2024-10-31

This following is a very brief summary of what happened in class on Halloween!

We investigated the exactness of the functor ${Hom}_{R} (M, -) : R -Mod \to Ab$ . We proved that for every short exact sequence in $R -Mod$

0 \to J \overset{f}{\to} K \overset{g}{\to} L \to 0

we have an exact sequence of abelian groups

0 \to {Hom}_{R} (M, J) \overset{f \circ -}{\to} {Hom}_{R} (M, K) \overset{g \circ -}{\to} {Hom}_{R} (M, L) .

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

We then asked some follow-up questions, the first being whether there are $R$ -modules $M$ for which the functor ${Hom}_{R} (, M -)$ is exact, i.e., retains exactness on the right. We rephrased such a property in terms of diagrams, with the short version being right exactness of this functor would be equivalent to "morphisms from $M$ pulling back along surjections." We called such modules projective.

We ended with an unproven (by us) fact about projective modules.

Next time we'll repeat this analysis with the hom-in functor ${Hom}_{R} (-, N)$ and a tensor product functor $M \otimes_{R} -$ . For now, keep it spooky!

$Math Professor with Jack-o-Lantern Head.png|400$

Concepts

Exact Sequences IV - Exact Sequences and Functors

References

Dummit & Foote: Section 10.5