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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.

In this article, we study the relative negative K-groups K −n (f ) of a map f : X → S of schemes. We prove a relative version of the Weibel conjecture i.e. if f : X → S is a smooth affine map of noetherian schemes with dim S = d then K −n (f ) = 0 for n > d + 1 and the natural map K −n (f ) → K −n (f × A r ) is an isomorphism for all r > 0 and n > d. We also prove a vanishing result for relative negative K-groups of a subintegral map.