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Here is a really cool illustration of the principle which Emerton was outlining. We know that the Picard group of projective space $(n-1)$-space over a field $k$ is $\mathbf{Z}$, \mathbf{Z}$($n \ge 2$), generated by$\mathcal{O}(1)$. This underlies the proof that the automorphism group of such a projective space is${\rm{PGL}}_n(k)$. But what is the automorphism group of$\mathbf{P}^n_A$\mathbf{P}^{n-1}_A$ for a general ring $A$? Is it ${\rm{PGL}}_n(A)$? That is, there is a natural map $${\rm{PGL}}_n(A) {\rm{PGL}} _n (A) \rightarrow {\rm{Aut}}_A(\mathbf{P}^n_A)$$ \rm{Aut}} _A (\mathbf{P}^{n-1} _A)$$(see my answer in the posting called something like "Is {\rm{PSL}}_2 = {\rm{PGL}} _2?") and we want to know if it is an isomorphism. It's a really important fact that the answer is yes. But how to prove it? It's a shame that this isn't done in Hartshorne. By an elementary localization (as in the last step of Emerton's answer), we may assume A is local. In this case we claim that {\rm{Pic}}(\mathbf{P}^n_A) {\rm{Pic}}(\mathbf{P}^{n-1}_A) is infinite cyclic generated by \mathcal{O}(1). Since this line bundle has the known A-module of global sections, it would give the desired result if true (since {\rm{PGL}}_n(A) = {\rm{GL}}_n(A)/A^{\times} for local A) by the same argument as in the field case. And since we know the Picard group over the residue field, we can twist to get to the case when the line bundle \mathcal{L} is trivial on the special fiber and then we can formulate the problem in two equivalent ways: (I)"lift" the generating section there to a generating section over A, or (II) prove that f _{\ast}(\mathcal{L}) is invertible in A with the natural map f^{\ast}(f _{\ast} \mathcal{L}) \rightarrow \mathcal{L} an isomorphism. How to do it? Step 0: The case when A is a field. Done. Step 1: The case when A is artin local, via (I): this goes via induction on the length, the case of length 0 being Step 0 and the induction resting on cohomological results for projective space over the residue field. Step 2: The case when A is complete local noetherian ring. This goes via (I) using Step 1 and the theorem on formal functions (formal schemes in disguise). Step 3: The case when A is local noetherian. This is faithfully flat descent for (II) from Step 2 applied over \widehat{A}. Step 4: The case when A is local: descent from the noetherian local case in Step 3 via direct limit arguments. "QED" 1 Here is a really cool illustration of the principle which Emerton was outlining. We know that the Picard group of projective space over a field k is \mathbf{Z}, generated by \mathcal{O}(1). This underlies the proof that the automorphism group of such a projective space is {\rm{PGL}}_n(k). But what is the automorphism group of \mathbf{P}^n_A for a general ring A? Is it {\rm{PGL}}_n(A)? That is, there is a natural map$${\rm{PGL}}_n(A) \rightarrow {\rm{Aut}}_A(\mathbf{P}^n_A) (see my answer in the posting called something like "Is ${\rm{PSL}}_2 = {\rm{PGL}} _2$?") and we want to know if it is an isomorphism. It's a really important fact that the answer is yes. But how to prove it? It's a shame that this isn't done in Hartshorne.

By an elementary localization (as in the last step of Emerton's answer), we may assume $A$ is local. In this case we claim that ${\rm{Pic}}(\mathbf{P}^n_A)$ is infinite cyclic generated by $\mathcal{O}(1)$. Since this line bundle has the known $A$-module of global sections, it would give the desired result if true (since ${\rm{PGL}}_n(A) = {\rm{GL}}_n(A)/A^{\times}$ for local $A$) by the same argument as in the field case. And since we know the Picard group over the residue field, we can twist to get to the case when the line bundle $\mathcal{L}$ is trivial on the special fiber and then we can formulate the problem in two equivalent ways: (I)"lift" the generating section there to a generating section over $A$, or (II) prove that $f _{\ast}(\mathcal{L})$ is invertible in $A$ with the natural map $f^{\ast}(f _{\ast} \mathcal{L}) \rightarrow \mathcal{L}$ an isomorphism. How to do it?

Step 0: The case when $A$ is a field. Done.

Step 1: The case when $A$ is artin local, via (I): this goes via induction on the length, the case of length 0 being Step 0 and the induction resting on cohomological results for projective space over the residue field.

Step 2: The case when $A$ is complete local noetherian ring. This goes via (I) using Step 1 and the theorem on formal functions (formal schemes in disguise).

Step 3: The case when $A$ is local noetherian. This is faithfully flat descent for (II) from Step 2 applied over $\widehat{A}$.

Step 4: The case when $A$ is local: descent from the noetherian local case in Step 3 via direct limit arguments.

"QED"