Dualizable and semi-flat objects in abstract module categories

TL;DR

This paper introduces the concepts of dualizable and semi-flat objects in abstract module categories, providing homological characterizations and unifying existing results across various algebraic contexts.

Contribution

It defines dualizable objects in abstract module categories and offers a homological description of their direct limit closure, extending known results to DG-modules, graded modules, and sheaves.

Findings

Dualizable objects characterized homologically.

Semi-flat DG-modules are direct limits of finitely generated semi-free modules.

Unified framework for modules over rings, DG-modules, and sheaves.

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 R{\"o}hrl, 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.