# On diagram-chasing in double complexes

Abstract: We construct, for any double complex in an abelian category, certain "short-distance" maps, and an exact sequence involving these, instances of which can be pieced together to give the "long-distance" maps and exact sequences of results such as the Snake Lemma. Further applications are given. We also note what the building blocks of an analogous study of triple complexes would be.

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### Cites background from "On diagram-chasing in double comple..."

...We follow: books by Aluffi [1] and Freyd [26] and papers by Bergman [10] and Krause [45] for ideas about abelian categories; and the books by Gelfand and Manin [33], Neeman [52], Weibel [64] and Zimmerman [67] for ideas about triangulated categories....

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...In [10] Bergman presented an idea which captures the notion of diagram chasing in an abelian category....

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4 citations

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### Cites background from "On diagram-chasing in double comple..."

...We conclude noting that, if the square/rectangle is actually a double complex, the salamander lemma (see [39]) uses a similar idea to our ”following the path π”....

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...The construction of the square (or, in the finite case, of the rectangle) is really similar to a double complex (for more details on double complexes, see [36-39])....

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### "On diagram-chasing in double comple..." refers background in this paper

...3, the vertical exactness assumption would simply take the form Im(δ1) = Ker(δ1), and the diagram obtained, (65), would reduce to what is called in [6] an exact couple: A - A ] A· ; (78)...

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### "On diagram-chasing in double comple..." refers background in this paper

...n the above proof fulﬁll the promise that “long” connections would be reduced to composites of “short” ones. The proofs of the next three lemmas continue this theme. Lemma 11 (Snake Lemma, [1, p.23], [4], [9, p.158], [11, p.50]). If in the commuting diagram at left below, both rows are exact, and we append a row of kernels and a row of cokernels to the vertical maps, as in the diagram at right, (16) ...

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