2024-29-10

This following is a very brief summary of what happened in class on 2024-10-29.

We began by defining morphisms of chain complexes and hence also morphisms of exact sequences. We noted that the commutativity condition in morphisms between exact sequences can result in some surprising conclusions, using the famous Short Five Lemma as an example. We'll return to "diagram lemmas" in glorious detail in a few weeks.

We then took our first steps towards understanding chain complexes (and exact sequences) by asking how they interact with functors. In particular, we set the stage to study the interaction between exact sequences and three functors:

• the hom-out functor ${\mathrm{Hom}}_R(M,-)$
• the hom-in functor ${\mathrm{Hom}}_R(-,N)$
• the tensor product functor $M{\otimes }_R-$

We noted that, since the set ${\mathrm{Hom}}_R(M,N)$ has a natural structure of an abelian group, the first two functors actually "factor through" the forgetful functor $U:\mathbf{\text{Ab}}\to \mathbf{\text{Set}}$. In other words, we can view them both as functors to the category of abelian groups. Because of this, it makes sense to ask whether the image of an exact sequence in the category $R\mathbf{\text{-Mod}}$ is another exact sequence in the category $\mathbf{\text{Ab}}$, or whether something goes wrong.

We ended by noting (but not proving!) that the hom-out functor is left exact but not exact. More on that next class!

Concepts

• Exact Sequences III - Morphisms of Exact Sequences
• Exact Sequences IV - Exact Sequences and Functors

References

• Dummit & Foote: Section 10.5