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Dualizable and semi-flat objects in abstract module categories

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Abstract

In this paper, we define what it means for an object in an abstract module category to be dualizable and we give a homological description of the direct limit closure of the dualizable objects. Our description recovers existing results of Govorov and Lazard, Oberst and Röhrl, and Christensen and Holm. When applied to differential graded modules over a differential graded algebra, our description yields that a DG-module is semi-flat if and only if it can be obtained as a direct limit of finitely generated semi-free DG-modules. We obtain similar results for graded modules over graded rings and for quasi-coherent sheaves over nice schemes.

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Acknowledgements

I would like to thank my advisor Henrik Holm for suggesting the topic and for his guidance. I thank Sergio Estrada for discussing and providing references for the case of quasi-coherent sheaves. I thank Luchezar L. Avramov for making available his unpublished manuscript on differential graded homological algebra, joint with Hans-Bjørn Foxby and Stephen Halperin, which has served as a key source of inspiration. Finally, it is a pleasure to thank the anonymous referee for his/her thorough and insightful comments that greatly improved the manuscript and strengthened Theorem 2 (and its applications) significantly.

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  1. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen Ø, Denmark

    Rune Harder Bak

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Bak, R.H. Dualizable and semi-flat objects in abstract module categories. Math. Z. 296, 353–371 (2020). https://doi.org/10.1007/s00209-020-02501-z

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