How to do diagram chasing effectively?

Question

I am trying to teach myself some homological algebra, and the book I am using is Aluffi's wonderful Algebra: Chapter 0, which introduces homology at the end of chapter 3.

I have spent a lot of time on this book, doing all the exercises because I am new to category theory. But when the author reaches the Snake Lemma, I felt quite lost. I can produce a proof but it is a quite painful experience.

I also searched here, and got a better understanding of the Snake after someone recommended Bergman's Salamander Lemma, but I am still uncomfortable chasing diagrams.

In particular, I am wondering whether there is some effective way to do diagram chasing. Is it just experience or I need to change my mindset?

Again I am trying to teach myself, so I cannot actually get those visual proofs.

Thanks!

Answer 1

The first rule about diagram chasing is: You do it once - and only once. Of course, the expression "diagram chasing" already suggests that one needs to somewhat annoyingly run wild across all arrows back and forth. But from my experience, you have essentially only one option what to do next at each step, starting with the given and trying to move at least somewhat towards the destination and taking into account where you have exactness and so on, i.e. even though the number of single steps may seem considerable, all in all the snake lemma is less complicated than the Fox, goose and bag of beans puzzle.

Answer 2

Here is how I like to think of the proof of the snake lemma. This metaphor doesn't prove the snake lemma, it just keeps track of what you want to check!

I want to get from the top right of the diagram (kernel) to the bottom left (cokernel). I move down one step, left one step (swimming upstream), down one step, left one step (swimming upstream), down one step. I have to justify swimming upstream.

The first time I swam upstream, it was clear I could do it (because the arrow I was swimming against was surjective), but I had to make a choice (i.e. a lift EXISTS but is not UNIQUE). So I have to remember at the end to check that my final answer did not depend upon my choice.

The second time I swam upstream, it was not clear I could do it, but on the other hand, it was clear that if I could, I wouldn't have to make any choices (because the arrow I was swimming against was injective - that is, a lift may fail to EXIST, but if it does, it is guaranteed to be UNIQUE). So I have to remember to check that I was in the image of this arrow.

Answer 3

Kashiwara gives a completely element-free proof of the Snake Lemma in his book "Categories and Sheaves" on page 297.