Recovering divisor classes via their (T)-adic filtrations Original Research Article

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 Πn ≥ 1 Cl(B/tnB), where the definition of class group is extended to include the rings B/tnB. 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.