A constructive comparison of the rings \(R(X)\) and \(R\langle X\rangle\) and application to the Lequain-Simis induction theorem
Let \(R\) be a commutative unitary ring. Denote by \(S\) (respectively \(U\)) the multiplicative subset of \(R[X]\) formed by monic polynomials (respectively primitive polynomials, i.e, polynomials whose coefficients generate the whole ring). Let \(R<X>=S^{-1}R[X]\) and \(R(X)=U^{-1}R[X]\). In this paper, the authors constructively prove that for any ring \(R\) with Krull dimension \(\leq d\), the ring \(R<X>\) locally behaves like the ring \(R(X)\) or a localization of a polynomial ring of type \((S^{-1}R)[X]\) with \(S\) a multiplicative subset of \(R\) such that the Krull dimension of \(S^{-1}R\leq d-1\). As an application, they give a simple and constructive proof of the Lequain-Simis induction theorem which is an important variation of the Quillen induction theorem.
scientific article; zbMATH DE number 5349020
scientific article