2024-10-11

This following is a very brief summary of what happened in class on 2024-10-11.

Picking up where we left off last class, we started with the official definition of the free module on a set $X$, as the set of all finite, formal $R$-linear combinations of the elements of $X$. We then proceeded to run through the checklist of all things required to officially conclude that the free module functor $F$ is left adjoint to the forgetful functor $U$. That included:

• Defining the $R$-module structure on $F(X)$
• Describing the arrow map for $F$, i.e., how each set map $f:X\to Y$ should be sent to an $R$-module morphism $F(f):F(X)\to F(Y)$
• Describing a set map $X\to U(F(X))$, sending each element $x$ to the "basis element" $e_x$ (which is the formal sum with coefficient $1_R$ for $x$ and $0_R$ for all other elements of $X$)
• Describing the set maps ${\varphi }_{X,M}$ that takes each module morphism $F(X)\to M$ to a set map $X\to U(M)$
• Describing the (inverse) set map ${\xi }_{X,M}$ that takes each set map $X\to U(M)$ to a module morphism $F(X)\to M$
• Describing the naturality conditions on our bijections

For most of these items, we explicitly defined the maps, but then simply noted what one would need to check. (I promise that none of those things we skipped actually checking are very devious or interesting!)

We ended by defining the submodule generated by a subset $X\subseteq U(M)$, as well as what it means for a subset to generate $M$, and for $M$ to be cyclic.

Concepts

• Free modules
• Examples of free modules
• Generators for modules and submodules

References

• Dummit & Foote: Section 10.3