Download Actions of Linearly Reductive Groups on Affine Pi Algebras by Nilolaus Vonessen PDF

By Nilolaus Vonessen

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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. 9, namely that MinRG = $(Min R). 15 below. 9 are satisfied.

Then RG = k[x] ^ k[x]. Note that x is regular in RG, but that x is a zero divisor in R. Hence the field of fractions of RG does not embed in the total ring of fractions of R. Moreover, note that the minimal prime ideals of R are Pi = xR and Pi — yR. But Pi n RG = xRG is not a minimal prime of RG. Concerning Gelfand-Kirillov dimension, note that GK(RG/(P1 n RG)) = 0 but GK(RG/(P2 n RG)) = 1. Finally, let us interpret the situation geometrically. The spectrum V of R is the union of the X- and Y-coordinate axes, and the spectrum W of RG is the X-axis.

Replacing R by R/I> we may therefore assume that the conductor C of the trace ring TR in R contains a non-zero fixed point. Now we are done by the previous paragraph. 10(a) and (b). 12 REMARKS. 1. 10 a little bit farther. , that R is reduced modulo its nil radical. 6. 10(b) to assume that the nil radical of R is prime. 2. Suppose that the characteristic of the ground field k is zero. 10(b) that G be connected: Let R be an afRne PI-algebraf and let G be a linearly reductive group acting rationally on R such that the fixed ring RG is left Noetherian.

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