By Nilolaus Vonessen
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Additional info for Actions of Linearly Reductive Groups on Affine Pi Algebras
This is contained in the last result of this paragraph. 14 PROPOSITION. Let R be a semiprime affine (left) Noetherian Pi-algebra over k which is Unite over its center C, and let G be a reductive group over k acting rationally on R. 2). Then the total ring of fractions Q(RG) ofRG exists, is Artinian, and is contained in Q(R). Let S(CG) be the Small set of CG. Then S(CG) consists actually of regular elements of R. As a consequence, Q(RG) = S(CG)~1RG and is thus a localization of RG at regular central fixed points.
The action of G on S extends componentwise to a rational action on R by fixing x and y. The fixed NlKOLAUS VONESSEN 42 ring is G _ fk[z] V 0 k[x,y]\ * M >/ ' The set S of regular elements of RG consists of all matrices in RG with non-zero entries on the diagonal. It is also the Small set of RG. It satisfies neither Ore condition: For example, ( ^ ) S n (*0\)RG = 0 and S ( ° J) n i? G (J°) = 0. 9 can be improved. This is contained in the last result of this paragraph. 14 PROPOSITION. Let R be a semiprime affine (left) Noetherian Pi-algebra over k which is Unite over its center C, and let G be a reductive group over k acting rationally on R.
6, it is sufficient to show that A/NA is affine. Since NB D i i s a nilpotent ideal of A, it is contained in N^. And since the extension A C B is centralizing, the ideal generated by NA in B is nilpotent. Hence NA C iV#, in fact NA = NB H A. So A/NA embeds into B/NB, and this ring extension satisfies the hypotheses of the lemma. So one may assume that both A and B are semiprime. ACTIONS ON PI-ALGEBRAS 29 Let C denote the conductor of B in A. Since B is Noetherian, it has a finite number of minimal prime ideals.