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Read e-book online Actions of Linearly Reductive Groups on Affine Pi Algebras PDF

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By Nilolaus Vonessen

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Choose a minimal prime P of R such that P D C = Q. Then QC\CG = Pn CG, and GK(C G /(Q n CG)) = GK(CG/{P = n CG)) GK(RG/(PnRG)). Here the second equality holds because RG is a finite CG-module. And the last GKdimension is by assumption independent of the choice of P G Min (R). It follows that there are no strict inclusion relations among the primes in (MinC) n CG. Since the latter set clearly contains MinC G , it follows that (MinC) n CG = MmCG. 9 applies to the action of G on C. 14 does not hold if R is not a finite module over its center.

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.

Clearly, p\ is a minimal prime ideal of RG, but p2 is not. 7 EXAMPLE. Let G be a connected linearly reductive group. Then there is a rational action ofG on an affine semiprime Noetherian Pi-algebra R finite over its center with a minimal prime ideal P of R such that $(P) — {pi,P2} where p\ is a minimal prime ideal ofRG but p2 is not. In particular, heightp\ ^ heightp 2. 21, and let x be a commuting indeterminate over 5. Let A = S[x]. We define an algebra R* by R ,_ / A -\mA mA\ A)' ACTIONS ON PI-ALGEBRAS 37 Then R1 is an affine Noetherian Pi-algebra finite over its center.

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