Roger Wiegand

In the 1980's, Sylvia Wiegand and I developed machinery for studying the torsion-free cancellation problem in dimension one: When does $M \oplus L \cong N \oplus L$ imply $M \cong N$? Here $M, N$ and $L$ are finitely generated torsion-free modules over a one-dimensional Noetherian domain $R$ with finite integral closure. Over the next twenty years, Guralnick, Klingler, Levy and others made considerable progress on the analogous question for mixed modules (finitely generated modules that are neither torsion nor torsion-free), but no general theory emerged. Recently, Wolfgang Hassler and I, intrigued by the nagging question of whether a ring $R$ as above having torsion-free cancellation actually has cancellation for {\it all} finitely generated modules, developed analogous machinery for handling the mixed (non-CM) case. Although most of the theory of torsion-free modules extends to the general case, the answer to the question above is ``no''. While cancellation is really a global question, most of the work is at the local level, and a key to our negative answer to the nagging question is the construction of indecomposable mixed modules of rank two over rings like $k[[t^2,t^3]]$. (At this point it is unknown whether this ring has indecomposables of arbitrarily large rank, but we suspect that it does.)