The limiting behavior on the restriction of divisor classes to hypersurfaces

Let A be an excellent local normal domain and {fn}n=1∞ a sequence of elements lying in successively higher powers of the maximal ideal, such that each hypersurfaceA/fnA satisfies R1. We investigate the injectivity of the maps Cl(A)→Cl((A/fnA)′), where (A/fnA)′ represents the integral closure. The first result shows that no non-trivial divisor class can lie in every kernel. Secondly, when A is, in addition, an isolated singularity containing a field of characteristic zero, dim A 4, and A has a small Cohen–Macaulay module, then we show that there is an integer N>0 such that if , then Cl(A)→Cl((A/fnA)′) is injective. We substantiate these results with a general construction that provides a large collection of examples.