We study consequences and applications of the folklore statement that every double complex over a field decomposes into so-called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences easy to understand. We describe a notion of ‘universal’ quasi-isomorphism and the behaviour of the decomposition under tensor product and compute the Grothendieck ring of the category of bounded double complexes over a field with finite cohomologies up to such quasi-isomorphism (and some variants).

中文翻译：

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关于双配合物的结构

我们研究了民间传说的后果和应用，即一个领域上的每个双复合体都分解成所谓的正方形和锯齿形。这一结果使有关相关上同调群和谱序列的问题变得容易理解。我们描述了“通用”准同构的概念和张量积下分解的行为，并在具有有限上同调的场上计算有界双复形的范畴的格罗腾迪克环，直到这种准同构（和一些变体）。