On diagram-chasing in double complexes

Abstract

Diagram-chasing arguments frequently lead to "magical" relations between distant points of diagrams: exactness implications, connecting morphisms, etc.. These long connections are usually composites of short "unmagical" connections, but the latter, and the objects they join, are not visible in the proofs. I try to remedy this situation. Given a double complex in an abelian category, we consider, for each object A of the complex, the familiar horizontal and vertical homology objects at A, and two other objects, which we name the "donor" A_{\box} and and the "receptor" ^{\box}A at A. For each arrow of the double complex, we prove the exactness of a 6-term sequence of these objects (the "Salamander Lemma"). Standard results such as the 3x3-Lemma, the Snake Lemma, and the long exact sequence of homology associated with a short exact sequence of complexes, are obtained as easy applications of this lemma. We then obtain some generalizations of the last of the above examples, getting various exact diagrams from double complexes with all but a few rows and columns exact. The total homology of a double complex is also examined in terms of the constructions we have introduced. We end with a brief look at the world of triple complexes, and two exercises.