Homework 2 corrections

Question 2.

One can define exterior powers over commutative rings + good properties. So if:

$P\oplus R^{n-1}\simeq R^n\Rightarrow P\simeq P{\otimes }_{R}R\simeq \underset{R}{\overset{1}{\wedge }}P{\otimes }_R\underset{R}{\overset{n-1}{\wedge }}{R}^{n-1}\simeq \underset{R}{\overset{n}{\wedge }}{R}^n\simeq R$

since rank of $P$ is one ( so constant ) over $R$ then $\underset{R}{\overset{k}{\wedge }}P=0\mathrm{\forall }k>1$. [ I have assumed in all this that $R$ is unital. ]

For reference you may look at the K-book chapter 1, pages: 4 (ex 1.6), 15, 16

For discussion of rank look at the K-book chapter 1, pages: 2, 9

Exercise. If $R$ has the invariant basis property (IBP), i.e. $R^n\simeq R^m\iff n=m$, then the rank of a stably free $R$-module $P$ defined by ${\text{rank}}_R(P)=m-n$ where $m,n\in \mathbb{\mathbb{N}}\cup 0$ are such that $P\oplus R^n\simeq R^m$ is well defined.