Dualizable and semi-flat objects in abstract module categories
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Mathematische Zeitschrift
Dualizable and semi-flat objects in abstract module categories Rune Harder Bak1 Received: 22 November 2016 / Accepted: 10 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Cotorsion pairs · Differential graded algebras and modules · Direct limit closure · Dualizable objects · Locally finitely presented categories · Semi-flat objects Mathematics Subject Classification Primary 18E15; Secondary 16E45 · 18G35
1 Introduction In the literature, one can find several results that describe how some kind of “flat object” in a suitable category can be obtained as a direct limit of simpler objects. Some examples are: 1. In 1968 Lazard [22], and independently Govorov [11] proved that over any ring, a module is flat if and only if it is a direct limit of finitely generated projective modules. 2. In 1970 Oberst and Röhrl [25, Thm 3.2] proved that an additive functor on a small additive category is flat if and only if it is a direct limit of representable functors. 3. In 2014 Christensen and Holm [5] proved that over any ring, a complex of modules is semi-flat if and only if it is a direct limit of perfect complexes (= bounded complexes of finitely generated projective modules). 4. In 1994 Crawley-Boevey [6] proved that over certain schemes, a quasi-coherent sheaf is locally flat if and only if it is a direct limit of locally free sheaves of finite rank. In 2014
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Rune Harder Bak [email protected] Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark
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Brandenburg [3] defined another notion of flatness and proved one direction for more general schemes. In Sect. 3 we provide a categorical framework that makes it possible to study results and questions like the ones mentioned above. It is this framework that the term “abstract module categories” in the title refers to. From a suitably nice (axiomatically described) class S of objects in such an abstract module category C , we define a notion of semi-flatness (with respect to S ). This definition depends only on an abstract tensor product, which is built into the aforementioned framework, and on a certain homological condition. We write lim S for − → the class of objects in C that can be obtained as a direct limit of objects from S . Our main result shows that under suitable assumptions, lim S is precisely the
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