By Nilolaus Vonessen
Ebook by means of Vonessen, Nilolaus
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Additional resources for Actions of Linearly Reductive Groups on Affine Pi Algebras
But condition (b) is not satisfied. One can see this as follows. Let Iu = C\g£G Mv9. Since P | a € G ( m i ) a = xA and HcrecC7712)** = 3^> ^ f°ll° w s * na * / A 7i = I xA \xyA A xA xyA A \ xA 1 , and xA J / yA I2 = I xyA \xyA yA yA yA A\ A J . A A xA + yA A\ A J . But Ji +12 is itself a maximal ideal, and R/(h -f I2) ~ A/(xA + yA) = fc. Hence the Pi-degree of Ji +I2 is 1, and Mi-GnM2-G C Specii?. 9. 16, we make the following assumptions. As before, R is an affine Pi-algebra, but G is now a linear algebraic group acting rationally on R.
Denote by Spec R/G the set of G-orbits in Spec P. Then $ induces a map $': Spec R/G —• Spec P G / $ . Main result in [Montgomery 81] is that $' is a bijection, and in fact a homeomorphism if one endows SpecP/G and S p e c P G / $ with the respective quotient Zariski topologies. This result corresponds to the fact that for a finite group H acting on an affine commutative algebra 5, Spec SH is a geometric (or strict) quotient of Spec 5, cf. 52]. For actions of linearly reductive groups on affine Pi-algebras, one cannot expect such a result.
Then RG = k[x] ^ k[x]. Note that x is regular in RG, but that x is a zero divisor in R. Hence the field of fractions of RG does not embed in the total ring of fractions of R. Moreover, note that the minimal prime ideals of R are Pi = xR and Pi — yR. But Pi n RG = xRG is not a minimal prime of RG. Concerning Gelfand-Kirillov dimension, note that GK(RG/(P1 n RG)) = 0 but GK(RG/(P2 n RG)) = 1. Finally, let us interpret the situation geometrically. The spectrum V of R is the union of the X- and Y-coordinate axes, and the spectrum W of RG is the X-axis.
Actions of Linearly Reductive Groups on Affine Pi Algebras by Nilolaus Vonessen