Subintegrality, invertible modules and Laurent polynomial extensions

Abstract

Let A⊆B be a commutative ring extension. Let $\mathcal I(A, B)$ be the multiplicative group of invertible A-submodules of B. In this article, we extend a result of Sadhu and Singh by finding a necessary and sufficient condition on an integral birational extension A⊆B of integral domains with dimA≤1, so that the natural map $\mathcal I(A,B) \rightarrow \mathcal I (A [X, X^{-1}],B [X, X^{-1}])$ is an isomorphism. In the same situation, we show that if dimA≥2, then the condition is necessary but not sufficient. We also discuss some properties of the cokernel of the natural map $\mathcal I(A,B) \rightarrow \mathcal I (A [X, X^{-1}], B [X, X^{-1}])$ in the general case.