Subintegrality, Invertible Modules and Laurent Polynomial Extensions

TL;DR

This paper characterizes when invertible modules over a ring extension remain invariant under Laurent polynomial extension, focusing on integral domains of dimension at most one, and explores properties of the associated cokernel.

Contribution

It extends Sadhu and Singh's result by providing a necessary and sufficient condition for isomorphism of invertible module groups in dimension one, and analyzes the case when dimension is higher.

Findings

The natural map is an isomorphism under specific conditions for dimension ≤ 1.

Necessary but not sufficient conditions are identified for higher dimensions.

Properties of the cokernel of the natural map are discussed in general cases.

Abstract

Let $A\subseteq B$ 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 $A\subseteq B$ of integral domains with $\dim A\leq 1$, 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 $\dim A\geq 2$ 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.