By Nilolaus Vonessen
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Additional resources for Actions of Linearly Reductive Groups on Affine Pi Algebras
G (J°) = 0. 9 can be improved. This is contained in the last result of this paragraph. 14 PROPOSITION. Let R be a semiprime affine (left) Noetherian Pi-algebra over k which is Unite over its center C, and let G be a reductive group over k acting rationally on R. 2). Then the total ring of fractions Q(RG) ofRG exists, is Artinian, and is contained in Q(R). Let S(CG) be the Small set of CG. Then S(CG) consists actually of regular elements of R. As a consequence, Q(RG) = S(CG)~1RG and is thus a localization of RG at regular central fixed points.
Denote by V and W the prime spectra of R and RG, respectively. Then $ is the map $: V —• W given by $ ( P ) = P H RG. Now Min RG = $(Min R) means that for all minimal prime ideals P of P, P D RG is a minimal prime ideal of RG. The minimal primes of a ring correspond to the irreducible components of its prime spectrum. Since $ is continuous, it maps every irreducible component of V into some irreducible component of W. Therefore MinP G = $ ( M i n P ) means that $ maps every irreducible component of V onto some irreducible component of W.
Let B = B/P and A = A/P D A. Let pi, . . ,p n be the prime ideals of A minimal over P C\ A. , the set of those elements of A which are regular modulo the nil radical of A. Finally, rk denotes Goldie rank, and GK denotes Gelfand-Kirillov dimension. Concerning Gelfand-Kirillov dimension, see [Borho and Kraft 76] or [Krause and Lenagan 85]. The prime ideal P of B is called (right) well-behaved over A if the following five properties hold: Homogeneity. For all b £ 0 in B, Regularity. The elements ofS are regular in B.
Actions of Linearly Reductive Groups on Affine Pi Algebras by Nilolaus Vonessen