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 proper foundation for completing a triangulated category.
翻译:本说明提出一种新的方法来完成三角隔热类别,该方法基于“Cauchy 序列”的概念。 我们将此应用于完美复合物的类别。 显示在右一致环上有限展示模块的封闭衍生类别是完美复合物类别的完成。 其结果延伸到了非动物无脊椎动物计划,并直接构建了独特性类别。 亚伯利亚类别平行完成理论与衍生类别完成情况相符。 有三个附录。 由Tobias Barthel编写的第一个附录讨论了环光谱的完美复合物的完成情况。 由Tobias Barthel和Henning Krause的第二个附录讨论了为分离的诺埃特里亚计划设计的完美组合。 由Barthel和Henning Krause的精细剂为分离的诺埃特里亚计划提供的关于连接衍生的一致夹层的描述作为完成情况。 Bernhard Kell的最后附录介绍了三角分类的变形增强概念,并为完成三角分类提供了适当的基础。