Subintegrality, invertible modules and Laurent polynomial extensions

Abstract

Let A ⊆ B be a commutative ring extension. Let 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 I(A, B) → 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 I(A, B) → I(A[X, X −1 ], B[X, X −1 ]) in the general case.

Relative Cartier divisors and Laurent polynomial extensions

“…The following result generalizes a result of Asanuma (see [11, 3.4]), who considered the case B = frac(A), as well as several results of the first author in [7]. Theorem 6.8.…”

“…The following result generalizes a result of Asanuma (see [11, 3.4]), who considered the case B = frac(A), as well as several results of the first author in [7]. (2) Now suppose that A is a 1-dimensional domain, and that A ⊂ B is an integral, birational and anodal extension.…”

Relative Cartier divisors and K-theory

“…Our first observation is that when f is Spec(B) → Spec(A) for a commutative ring extension A ֒→ B, Pic (f ) is isomorphic to the relative Cartier divisor group I(f ), defined in [13] as the group of invertible A-submodules of B under multiplication and studied in [15,14,16]. This definition of I(f ) also makes sense (and we still have I(f ) ∼ = Pic (f )) for scheme maps f : X → S for which O × S → f * O × X is an injection of sheaves.…”