§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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