Consider $GL\_2$ as the affine group scheme with coordinate ring 
${\mathbb Z}[x\_1,x\_2,x\_3,x\_4,y]/(\det\left(\begin{array}{cc}x\_1& x\_2\\ x\_3& x\_4\end{array}\right)y-1)$.
The group scheme $PGL\_2$ is then given by the subring $S$ of $GL\_1$-invariants, which is the subring generated by all monomials of the form $x\_ix\_iy^2$. 
In this way, we define $PGL\_2( R )=Hom(S,R)$ for any ring $R$.
By Hilbert 90 we know that the sequence
$$
0\to GL\_1( R )\to GL\_2( R )\to PGL\_2( R )\to 1
$$
is exact if $R$ is a field. From that one can derive that it stays exact if $R$ is factorial. But what for a general commutative ring with unit? Is it always exact? If not, is there a handy description of all rings for which it is?



Is GL2( R ) - > PGL2( R ) surjective?
The answer is no. Here is an explicit example. Let $R=\mathbb{Z}[\sqrt{-5}]$.


Consider the matrix $$\left(\begin{array}{cc}1+\sqrt{-5}& 2\\ 2& 1-\sqrt{-5}\end{array}\right).$$


It represents an element of $PGL\_2(R)$ that is not in the image of $GL\_2(R)$.


The motivation for this example is that the ideal $(2,1+\sqrt{-5})$ is not principal in $R$ and this should be relevant because the next term in the long exact sequence is $H^1(R,\mathbb{G}\_m)$ (and so the sequence written in the question will be exact whenever this $H^1$ vanishes).


I would love to see a more conceptual proof of the failure of $GL\_2(R)\to PGL\_2(R)$ to be surjective.


