2024-10-28

This following is a very brief summary of what happened in class on 2024-10-28.

After looking at some suggestive examples, we defined the notion of an exact sequence, at least in the category of $R$ -modules. We then revisited our original examples and considered several more. Highlights included the following:

a morphism $f : X \to Y$ is injective if and only if the sequence $0 \to X \overset{f}{\to} Y$ is exact at $X$
a morphism $f : X \to Y$ is surjective if and only if the sequence $X \overset{f}{\to} Y \to 0$ is exact at $Y$
whenever we have a submodule $N \subseteq M$ , we have an associated short exact sequence $0 \to N \to M \to M / N \to 0$
for any pair of $R$ -modules $M_{1}, M_{2}$ we have a short exact sequence $0 \to M_{1} \to M_{1} \oplus M_{2} \to M_{2} \to 0$
for any morphism $f : X \to Y$ we have the associated short exact sequence $0 \to \ker (f) \to X \to im (f) \to 0$

We ended by introducing the more general idea of a chain complex. As we'll soon see, functors between categories will be able to take chain complexes to chain complexes, whereas they will often take exact sequences to non-exact chain complexes. It's actually this "failure of exactness" that will make things interesting!

Concepts

References

Dummit & Foote: Section 10.5