homological-algebra

The five lemma is one of the basic lemmas of homological algebra, useful for example in the construction of the connecting homomorphism in the homology long exact sequence. Let be an abelian category. Consider a commutative diagram in of the form where the top and bottom rows are exact sequences. For simplicity we denote all the differentials in both exact sequences by . on terminology The weak f…

The notion of spectral sequence is an algorithm or computational tool in homological algebra and more generally in homotopy theory which allows to compute chain homology groups/homotopy groups of bi-graded objects from the homology/homotopy of the two graded components. Notably there is a spectral sequence for computing the homology of the total complex of a double complex from the homology of it…