By Nilolaus Vonessen

ISBN-10: 0821824775

ISBN-13: 9780821824771

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

Example text

Let R be an affine prime (left) Noetherian Pl-aJgebra, and let G be a linearly reductive group acting rationally on R. Then the action ofG extends to a rational action on the trace ring TR of R. Consider the following diagram of ring extensions: (TR)G / I TGRG I \ RG I TG Here all algebras are affine and Noetheriant and all extensions are finite. Moreover, RG Q TG R G is a centralizing extension, and TG C TGRG C (TR)G is an interme- diate centralizing extension". 4, the action of G on R extends to a rational action on the trace ring TR under which the commutative trace ring T is invariant.

It follows that if P is a minimal prime ideal of P, then C\geG Pg = 0. Hence P n RG = 0. 9 applies in this case. 2. Suppose that R is semiprime. Let Pi, . . , 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.

Given a connected linear algebraic group G which is not unipotent, there is a rational action of G on an affine prime PIalgebra R such that equidimensionality and homogeneity do not hold. Moreover, R has a prime ideal P such that $(P) — {p\, pi} where p\ is a minimal prime ideal ofRG, but pi is not. In particular, heightp\ ^ heightp2. Moreover, PldegR — PIdegR/P. 13, we constructed an affine prime Pi-algebra R with a rational action of G such that *. ». where x is an indeterminate. The minimal prime ideals of RG are q\ = k[x] and q2 = &, embedded into RG in the obvious way.

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