This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category.

中文翻译：

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完成完美的复合体

本笔记提出了一种基于柯西序列概念来完成三角化范畴的新方法。我们将此应用于完美复合体的类别。结果表明，右相干环上有限呈现模的有界导出范畴是完美配合范畴的完备。结果扩展到非仿射诺特方案并产生奇点范畴的直接构造。阿贝尔范畴的平行完成理论与派生范畴的完成是兼容的。共有三个附录。Tobias Barthel 的第一个讨论了环形光谱的完美配合物的完成。Tobias Barthel 和 Henning Krause 的第二篇论文为分离的诺特方案提炼了对连贯滑轮的有界派生类别的描述作为完成。Bernhard Keller 的最后一个附录介绍了三角类别的形态增强的概念，并为完成三角类别提供了基础。