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Ryan Reich
Ryan Reich
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This is rather a comment to Steven Landsburg/ Vaserstein answer, but too long for a comment.

Finding a quotient of $\mathbf Z[X]$ with $E_2 \neq \mathrm{SL}_2$ is easy. Actually, $\mathbf Z[X]$ will do. For example, Cohn proved that the matrix : $$\begin{bmatrix} 1+2x & 4 \\ -x^2 & 1-2x \end{bmatrix} $$$$\begin{bmatrix} 1+2x & 4 \cr -x^2 & 1-2x \end{bmatrix} $$ is in $\mathrm{SL}_2$ but not in $E_2$.

If one prefers an example in the spirit of the one suggested by Vaserstein, then Cohn (again!) shjowed that in the ring of integers of $\mathbf Q[\sqrt{-19}]$, with $\theta= \frac{1+\sqrt{-19}}{2}$, the matrix $$ \begin{bmatrix} 3- \theta & 2+ \theta \\ -3-2\theta & 5-2\theta \end{bmatrix} $$$$ \begin{bmatrix} 3- \theta & 2+ \theta \cr -3-2\theta & 5-2\theta \end{bmatrix} $$ is in $\mathrm{SL}_2$ but not in $E_2$.

(I took these two examples of matrices from the book of T.Y. Lam on "Serre's problem on projective module, Chapter I.9.)

With these examples, it should be straightforward to make explicit a unimodular row, but I have to admit that I don't understand Vaserstein's argument. (Is it well known that $\mathrm{sr}(A)= 2 $ implies $\mathrm{SL}_2(A)= E_2(A)$?)

This is rather a comment to Steven Landsburg/ Vaserstein answer, but too long for a comment.

Finding a quotient of $\mathbf Z[X]$ with $E_2 \neq \mathrm{SL}_2$ is easy. Actually, $\mathbf Z[X]$ will do. For example, Cohn proved that the matrix : $$\begin{bmatrix} 1+2x & 4 \\ -x^2 & 1-2x \end{bmatrix} $$ is in $\mathrm{SL}_2$ but not in $E_2$.

If one prefers an example in the spirit of the one suggested by Vaserstein, then Cohn (again!) shjowed that in the ring of integers of $\mathbf Q[\sqrt{-19}]$, with $\theta= \frac{1+\sqrt{-19}}{2}$, the matrix $$ \begin{bmatrix} 3- \theta & 2+ \theta \\ -3-2\theta & 5-2\theta \end{bmatrix} $$ is in $\mathrm{SL}_2$ but not in $E_2$.

(I took these two examples of matrices from the book of T.Y. Lam on "Serre's problem on projective module, Chapter I.9.)

With these examples, it should be straightforward to make explicit a unimodular row, but I have to admit that I don't understand Vaserstein's argument. (Is it well known that $\mathrm{sr}(A)= 2 $ implies $\mathrm{SL}_2(A)= E_2(A)$?)

This is rather a comment to Steven Landsburg/ Vaserstein answer, but too long for a comment.

Finding a quotient of $\mathbf Z[X]$ with $E_2 \neq \mathrm{SL}_2$ is easy. Actually, $\mathbf Z[X]$ will do. For example, Cohn proved that the matrix : $$\begin{bmatrix} 1+2x & 4 \cr -x^2 & 1-2x \end{bmatrix} $$ is in $\mathrm{SL}_2$ but not in $E_2$.

If one prefers an example in the spirit of the one suggested by Vaserstein, then Cohn (again!) shjowed that in the ring of integers of $\mathbf Q[\sqrt{-19}]$, with $\theta= \frac{1+\sqrt{-19}}{2}$, the matrix $$ \begin{bmatrix} 3- \theta & 2+ \theta \cr -3-2\theta & 5-2\theta \end{bmatrix} $$ is in $\mathrm{SL}_2$ but not in $E_2$.

(I took these two examples of matrices from the book of T.Y. Lam on "Serre's problem on projective module, Chapter I.9.)

With these examples, it should be straightforward to make explicit a unimodular row, but I have to admit that I don't understand Vaserstein's argument. (Is it well known that $\mathrm{sr}(A)= 2 $ implies $\mathrm{SL}_2(A)= E_2(A)$?)

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Oblomov
Oblomov
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This is rather a comment to Steven Landsburg/ Vaserstein answer, but too long for a comment.

Finding a quotient of $\mathbf Z[X]$ with $E_2 \neq \mathrm{SL}_2$ is easy. Actually, $\mathbf Z[X]$ will do. For example, Cohn proved that the matrix : $$\begin{bmatrix} 1+2x & 4 \\ -x^2 & 1-2x \end{bmatrix} $$ is in $\mathrm{SL}_2$ but not in $E_2$.

If one prefers an example in the spirit of the one suggested by Vaserstein, then Cohn (again!) shjowed that in the ring of integers of $\mathbf Q[\sqrt{-19}]$, with $\theta= \frac{1+\sqrt{-19}}{2}$, the matrix $$ \begin{bmatrix} 3- \theta & 2+ \theta \\ -3-2\theta & 5-2\theta \end{bmatrix} $$ is in $\mathrm{SL}_2$ but not in $E_2$.

(I took these two examples of matrices from the book of T.Y. Lam on "Serre's problem on projective module, Chapter I.9.)

With these examples, it should be straightforward to make explicit a unimodular row, but I have to admit that I don't understand Vaserstein's argument. (Is it well known that $\mathrm{sr}(A)= 2 $ implies $\mathrm{SL}_2(A)= E_2(A)$?)