A filtered complex is a chain complex equipped with a filtration, which is a nested sequence of subcomplexes that allows for the gradual approximation of the whole complex. This concept is crucial in homological algebra as it enables the use of spectral sequences to extract information about the homology of the complex through its filtered pieces. The filtration helps in organizing and simplifying complex relationships between different elements of the complex, making it easier to study their properties and interconnections.
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