Snake lemma variations

TL;DR

This paper explores variations of the Snake Lemma, including the Kernel and Cokernel Complex Lemmas, highlighting their roles in homological algebra and representation theory.

Contribution

It introduces and analyzes new variations of the Snake Lemma, connecting kernel and cokernel complexes with applications in functor support and homology.

Findings

Kernel Complex Lemma establishes isomorphic homology in exact diagrams.

Cokernel Complex Lemma helps determine support of finitely presented functors.

Snake Lemma is derived as a consequence of these two lemmas.

Abstract

The Kernel Complex Lemma states that given commutative diagram with exact rows and exact columns which covers the region under a $\Gamma$-shape, then the kernel sequence on the top and the kernel sequence at the left have in each position isomorphic homology. The dual version, the Cokernel Complex Lemma, has been used to determine the support of a finitely presented functor on the Auslander-Reiten quiver. We note that the Snake Lemma is a consequence of the two results combined.