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.

中文翻译：

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抽象模块类别中的可对偶和半平面对象

在本文中，我们定义了抽象模块类别中的对象可对偶的含义，并给出了可对偶对象的直接极限闭包的同调描述。我们的描述恢复了 Govorov 和 Lazard、Oberst 和 Röhrl 以及 Christensen 和 Holm 的现有结果。当应用于微分梯度代数上的微分梯度模时，我们的描述产生一个 DG 模是半平坦的，当且仅当它可以作为有限生成的半自由 DG 模的直接极限获得。对于分级环上的分级模块和良好方案上的准相干滑轮，我们获得了类似的结果。