Finite generation of class groups of rings of invariants

Abstract

Let R be a normal affine domain over the algebraically closed field k, and let G be a connected algebraic group acting rationally on R. It is shown that the divisor class group of RG is a homomorphic image of an extension of a subgroup of the class group of R by a subquotient of the character group of G. In particular, if R has finitely generated class group, so does RG. The object of this note is to establish the following theorem: Let R be a normal affine domain over the algebraically closed field k, and let G be a connected algebraic group acting rationally on R. Then if R has a finitely generated divisor class group, then so does RG. (If K is the quotient field of R, then RG is R n KG, so RG is a Krull domain and hence has a divisor class group.) The following conventions are adopted: k is the fixed algebraically closed base field. For a commutative k-algebra A, U(A) denoLes the group of units of A and Uk(A) = U(A)/k*. We begin with some observations regarding group actions and units. PROPOSITION 1. Let R be an integral domain k-algebra with quotient field K suvh that Uk (R) is a finitely generated group, and let G be a connected algebraic group acting as k-algebra automorphisms of R, such that every unipotent subgroup of G acts rationally on R. Then: (a) Every f in U(R) is a semi-invariant for G. (b) If f is in K such that g(f )/f e U(R) for all g e G, then f is a semiinvariant for G. PROOF. First we consider the case where G is unipotent and R is the coordinate ring of the affine k-variety V. If f is a nonvanishing function on V and v an element of V, then g -> f (gv) is a nonvanishing function on G, hence constant since G is unipotent. Thus f is an invariant. In general R is a direct limit of such coordinate rings, and hence every unit of R is invariant under every unipotent subgroup of G. Now we can establish (a). We need to know that G acts trivially on Uk(R), and by the above paragraph it is enough to treat the case G = Gm. Now Uk (R) is a finitely generated free abelian group, and the action of Received by the editors December 29, 1975. AMS (MOS) subject classifications (1970). Primary 13A05; Secondary 20G 15. Copyright CD 1977, American Mathematical Society