By Nilolaus Vonessen

Ebook by means of Vonessen, Nilolaus

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

**Example text**

Hence NA C iV#, in fact NA = NB H A. So A/NA embeds into B/NB, and this ring extension satisfies the hypotheses of the lemma. So one may assume that both A and B are semiprime. ACTIONS ON PI-ALGEBRAS 29 Let C denote the conductor of B in A. Since B is Noetherian, it has a finite number of minimal prime ideals. Denote by Q the intersection of those minimal prime ideals of B which contain the conductor C, and let Pi, . . , Pn be the remaining minimal primes of B. Note that such Pi exist since B is semiprime and C ^ 0.

13 T H E O R E M . Let R be an afRne Pi-algebra, group acting rationally Pi-degree d such that 9(M1)ni(M2) ^ 0. T i e n ~M^GnM^G contains a maximal ideal of Pi-degree PROOF. and let G be a linearly on R. Let Mi and M2 be maximal ideals ofR reductive with the same IfW^GnJl^ ^ 0. d, then $ ( M i ) = $ ( M 2 ) . 12, Afi-Gfl Af 2 -G ^ 0. Define Iv - C\g£G M,,9'. Since G is linearly reductive, we may factor out R by 7i n JT2, and thus assume that Mi n M2 H RG — h n I2 n RG = 0. Then RG embeds into R/Mx G dimensional fc-algebra.

NlKOLAUS VONESSEN 50 Iv n RG = Pu H RG. In particular, e G / i , but 6 ^ / 2 - By (b), there is a maximal ideal M of R of Pi-degree d containing I\ -f I2. Let Q be a prime ideal of R minimal over I2 and contained in M. 19). Hence also I2 = f]geG Qg> s o t n a t e is not contained in Q. We conclude that M/Q is a maximal ideal of highest Pi-degree of the affine prime Pi-algebra R/Q, although M/Q contains a non-zero idempotent. 7. It follows that $(1*2) Q ${Pi)- By symmetry, equality holds. 5 in view of this lemma.