By Nilolaus Vonessen

ISBN-10: 0821824775

ISBN-13: 9780821824771

Ebook by means of Vonessen, Nilolaus

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

**Sample text**

Suppose that for some minimal prime ideal P of P, P Pi RG is not a minimal prime ideal. Then P D RG is not contained in X, so it contains an RG-regular element a. But a is clearly not regular in P, a contradiction to the assumption that Q(RG) C Q(P). 7). 7: The image of ( 0 x) G Rf is regular in RG but not in R. We now give another example of this phenomenon which involves a commutative ring. 12 EXAMPLE. A commutative affine semiprime algebra R with a rational action of G m suci that the total ring of fractions of the fixed ring RG exists but is not contained in the total ring of fractions of R.

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. We will need here a technical result which we will prove only in §8. 15 EXAMPLE. Let G be a connected linearly reductive group. Then there is a rational action ofG on an affine prime Noetherian Pi-algebra R such that the total ring of fractions ofRG cannot be realized by inverting the central fixed points of R. 4, choose an affine domain A over k with subalgebras A\ and A2 such that -A is a finite module over both A\ and A2 but such that A is not finite over A\ Pi A2.

G of fractions of R . 10]. We will use this freely. Since P„ H 5 = (P„ n RG) n 5 = 0, it follows that S^P,, l l ideal of S~ R* And since S' Pu 1 is a proper prime 1 n R = Pv, RjPv embeds into S- R/S'1P1/. l the Pi-degree of both S" Pi and S~ P2 is d. Moreover, we have that Min# 1 l G! Min(5- E ) is a bijection. And since S- (Pv that also $R{PP) G X Hence G —+ X G D R ) - S" PV n S~ R \ it follows —• #5-i^(5~ 1 P 1 / ) is a bijection. Therefore it suffices to show that *5-»Ji(5- 1 Pi) = * 5 - » « ( 5 - 1 f t ) .