2025-10-24

This following is a very brief summary of what happened in class on 2025-10-24.

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 ${\mathrm{Hom}}_{R\text{-}\mathbf{\text{Mod}}}(M,-)$
• the hom-in functor ${\mathrm{Hom}}_{R\text{-}\mathbf{\text{Mod}}}(-,N)$
• the tensor product functor $M{\otimes }_R-$

We noted that, since the set ${\mathrm{Hom}}_{R\text{-}\mathbf{\text{Mod}}}(M,N)$ 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 then investigated the exactness of the functor ${\mathrm{Hom}}_{R\text{-}\mathbf{\text{Mod}}}(M,-):R\mathbf{\text{-Mod}}\to \mathbf{\text{Ab}}$. We proved that for every short exact sequence in $R\mathbf{\text{-Mod}}$

we have an exact sequence of abelian groups

Because of the above property, we say the functor ${\mathrm{Hom}}_{R\text{-}\mathbf{\text{Mod}}}(M,-)$ is left exact.

Next time we'll ask some follow-up questions, the first being whether there are $R$-modules $P$ for which the functor ${\mathrm{Hom}}_{R\text{-}\mathbf{\text{Mod}}}(P,-)$ is exact, i.e., retains exactness on the right.

Concepts

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

References

• Dummit & Foote, Abstract Algebra: Section 10.5