Integer-valued polynomials on Krull rings

Abstract

If R R is a subring of a Krull ring S S such that R Q R_{Q} is a valuation ring for every finite index Q = P ∩ R Q=P\cap R , P P in Spec 1 ( S ) ^{1}(S) , we construct polynomials that map R R into the maximal possible (for a monic polynomial of fixed degree) power of P S P PS_{P} , for all P P in Spec 1 ( S ) ^{1}(S) simultaneously. This gives a direct sum decomposition of Int ( R , S ) (R,S) , the S S -module of polynomials with coefficients in the quotient field of S S that map R R into S S , and a criterion when Int ( R , S ) (R,S) has a regular basis (one consisting of 1 polynomial of each non-negative degree).