2024-12-05

This following is a very brief summary of what happened in class on 2024-12-05.

We picked up our study of chain complexes, recalling their definition and defining morphisms of chain complexes. We noted that the category of chain complexes is itself an abelian category, so it made sense to talk about chain complexes in that category. This led to the idea of a double complex.

We then analyzed the various ways to measure exactness in a chain complex. We noted that there are "horizontal" and "vertical" homology objects, as well as two related quotient objects arising from certain "diagonal" arrows. We called these new objects the donor and receptor objects. We then noted the existence of intramural and extramural maps.

We finally brought all of these objects together in the (soon-to-be?) celebrated Salamander Lemma. While we didn't prove that lemma, we did see how it immediately leads to situations in which the extramural maps are isomorphisms.

Tomorrow we will start by noting situations in which the Salamander Lemma guarantees intramural maps are isomorphisms, and then we'll set the Salamander Lemma loose on a bunch of named diagram lemmas. Be warned, Snake Lemma! The Salamander Lemma is coming for you!

Concepts

References