By Nilolaus Vonessen

ISBN-10: 0821824775

ISBN-13: 9780821824771

Ebook via Vonessen, Nilolaus

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

**Example text**

Therefore multiplication by x induces an injective RG-module homomorphism TGRG «-> RG. Hence TGRG is a Noetherian left i£G-module. As seen above, it follows that RG is affine and two-sided Noetherian. But more can be said in this case. Clearly TG — T. Since TR is a finite T-module, it follows that TR is in fact a finite ^-module. Hence also R is a finite jR^-module and is therefore left Noetherian. 1. 9 PROPOSITION. Let R be an affine prime Pl-algebra1 and let G be a linearly reductive group acting rationally on R.

Let A C B be a right restricted extension of right Noetherian affine k-algebras where k is a Held. Suppose that the Gelfand-Kirillov dimension of A is Unite, that B is semiprime, and that for all minimal prime ideals P of B the following condition holds: If p is a prime ideal in A minimal over P fl A, then p is a minimal prime ideal of A. Then the Small set S of A consists of regular elements ofB and satisfies the right Ore conditions for A and B. As a consequence, AS'1 = Q(A) is the total ring of right fractions of A, is right Artinian, and is contained in the total ring of right fractions Q(B) of B.

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.

### Actions of Linearly Reductive Groups on Affine Pi Algebras by Nilolaus Vonessen

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