Picard groups of some rings of invariants

Abstract

Let A = K[X, ,..., X,], the polynomial ring of n variables over a field K, G a finite group of K-automorphisms acting on A so that the order of G is relatively prime to the characteristic of K, AC the ring of invariants. Little is known about AC. G. C. Shephard and J. A. Todd proved that if G is a subgroup of the general linear group and AC is a polynomial ring, then G is a finite reflection group [lo]. C. Chevalley proved the converse, namely, if G is a finite reflection group, then AC is a polynomial ring [4]. What happens if G is not necessarily a subgroup of the general linear group ? T. Igarashi, a student of M. Miyanishi in Osaka University, proved in March 1976 that in case of K being algebraically closed and n = 2, then AC is regular if and only if AC is a polynomial ring [5]. D. Anderson under the same assumption as that of T. Igarashi proved that every projective module over AC is.stably isomorphic to the direct sum of a free module and a projective module of rank one, and then apply Murthy and Swan’s Thcorcm [8], these two are in fact isomorphic [l]. We shall prove in this article Pic(AG) is trivial without assuming anything of the field K or the order of G. Our main result is Theorem 5.3. Hence, if AC is regular, it is factorial by our main result. Part of this article forms the second part of the author’s Ph. D. thesis [7] in the University of Chicago. The first part of the thesis will appear soon with the title, Projective Modules of Some Polynomial Rings.