§0. INTRODUCTION Let k be an algebraically closed field, and let R be an affine (i.e., finitely gen- erated) A -algebra satisfying a polynomial identity. Let G be a linear algebraic group acting rationally on R. We study the relationship between R and the fixed ring RG, trying to extend both the theory of actions of finite groups on non-commutative rings and commutative invariant theory. The flavor of the subject developed here is perhaps best caught in the following theorem, which summarizes many results obtained in this paper. THEOREM. (8.1) Suppose that R is left Noetherian and G linearly reductive. Then the following properties hold. (1) RG is Noetherian. (2) RG is affine. (3) (Lying over) For every prime ideal p ofRG} there is a prime P of R such that p is a minimal prime over P n RG. (4) (Separation by invariants) If Ii and I2 are G-stable ideals ofR such that R = h+I2, thenRG = IG+IG. Conversely suppose that G is a linear algebraic group which is not linearly reductive. Then there is an affine prime Noetherian Pi-algebra R with a rational action of G such that none of the above properties hold. This result gives, of course, several characterizations of linearly reductive groups and we will obtain others (albeit more technical ones) in §8. Its main importance lies, however, in the fact that actions of linearly reductive groups have these properties (al- though (1) and (4) are not especially deep they can be easily deduced by elementary arguments using the Reynolds operator). Let us note two particular consequences of this theorem. Firstly, it implies that the theory developed in this paper is essentially a characteristic zero theory. To paraphrase Mumford in the introduction to the first edition of his book on geometric invariant theory, this assumption is hidden in the fact that the group G is in most cases assumed to be linearly reductive. By a result of Nagata (see 2.8), only relatively few groups are linearly reductive in prime characteristic. In fact, the only connected linearly reductive groups in characteristic p are the tori Gm x x Gm, where Gm denotes the multiplicative group of the ground field k. 1
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