Michael Robinson (02/19/24): The snake lemma and indecomposable chain maps
Title: The snake lemma and indecomposable chain maps
Abstract:
The snake lemma is usually proven by a magnificent diagram chase. While theatrically satisfying, it is perhaps a bit opaque why the statement of the snake lemma is true at all. The crux of the matter is the behavior of the connecting homomorphism, and how it is induced by the rest of the short exact sequence. This talk will try to shed some light on the connecting homomorphism, by looking at the important-but-special case of chain complexes of vector spaces.
We will rely upon a complete (finite!) representation of the chain maps involved, due to Escolar and Hiraoka. This characterization turns out to be broadly useful to the algorithmic study of homological algebra, and also dramatically simplifies many standard proofs. This characterization reveals that there are nine "easy" indecomposable chain maps, and one exceptional indecomposable. That exceptional indecomposable turns out to be key in understanding when the connecting homomorphism is nontrivial.