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added edit to clarify things

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edited Feb 16, 2018 at 17:15

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After spending considerable time trying to construct a counterexample, I turned to Google and found the book "Rings Related to Stable Range Conditions" by Huanyin Chen, which claims, on page 338, that the stable range of ${\mathbb Z}[X]$ is in fact 2. This would explain my inability to find a counterexample (though not, perhaps, my willingness to put aside other urgent projects in order to look for one). However, I've not been able to understand his argument.

There is clearly a typo where he says "Clearly,${\mathbb Z}[x]$ is a euclidean integral domain; hence it is a Dedekind domain". Presumably he means ${\mathbb Z}$ instead of ${\mathbb Z}[x]$? But then I cannot fully follow the rest of the argument, possibly because it references Example 12.1.14, which is not part of the preview available on either Amazon or Google books.

In any event, though you probably already know this, the simplest class of nontrivial unimodular rows over ${\mathbb Z}[X]$ consists of those of the form $(1+aX,bX^m,cX^m)$ where $b,c$ and $m$ are arbitrary and some power of $a$ lives in the ideal $(b,c)$. I tried to find $a,b,c$ for which this row was provably not reducible, but I was insufficiently clever to pull this off.

Edit This claim that the stable range of ${\mathbb Z}[X]$ is 2 is not correct according to this other answer of mine.

After spending considerable time trying to construct a counterexample, I turned to Google and found the book "Rings Related to Stable Range Conditions" by Huanyin Chen, which claims, on page 338, that the stable range of ${\mathbb Z}[X]$ is in fact 2. This would explain my inability to find a counterexample (though not, perhaps, my willingness to put aside other urgent projects in order to look for one). However, I've not been able to understand his argument.

There is clearly a typo where he says "Clearly,${\mathbb Z}[x]$ is a euclidean integral domain; hence it is a Dedekind domain". Presumably he means ${\mathbb Z}$ instead of ${\mathbb Z}[x]$? But then I cannot fully follow the rest of the argument, possibly because it references Example 12.1.14, which is not part of the preview available on either Amazon or Google books.

In any event, though you probably already know this, the simplest class of nontrivial unimodular rows over ${\mathbb Z}[X]$ consists of those of the form $(1+aX,bX^m,cX^m)$ where $b,c$ and $m$ are arbitrary and some power of $a$ lives in the ideal $(b,c)$. I tried to find $a,b,c$ for which this row was provably not reducible, but I was insufficiently clever to pull this off.

After spending considerable time trying to construct a counterexample, I turned to Google and found the book "Rings Related to Stable Range Conditions" by Huanyin Chen, which claims, on page 338, that the stable range of ${\mathbb Z}[X]$ is in fact 2. This would explain my inability to find a counterexample (though not, perhaps, my willingness to put aside other urgent projects in order to look for one). However, I've not been able to understand his argument.

There is clearly a typo where he says "Clearly,${\mathbb Z}[x]$ is a euclidean integral domain; hence it is a Dedekind domain". Presumably he means ${\mathbb Z}$ instead of ${\mathbb Z}[x]$? But then I cannot fully follow the rest of the argument, possibly because it references Example 12.1.14, which is not part of the preview available on either Amazon or Google books.

In any event, though you probably already know this, the simplest class of nontrivial unimodular rows over ${\mathbb Z}[X]$ consists of those of the form $(1+aX,bX^m,cX^m)$ where $b,c$ and $m$ are arbitrary and some power of $a$ lives in the ideal $(b,c)$. I tried to find $a,b,c$ for which this row was provably not reducible, but I was insufficiently clever to pull this off.

Edit This claim that the stable range of ${\mathbb Z}[X]$ is 2 is not correct according to this other answer of mine.

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created Jun 5, 2013 at 22:37

Steven Landsburg

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After spending considerable time trying to construct a counterexample, I turned to Google and found the book "Rings Related to Stable Range Conditions" by Huanyin Chen, which claims, on page 338, that the stable range of ${\mathbb Z}[X]$ is in fact 2. This would explain my inability to find a counterexample (though not, perhaps, my willingness to put aside other urgent projects in order to look for one). However, I've not been able to understand his argument.

There is clearly a typo where he says "Clearly,${\mathbb Z}[x]$ is a euclidean integral domain; hence it is a Dedekind domain". Presumably he means ${\mathbb Z}$ instead of ${\mathbb Z}[x]$? But then I cannot fully follow the rest of the argument, possibly because it references Example 12.1.14, which is not part of the preview available on either Amazon or Google books.

In any event, though you probably already know this, the simplest class of nontrivial unimodular rows over ${\mathbb Z}[X]$ consists of those of the form $(1+aX,bX^m,cX^m)$ where $b,c$ and $m$ are arbitrary and some power of $a$ lives in the ideal $(b,c)$. I tried to find $a,b,c$ for which this row was provably not reducible, but I was insufficiently clever to pull this off.