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

By Nilolaus Vonessen

ISBN-10: 0821824775

ISBN-13: 9780821824771

Publication via Vonessen, Nilolaus

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Clearly TG — T. Since TR is a finite T-module, it follows that TR is in fact a finite ^-module. Hence also R is a finite jR^-module and is therefore left Noetherian. 1. 9 PROPOSITION. Let R be an affine prime Pl-algebra1 and let G be a linearly reductive group acting rationally on R. Suppose that the center ofR is pointwise fixed and that RG is left Noetherian. Then R is a finite left RG-module. In particular, both R and RG are two-sided Noetherianf and RG is affine. 4). 3. But now back to the question whether the fixed ring is affine if it is Noetherian.

Then and p° = (ia )fl /(Jinia )°=^ ("^oew. It follows that $(P) = {0 © k[as], A; © (x)}. Here the first prime is minimal, but the second prime is not. • For later use, we include here the following lemma which is a consequence of equidimensionality. It will turn out to be a very powerful tool in proofs. 1. 17(a). 8 LEMMA. Let G be a reductive group acting rationally on an affine Pi-algebra R. Suppose that R is a finite module over a G-invariant central subalgebra C of R. Then ifp G $(P) , pH CG = P n CG.

Let R be an affine Pi-algebra with a group G of automorphisms. Let Pi and P2 be prime ideals of R such that $ ( i \ ) H $(^2) 7^ $• Suppose that (a) Pi and P2 iave the same Pi-degree d, (b) Pi-GC) P2'G is non-empty and contains a prime ideal of Pi-degree d, (c) RG is Artinian. Then *(Px) = $(P 2 ). Here Pv-G denotes the closure of the orbit of P„ in Specil. Suppose $(P 2 ) 2 *(-Pi). Then *(Pi) ^ SpecR G , so that Pifli2 G is not nilpotent. Moreover, y/Px n RG ^ y/P2 H RG. 6, PiC\RG contains a non-zero idempotent e which is not contained in P 2 D RG.

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

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