By Nilolaus Vonessen

Booklet through Vonessen, Nilolaus

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**Extra info for Actions of Linearly Reductive Groups on Affine Pi Algebras**

**Sample text**

The algebra B is also Noetherian, and the ring extension RG C B is finite and centralizing. 3) implies that RG itself is affine. 4. 8. 5 which deals with the case that the Noetherian Pi-algebra R is prime. Let us examine the situation in case that R is not Noetherian. Let R be an affine prime Pi-algebra, and let G be a linearly reductive group acting rationally on R. 4, the action of G extends to a rational action on the trace ring TR of R under which the commutative trace ring T is invariant. 2, both T and TR are affine and Noetherian, and TR is a finite module over T.

Therefore RG/(M D RG) is Artinian, so that $(M) consists actually of maximal ideals of RG. Hence $ restricts to a correspondence $ m : M a x P —oMaxR G given by $m(M) = $(M) = {me Max RG\ m DMn RG}. If R is commutative, then $ and $ m are actually maps: Because in that case the intersection of a prime ideal P of R with RG is a prime ideal of RG. For noncommutative P, this is in general not true, as trivial examples show. 3). In general, it is difficult to say much about $ and $ m . 2 PROPOSITION.

By assumption, each p ; is a minimal prime ideal of A. Let A = A/N where N = \/0- Then we can form AS , and A/pj embeds into AS /pjS . Hence 5 is regular in A/pj = A/pj. Since A/(pi H • • • Opm) embeds into 0 A/pj, S is regular in A/(pi n - - . f l p m ) = A/y/P n A. So we showed that the Small set 5 of A is regular in B. 1] implies that 5 is a right Ore set for both A and B. 8], AS'1 is right Artinian. Therefore AS~X is the total ring of right fractions of A and it is contained in the total ring of right fractions of B.