Reduced class groups grafting relative invariants

Abstract

Let ( X , T ) be a regular stable conical action of an algebraic torus on an affine normal conical variety X defined over an algebraically closed field of characteristic zero. We define a certain subgroup of Cl ( X / / T ) and characterize its finiteness in terms of a finite T-equivariant Galois descent X ˜ of X. Consequently we show that the action ( X , T ) is equidimensional if and only if there exists a T-equivariant finite Galois covering X → X ˜ such that ( X ˜ , T ) is cofree. Moreover the order of Gal ( X / X ˜ ) is controlled by a certain subgroup of Cl ( X ) . The present result extends thoroughly the equivalence of equidimensionality and cofreeness of ( X , T ) for a factorial X. The purpose of this paper is to evaluate orders of divisor classes associated to modules of relative invariants for a Krull domain with a group action. This is useful in studying on equidimensional torus actions as above. The generalization of R.P. Stanleyʼs criterion for freeness of modules of relative invariants plays an important role in showing key assertions.