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Let $K$ be a $k$-algebra, consider $o(n,K)$ the set of orthogonal matrices of size $n^2$ over $K$ ( i.e $M M^T = I $). Let´s do the case $n=2$. Let's consider the ring $$ A = k\left[ {x,y,z,w} \right]/\left( {x^2 + y^2 - 1,xz + yw,z^2 + w^2 - 1} \right). $$

Where the system is from the equation $M M^T = I$ with $M$ a matrix with entries $x$, $y$, $z$, $w$ , and then equaling to zero. I have to prove that there exist a natural bijection between:

$o(2,K)$ and $\operatorname{Hom}_k( A , K )$ where $\operatorname{Hom}_k( A , K )$ denotes the homomorphism of $k$-algebras $f\colon A \to K$ i.e homomorphism of rings, that fix $k$. Only guessing I think that the map it´s that it take a matrix, and send to the homomorphism, that send each coordinate $ T_i $ of $A$, to one of the coordinates of the orthogonal matrix.

But I can't prove neither injectivity nor surjectivity.

EDITED: Thanks!

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  • $\begingroup$ Am I right in thinking that where you wrote "set of matrix" (twice), you intended the plural, "set of matrices"? $\endgroup$
    – joriki
    Apr 24, 2012 at 4:36
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    $\begingroup$ This is a general property: if $A=k[x_1, \dots, x_n]/(F_1,...,F_m)$, then the set of $x=(x_1, \dots, x_n)\in K^n$ such that $F_1(x)=\cdots=F_m(x)=0$ is canonically in bijection with the set $Hom_k(A, K)$ of $k$-algebra homomorphisms from $A$ to $K$. It is simpler to do the general case rather the specific situation. $\endgroup$
    – user18119
    Apr 25, 2012 at 13:06

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