By Nilolaus Vonessen
Publication by means of Vonessen, Nilolaus
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Extra info for Actions of Linearly Reductive Groups on Affine Pi Algebras
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. Suppose that R is prime, and that PIdeg RG — PIdeg R. Then RG is afRne and prime. Here the Pi-degree of RG is defined as the maximum of the Pi-degrees of the prime homomorphic images of RG. We need the following result, which one can deduce from Regev's conjecture which has now been proved by Formanek, see [Formanek 87, §6 and §10]: Let A C B be Pi-algebras with B prime.
8 which is very technical. It does not make any assumptions on the group G, but it requires that the fixed ring RG is Artinian. In the second half of this paragraph we will study these questions for actions of reductive and linearly reductive groups. Reductivity will mainly be needed to reduce to the case that RG is Artinian: This is done by factoring out R by some large (^-invariant ideal I, and the reductivity of G yields under suitable hypotheses that (R/I)G and RG/(InRG) are closely related.
Define (A R = I xA \xyA A A yA A\ A A) . Then R is an affine prime Noetherian Pi-algebra finite over its center A — A-I3. Let G be the multiplicative group G m . It acts on A as follows: If a G G and p(x\ y) £ A, then p(z, y)a = p(axyay). Since xA% yA, and xyA are G-stable ideals of A, the action of G on A extends to a componentwise action of G on R. Since AG = fc, RG = Note that again RG is not semiprime although R is prime. Let mi be a maximal ideal of A containing x but not y. Similarly, let m 2 be a maximal ideal of A containing y but not x.