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2019 Picard groups of some module categories
László Fuchs, Peter Vámos
J. Commut. Algebra 11(4): 547-572 (2019). DOI: 10.1216/JCA-2019-11-4-547

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 $R$.

Citation

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László Fuchs. Peter Vámos. "Picard groups of some module categories." J. Commut. Algebra 11 (4) 547 - 572, 2019. https://doi.org/10.1216/JCA-2019-11-4-547

Information

Published: 2019
First available in Project Euclid: 7 December 2019

zbMATH: 07147396
MathSciNet: MR4039982
Digital Object Identifier: 10.1216/JCA-2019-11-4-547

Subjects:
Primary: 13A15, 13F05
Secondary: 13C12, 13D07, 14C22, 16D90

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.11 • No. 4 • 2019
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