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

Let be a commutative ring with 1 and let be a free unital -bimodule such that . Prove that the th exterior power is a free -module of rank .

Suppose is a free generating set of .

Recall from this previous exercise that the th tensor power is free with free generating set .

First we will construct a generating set for . Certainly is generated by the set of all simple -tensors . Note that, by multilinearity, we may assume that each comes from the free generating set : say . Moreover, up to a possible multiplication by -1, we can say that . We claim that these are all distinct. To see this, for each increasing choice function , define by letting if is a permutation of (namely the unique permutation ), and 0 otherwise, and extend linearly. Certianly for increasing choice functions , , so that we have an induced homomorphism . Moreover, is zero on all of the except the one whose indices are given by . Thus the elemens of our generating set are distinct. There are distinct ways to choose the indices , and so we have a generating set of of order .

Note that any nontrivial linear combination in induces a nontrivial linear combination in the free basis on ; thus our basis is free.

Hence is a free -module having free rank .

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