## The rank of the kth exterior power of a free module

Let $R$ be a commutative ring with 1 and let $M$ be a free unital $(R,R)$-bimodule such that $rm = mr$. Prove that the $k$th exterior power $\bigwedge^k(M)$ is a free $R$-module of rank ${n \choose k}$.

Suppose $E = \{e_i\}_{i=1}^n$ is a free generating set of $M$.

Recall from this previous exercise that the $k$th tensor power is free with free generating set $\{e_{i_1} \otimes \cdots \otimes e_{i_k} \ |\ i : k \rightarrow n\}$.

First we will construct a generating set for $\bigwedge^k(M)$. Certainly $\bigwedge^k(M)$ is generated by the set of all simple $k$-tensors $m_1 \wedge \cdots \wedge m_k$. Note that, by multilinearity, we may assume that each $m_i$ comes from the free generating set $E$: say $e_{i_1} \wedge \cdots \wedge e_{i_k}$. Moreover, up to a possible multiplication by -1, we can say that $i_1 < i_2 < \cdots < i_k$. We claim that these are all distinct. To see this, for each increasing choice function $\lambda : k \rightarrow n$, define $\varphi_\lambda : \mathcal{T}^k(M) \rightarrow R$ by letting $\varphi_\lambda(e_{j_k} \otimes \cdots \otimes e_{j_k}) = \epsilon(\sigma)$ if $\mathsf{im}\ j$ is a permutation of $\mathsf{im}\ \lambda$ (namely the unique permutation $\sigma$), and 0 otherwise, and extend linearly. Certianly for increasing choice functions $\lambda$, $\mathcal{A}^k(M) \subseteq \varphi_\lambda$, so that we have an induced homomorphism $\bigwedge^k(M) \rightarrow R$. Moreover, $\varphi_\lambda$ is zero on all of the $e_{i_1} \wedge \cdots \wedge e_{i_k}$ except the one whose indices are given by $\lambda$. Thus the elemens of our generating set are distinct. There are ${n \choose k}$ distinct ways to choose the indices $i_j$, and so we have a generating set $B$ of $\bigwedge^k(M)$ of order ${n \choose k}$.

Note that any nontrivial linear combination $\sum \alpha_i z_i = 0$ in $\bigwedge^k(M)$ induces a nontrivial linear combination in the free basis on $\mathcal{T}^k(M)$; thus our basis is free.

Hence $\bigwedge^k(M)$ is a free $R$-module having free rank ${n \choose k}$.