Definition of Exterior Power in Rotman

Question

In Rotman's An Introduction to Algebraic Topology, he defines the exterior power as:

If $M$ is an $A$-module and $p\ge0$, then the $p$th exterior power of $M$, denoted by $\bigwedge^pM$, is the abelian group with the following presentation:

Generators: $A\times M\times\dots\times M$ ($p$ factors $M$).

Relations: Some list of relations (I can type them up/screenshot them if needed)

He goes on to say:

If $F$ is the free abelian group with basis $A\times M\times\dots\times M$ and if $S$ is the subgroup of $F$ generated by the relations, then the coset $(a,m_1,\dots,m_p)+S$ is denoted by $am_1\wedge\dots\wedge m_p$. Thus every element of $\bigwedge^pM$ has an expression (not necessarily unique) of the form $\sum_ja_jm_1^j\wedge\dots\wedge m_p^j$ where $a_j\in A$ and $m_i^j\in M$.

I'm a bit confused by how $S$ is different from $\bigwedge^pM$. They seem to be defined the same way. In particular, $F$ seems to just be $\bigwedge^pM$ without the relations, so by adding in the relations (which gives us the exterior power), we should also get $S$. Obviously, this is wrong, but if someone could explain why, that'd be great.

Thanks!

Answer 1

$S$ is the submodule (of $F$) generated by the relations while $\bigwedge^p M$ is the quotient module $F/S$.

Answer 2

Perhaps read up on group presentations. In general you get a free group on the generators, and then you mod out by the normal subgroup generated by the relations. Looking at some examples would be a good idea.

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Take the dihedral group. A presentation is $D_n=\langle r,s\mid r^n,s^2,(rs)^2\rangle $.

So, you start with the free group on two generators, and mod out by the subgroup generated by the three relations.

What you get is the symmetries of a regular $n$-gon. (This group isn't abelian.)

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Somewhat more simply, consider the cyclic group of order $n$. It has presentation $\langle a\mid a^n\rangle $.

Here $F=\Bbb Z$ and $S=n\Bbb Z$.

Each element of $F/S$ is a coset. For instance, in $\Bbb Z/n\Bbb Z$, each coset is represented by an integer. To address your question above, there is a difference between a coset and its representative. The representatives of the $S$ cosets come from $F$. This seems to be the source of your confusion. Take $\Bbb Z_3$. The coset $[2]=\{2+3k\mid k\in\Bbb Z\}$, which is quite distinct from $2$ itself.

You are working in the category of abelian groups, so this link is relevant.