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

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By Nilolaus Vonessen

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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.

PROOF. Denote by Z the center of P, and by K the total ring of fractions of Z. 7]. Denote by KA the subalgebra of KB generated by K and A. Then KA is a finite IT-module and as such Artinian. Let Pi, . . , Pn be the prime ideals of KA, Then their intersection N is the nil radical of KA and is nilpotent. Since A C KA is a centralizing extension, the Pi O A are prime ideals of A. And since Pi • • • Pn C N is nilpotent, every prime ideal of A contains one of the Pi D A. It follows that A has only a finite number of minimal prime ideals.

P n be the minimal prime ideals of R. Suppose that GK(P G /(P; fl RG)) is independent of i. Denote this number by d. 2]. Equidimensionality implies that for all primes p in $(Pi) U • • • U $(Pn), GK(RG/p) = d. Hence there are no strict inclusion relations among the primes in this set. 9 are satisfied. 3. Let Pi and Pi be prime ideals of R such that $(Pi) and $(P2) have non- NlKOLAUS VONESSEN 40 empty intersection. Let Iv — HOGG ^V9 ar*d suppose that 7 x 0 / 2 = 0- Then 72 is in particular semiprime.

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