2024-11-01

This following is a very brief summary of what happened in class on 2024-11-01.

After recapping the situation of the functor ${\mathrm{Hom}}_R(M,-):R\mathbf{\text{-Mod}}\to \mathbf{\text{Ab}}$ and projective modules, we repeated the analysis (skipping any proofs) for the functors ${\mathrm{Hom}}_R(-,N):(R\mathbf{\text{-Mod}}{)}^{\text{op}}\to \mathbf{\text{Ab}}$ and $M{\otimes }_R-:R\mathbf{\text{-Mod}}\to \mathbf{\text{Ab}}$, the latter under the assumption $R$ is commutative (so that we could use the Rapha construction to put $(R,R)$-bimodule structures on $R$-modules, as needed).

We noted the following:

• The functor ${\mathrm{Hom}}_R(-,N)$ is left exact. Modules $I$ for which this functor is exact are called injective.
• A module $I$ is injective if and only if it has the following property: whenever $I$ is a submodule of an $R$-module $M$, it is (isomorphic to) a direct summand of $M$
• The functor $M{\otimes }_R-$ is right exact. Modules $M$ for which this functor is exact are called flat.
• Projective $R$-modules are flat.

This ends our initial survey into how functors can interact with exact sequences. There is a ton more to the story, but we'll leave it for now.

Concepts

• Exact Sequences IV - Exact Sequences and Functors

References

• Dummit & Foote: Section 10.5