§1. STATEMENTS OF RESULTS Let R be an affine fc-algebra satisfying a polynomial identity, and let G be a linear algebraic group acting rationally on R. We will discuss properties of the action of G on # , and especially the relationship between R and the fixed ring RG. To avoid awkward and lengthy formulations of results, we assume throughout this paragraph that G is linearly reductive, and we will "star" a theorem if it holds in prime characteristic also for actions of reductive groups under the additional assumption that R is a finite module over its center. We note that in the latter case both the algebra R and its center are affine and Noetherian. We consider first the old problem whether or better, under which hypotheses the fixed ring RG is affine. This is certainly not always the case. A famous result of Hilbert asserts that RG is affine over k if R is commutative and G is a linearly reductive group acting rationally on R (see 2.13). Nagata generalized this in prime characteristic to the case of (geometrically) reductive groups (2.16). If R is commutative and G is finite, RG is also affine this is a classical result of E. Noether (see [Montgomery 82]). This latter result can be extended to the case that R is a finite module over its center [J0ndrup 86], [Montgomery and Small 86]. If R is non- commutative and G is finite, RG need not be affine. But in case that R is Noetherian and that the order of the group G is invertible in &, [Montgomery and Small 81] showed that RG is affine here R need not be a Pi-algebra. More about this question for finite group actions can be found in the survey paper [Montgomery 82]. The first major result of this paper is the generalization of Hilbert's theorem to the case of affine, Noetherian Pi-algebras. *1.1 THEOREM. (4.4) Suppose that R is left Noetherian. Then RG is affine and Noetherian. The main tool involved in the proof of this result is the trace ring associated to an affine prime PI-algebra. One of the first steps of the proof is to show that a rational action of a linear algebraic group extends to a rational action on the trace ring this is done in §3. The method of using the trace ring to show that RG is affine (in the case of a finite group) is due to [Montgomery and Small 81]. Crucial for this approach is their generalized Artin-Tate lemma (see 4.3). It is not possible to just drop in Theorem 1.1 the assumption that R is left 4
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