Reduced Ramification Indices of Quotient Morphisms under Torus Actions

According to Hilbertʹs theory on ramifications in number theory, the p′-part of the order of the inertia group of a prime divisor of a normal domain of characteristic p with a finite Galois group is equal to the p′-part of its reduced ramification index over its restriction to the fixed subring under the action of the Galois group. This classical correspondence plays a fundamental role in the study of relative invariants of finite groups on an affine factorial variety in arbitrary characteristics. The purpose of this paper is to extend the relationship stated above to one in the case of regular actions of affine algebraic groups and especially in the torus actions. Our generalization is also regarded as a criterion for finite extensions of tori to be central in terms of invariant theory and seems to be useful in extending the studies on relative invariants of finite groups and related materials and on torus invariants.