This question is a follow up to a question I asked some time ago,

Automorphism of $\mathbb{P}_A^n$

From the nice answers given, there is an exact sequence

$$1 \to GL(n+1,A)/A^* \to Aut(\mathbb{P}_A^n) \overset{\lambda}{\to} Pic(A)$$

Therefore, the obstruction for $PGL_n(A) = GL(n+1)/A^*$ is the vanishing of the boundary map $\lambda$.

On the other hand, I have seen that $PGL(n) \cong \operatorname{Spec} k[x_{ij},det(x_{ij}]^{\mathbb{G}_m}$. However, isn't this just the same as the affine scheme representing $\mathbb{GL}(n+1)/\mathbb{G}_m$ ?

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    $\begingroup$ Yes. Taking $A$-points doesn’t commute with taking quotients. $\endgroup$ – Qiaochu Yuan Aug 12 at 2:58
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    $\begingroup$ Oh ok. $(GL(n+1)/ \mathbb{G}_m)(A) \neq GL(n+1)(A)/\mathbb{G}_m(A)$. $\endgroup$ – user7090 Aug 12 at 3:03

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