The limiting behavior on the restriction of divisor classes to hypersurfaces

✍ Scribed by Sandra Spiroff

• Book ID: 104152695
• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 269 KB
• Volume: 186
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(03)00123-3

✦ Synopsis

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 hypersurface A=fnA satisÿes R1. We investigate the injectivity of the maps Cl(A) → Cl((A=fnA) ), where (A=fnA) represents the integral closure. The ÿrst result shows that no non-trivial divisor class can lie in every kernel. Secondly, when A is, in addition, an isolated singularity containing a ÿeld 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 fn ∈ m N , then Cl(A) → Cl((A=fnA) ) is injective. We substantiate these results with a general construction that provides a large collection of examples.