Dec 23, 2023 · Both the snake lemma and the kernel-cokernel sequence are instances of the long exact sequence of homotopy groups induced by a fiber sequence of anima, which is really the fundamental thing that is the source of almost all long exact sequences.

Jun 14, 2019 · In proving the control theorem the Snake Lemma is used in undertaking an analysis of the restriction maps of the local cohomology groups (on the right) and that of the global cohomology group (in the middle). One uses the Snake Lemma to get information about the restriction map on the Selmer group (which is the map on the left).

Jan 24, 2011 · The snake lemma doesn't make sense in the category of groups in general, because maps don't always have cokernels. If you assume that the three vertical maps in your snake lemma diagram do have cokernels, that is, that their images are normal subgroups of their targets, then the snake lemma is true with the same proof as in the abelian category case …

The salamander lemma is a lemma in homological algebra from which a number of theorems quickly drop out, some of the more famous ones include the snake lemma, the five lemma, the sharp 3x3 lemma (generalized nine lemma), etc.

What is the history behind the colorful name of this result? Cartan-Eilenberg states it without any particular fanfare.

This lemma has been presented to me several times in slightly different contexts, yet the proof has always been the same technical diagram chase and no further intuition behind the statement was provided. So my question is: do you have some intuition when thinking about the five lemma?

May 4, 2020 · My interpretation of Pedro's answer is something like """ part of the point of using spectral sequences is to make applications of the snake lemma invisible, and when working with spectral sequences it is already common to "apply finitely many snake lemmas" without comment.

But some proofs, such as that of five lemma (as far as I am able to reconstruct now), require a small amount of diagram chasing in addition to using snake lemma. There seems to be an alternate approach using the salamander lemma, as given in the references in this question. Salamander lemma can be used to prove the other important lemmas.

With all rows and columns exact. (This diagram lives in an abelian category). Show that there exists an exact sequence A1 → B1 → C1 A 1 → B 1 → C 1 making the diagram commute. Sure, it's not too hard with elements, I mean, it's just part of the snake lemma. However, proving it with universal properties is another story. By the universal property of the kernel, there are natural …

Jun 27, 2024 · Of course I can resort to R R -modules and do a good ol' diagram-chasing proof to check that (2) (2) is exact. My concern is: Can we prove exactness of (2) (2) from some of the diagram lemmas? (Snake, salamander, n × n n × n, four, five...) Or any indirect method that does not require chasing elements or morphisms.