Let R be a commutative ring with unit. We endow the categories of filtered complexes and of bicomplexes of R-modules, with cofibrantly generated model structures, where the class of weak equivalences is given by those morphisms inducing a quasi-isomorphism at a certain fixed stage of the associated spectral sequence. For filtered complexes, we relate the different model structures obtained, when we vary the stage of the spectral sequence, using the functors shift and décalage.

中文翻译：

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模型类别结构和谱序列

让R是一个有单位的交换环。我们赋予过滤复合物和双复合物的类别R-模块，具有共同生成的模型结构，其中弱等价类由在相关光谱序列的某个固定阶段诱导准同构的那些态射给出。对于过滤的复合物，当我们使用函子 shift 和 décalage 改变光谱序列的阶段时，我们将获得的不同模型结构联系起来。