Picard groups of some module categories

Abstract

The groups of equivalence classes of auto-equivalences of the categories of torsion and torsion-free modules over an integral domain $R$ are determined. For the category of torsion-free modules, this group is isomorphic to the Picard group $\operatorname {Pic} R$ of $R$. We prove that for the category of torsion $R$-modules, this group is isomorphic to a group of certain divisible torsion groups called ``cyclones'' where addition is given in terms of the torsion product. In particular, this group is abelian and contains the direct product of the Picard group of the $R$-completion $\widetilde R$ of $R$ and of the group of ``clones'' of $Q/R$, where $Q$ denotes the quotient field of $R$. The structure of the latter group is described in terms of the inverse system of the unit groups $U(R/Rr)$ of the proper quotients $R/Rr$ ($r \in R$) of $R$.