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
Hence the sets $ ( P ) partition SpecP^, and one can consider the set of equivalence classes S p e c P G / $ . 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.
Since Ri is a homomorphic image of RG, each Mi is a Noetherian right P^-module. And since Pi fl • • • fl P n = 0, M embeds into 0 M * . Hence also M = RGxRG is a Noetherian right PG -module. 4. Let us now prove the general case. 9. 2). 9] that RG is a Noetherian right jRG-module. By the previous step, RG xRG is a finite RG -module and therefore also a finite RG module. Hence RGxRG is a finite # G -module. • Borho theory yields now many interesting results, of which the following corollary is a first example.
Note that N Pi RG is a nilpotent ideal since N is. 6 implies that also RG is affine. Therefore we may assume that R is semiprime. 8]. 9. Assume that RG is affine. Since RG is Noetherian and IG/G0]'1 £ fc, it follows by the theorem in [Montgomery and Small 81] that also RG = (RG°)GIG° is affine. Therefore we may assume that G is connected. Since R is Noetherian (or since R is an affine Pi-algebra over a field), R has only a finite number of minimal prime ideals, say Pi, . . , P n . 6). Hence we can define a rational action of G on ©™=1 R/Pi, by letting the action on each summand be the one induced from R.