group acting on a factorial Gorenstein algebra
MaRDI profile type: Publication
cites work: Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay
cites work: Q3686804
cites work: Q4774141
cites work: A note on factorial rings
cites work: REGULAR ELEMENTS IN SPACES OF LINEAR REPRESENTATIONS OF REDUCTIVE ALGEBRAIC GROUPS
cites work: A FINITENESS THEOREM FOR REPRESENTATIONS WITH A FREE ALGEBRA OF INVARIANTS
cites work: Hilbert functions of graded algebras
description: scientific article; zbMATH DE number 3937335