DESCENT OF DIVISOR CLASS GROUPS OF KRULL DOMAINS

Abstract. For a pair of Krull domains (A,B) such that Q(B)/Q(A) is a field extension of their quotient fields and Q(A) ∩B = A, we study on the relation between the divisor class groups Cl(A) and Cl(B) by A. Magid’s diagram showing finite generation of class groups of rings of invariants (cf. [4]). We define the descent properties with the existence of the canonical morphism of class groups in the sense of Magid and obtain a ladder property of these descents. This can be applied to regular actions of algebraic tori on affine normal varieties and characterizes freeness of monomials of prime relative invariants on these varieties. Furthermore we define certain subgroups of class groups of normal domains and their invariant subrings which determine a class of modules of relative invariants to be free