Recovering divisor classes via their ( T)-adic filtrations

Abstract

Continuing the investigation of the canonical map of divisor class groups j ∗ : Cl(B) → Cl( B tB ) , we consider what happens if ker j ∗ is nonzero. The first result gives an injection of Cl( B) into Π n ≥ 1 Cl( B t nB ) , where the definition of class group is extended to include the rings B t nB . We also get an action of ker j ∗ on small MCM B-modules, resulting, in the case of a ring A without DCG and with a small MCM module C, in infinitely many distinct small MCM A[[ T]]-modules lifting C.