The behavior on the restriction of divisor classes to sequences of hypersurfaces.

摘要

Let (A, m ) be a normal, local domain and let f be a prime element such that A/f A is a normal hypersurface. Then there is a group homomorphism j*: Cl(A) → Cl(A/f A). It is well-known that j* need not be injective. However, consider a sequence fninfinity n=1 of primes such that An = A/ fnA is normal and limn →infinity fn = 0 in the m -adic topology. Two questions emerge. (1) Must it be the case that ⋂n=1infinity Ker(Cl(A) → Cl(An)) is trivial? (2) Are there situations where an integer N > 0 exists such that Cl(A) → Cl(An) is monic if fn ∈ mN ; i.e, the result in (1) above will be reached in a predictable finite number of steps?; We take statements (1) and (2) as "principles" which govern the behavior of the group homomorphism "restriction" Cl( A) → Cl(An) from the divisor class group of the ambient ring to that of the hypersurface.; Let A be an excellent local normal domain and fninfinity n=1 a sequence of prime elements lying in successively higher powers of the maximal ideal, such that each hypersurface A/ fnA satisfies R1. We establish the map of divisor classes jn*: Cl(A) → Cl((A/fnA)'), where (A/fnA)' represents the integral closure, and investigate the injectivity of jn*. The first result shows that no nontrivial divisor class can lie in every kernel. Secondly, when A is 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 fn ∈ mN ⇒ jn* is injective. We substantiate these results with a general construction that provides a large collection of examples.