Divisorial free modules of relative invariants on Krull domains

Abstract

An effective construction of relative invariants plays an important role in the study of finite reflection groups (e.g., [J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math., vol. 29, Cambridge Univ. Press, Cambridge, 1990]). Using a combinatorial method, R.P. Stanley (cf. [R.P. Stanley, Relative invariants of finite groups generated by pseudo-reflections, J. Algebra 49 (1977) 134–148; Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.) 1 (1979) 475–511]) generalized such classical result to a criterion for a module Sym ( V ) χ of invariants of G relative to a character χ to be Sym ( V ) G -free of rank one in the case where G is any finite complex subgroup of GL ( V ) . His criterion is useful in invariant theory of finite groups and in combinatorics. In this paper we will study on relative invariants of a group G consisting of automorphisms of a Krull domain R from the view point of a generalized partial result on ramifications in number theory mentioned in [H. Nakajima, Reduced ramification indices of quotient morphisms under torus actions, J. Algebra 242 (2001) 536–549]. We will give a criterion for R χ to be a free R G -module of rank one for a 1-cocycle χ of G in the unit group U ( R ) , and consequently establish a criterion for module of relative invariants of a finite central extension of algebraic tori to be free, in terms of local characters, which is similar to one in [H. Nakajima, Relative invariants of finite groups, J. Algebra 79 (1982) 218–234; R.P. Stanley, Relative invariants of finite groups generated by pseudo-reflections, J. Algebra 49 (1977) 134–148].