Homework 2 corrections

Question 2.

We should assume that the module is not simply stably free, but that it is also as good as possible in that regards. That is, $P\oplus R=R^{2}~$.

Question 4. We do not have usually left exactness in the stated short exact sequence, so that part should be dropped.

Look at the K-book chapter 2 page 14 exercise 2.3 ( Excision for $K_{0}~$ and how to make a non-unital ring unital)

In the exact sequence $K_{0}(R,I)\xrightarrow {\pi _{2*}} K_{0}(R)\to K_{0}(R/I)~$

$\pi _{2}:D_{R}(I)\to R~$ is given by $\pi _{2}(x,y)=y~$ and $\pi _{2*}[-]=[R\otimes _{D_{R}(I)}^{\pi _{2}}-]~$

Question 1.

$(a)\Rightarrow (b)~$ is fine.

but $(b)\Rightarrow (a)~$ should be modified to:

if $R^{n}\xrightarrow {f} R\to 0~$ is an exact sequence then $f~$ is given by a unimodular row.

The simple example that makes statement of $(b)\Rightarrow (a)~$ false is :

$0\to \mathbb {Z} \xrightarrow {\times 2} \mathbb {Z} ~$

Question 2.

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

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

since rank of $P~$ is one ( so constant ) over $R~$ then $\wedge _{R}^{k}P=0\;\forall \;k>1~$. [ 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. $R^{n}\simeq R^{m}\Leftrightarrow n=m~$, then the rank of a stably free $R~$-module $P~$ defined by ${\text{rank}}_{R}(P)=m-n~$ where $m,n\in \mathbb {N} \cup {0}~$ are such that $P\oplus R^{n}\simeq R^{m}~$ is well defined.

Question 3.

$(a)\nRightarrow (b)~$ ===> So one needs to add more assumptions.

example. $(\mathbb {Z} \oplus \mathbb {Z} /2\mathbb {Z} )\otimes _{\mathbb {Z} }2\mathbb {Z} \simeq \mathbb {Z} ~$ but $\mathbb {Z} \oplus \mathbb {Z} /2\mathbb {Z} ~$ is not a projective $\mathbb {Z} ~$-module.

This question is the exercise 3.1 of the K-book chapter 1 on page 23, also look at pages 15-23 for the review of line bundles and the picard group.