2014/04/25 by Vivek Sadhu, Sadhu, Vivek · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Meromorphic and Entire Functions #Rings, Modules, and Algebras #math.AC #msc:13B02 #msc:13F45
paper · pdf · doi:10.48550/arxiv.1404.6498
13 pages, Some changes made due to referee report, To appear in Proc. Indian Acad. Sci. (Math. Sci)
arxiv created 2014/10/31 · arxiv updated 2014/11/03
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 dim A≤ 1, so that the natural map \mathcal I(A,B) → \mathcal I (A [X, X-1],B [X, X-1]) is an isomorphism. In the same situation, we show that if dim A≥ 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) → \mathcal I (A [X, X-1],B [X, X-1]) in the general case.