automorphism of the projective space $\mathbb{P}_A^n$

Solution 1:

Ch. 0, §5 of Mumford's Geometric Invariant Theory (3rd ed.) answers 1) and 4) well, I think. Let $PGL(n+1) = \{\operatorname{det} (a_{ij}) \ne 0\} \subset \operatorname{Proj}\mathbb{Z}[a_{00},\ldots,a_{nn}]$ denotes the projective general linear group which acts on $\mathbb{P}^n_\mathbb{Z}$ in the usual way. Let $\mathscr{PGL}(n+1)$ denote the functor $S \mapsto \operatorname{Hom}(S,PGL(n+1))$ in the category of noetherian schemes. Let $\operatorname{Aut}(\mathbb{P}^n)$ denote the functor sending $S$ to the group of automorphisms $\operatorname{Aut}_S(\mathbb{P}^n_S)$ of $\mathbb{P}^n_S$ over $S$.

Theorem. The functors $\mathscr{PGL}(n+1)$ and $\operatorname{Aut}(\mathbb{P}^n)$ are isomorphic.

I don't know of a better description in general (i.e. for non-noetherian schemes), but this seems to answer 4).

Moreover, his proof answers 1): the invertible sheaves on $\mathbb{P}^n_S = \mathbb{P}^n_\mathbb{Z} \times S$ are all of the form $p_1^*(\mathcal{O}_{\mathbb{P}^n_\mathbb{Z}}(k)) \otimes p_2^*(L)$ where $L$ is an invertible sheaf on $S$, and $p_1,p_2$ are the projection morphisms. So, if $\alpha \in \operatorname{Aut}_S(\mathbb{P}^n_S)$, then $\alpha^*(\mathcal{O}_{\mathbb{P}^n_S}(1)) \cong p_1^*(\mathcal{O}_{\mathbb{P}^n_\mathbb{Z}}(k)) \otimes p_2^*(L)$ for some $k$ and $L$. Then, since $\alpha$ is an automorphism over $S$, we must have

$$\bigoplus_{i=0}^n X_i \cdot \mathcal{O}_S \cong p_{2*}p_1^*\mathcal{O}_{\mathbb{P}^n_\mathbb{Z}}(1) \cong p_{2*}(p_1^*(\mathcal{O}_{\mathbb{P}^n_\mathbb{Z}}(k)) \otimes p_2^*(L)) \cong \bigoplus_{r_0+\cdots+r_n=k}^n (X_0^{r_0}\cdots X_n^{r_n}) \cdot L$$

so $k=1$ and in fact by [EGAII, 4.2] every isomorphism $\alpha$ can be obtained by choosing an invertible sheaf $L$ and an isomorphism $\bigoplus_{i=0}^n X_i \cdot \mathcal{O}_S \overset{\sim}{\to} \bigoplus_{i=0}^n X_i \cdot L$. What's nice is that before, $PGL(n+1)$ consisted of $(n+1) \times (n+1)$ matrices of sections of $\mathcal{O}_{\mathbb{P}^n_\mathbb{Z}}(1)$ with non-vanishing determinant; now the isomorphisms $\bigoplus_{i=0}^n X_i \cdot \mathcal{O}_S \overset{\sim}{\to} \bigoplus_{i=0}^n X_i \cdot L$ are given by $(n+1) \times (n+1)$ matrices of sections of $L$ with non-vanishing determinant.

In particular, to answer your comment on Qiaochu's answer, it looks like $S$ having trivial Picard group is sufficient went to answer 3) affirmatively, but I'm not convinced it's necessary: by the proof it looks like what we need is that for all invertible sheaves $L$ on S such that $\bigoplus_{i=0}^n X_i \cdot \mathcal{O}_S$ and $\bigoplus_{i=0}^n X_i \cdot L$ are isomorphic, we also have that $\mathcal{O}_S$ and $L$ are isomorphic.

Solution 2:

2) No. Here is a topological analogy, which I learned from MO: instead of commutative rings, let's think about C*-algebras of continuous functions on compact Hausdorff spaces $X$. Then $\text{GL}_n(C(X))$ is equivalently the space of continuous functions $X \to \text{GL}_n(\mathbb{C})$, for example. The "correct" definition of $\text{PGL}_n(C(X))$, e.g. the one that makes it representable by a space, is therefore the space of continuous functions $X \to \text{PGL}_n(\mathbb{C})$.

This is not the same as $\text{GL}_n(C(X))$ mod its center; the discrepancy is described up to homotopy by an exact sequence

$$H^{0}(X, \mathbb{C}^{\times}) \to H^{0}(X, \text{GL}_n(\mathbb{C})) \to H^{0}(X, \text{PGL}_n(\mathbb{C})) \to H^{1}(X, \mathbb{C}^{\times}) \to \cdots$$

associated to the short exact sequence

$$1 \to \mathbb{C}^{\times} \to \text{GL}_n(\mathbb{C}) \to \text{PGL}_n(\mathbb{C}) \to 1$$

of topological groups, where by $H^0(X, G)$ I mean homotopy classes of continuous maps $X \to G$ and by $H^1(X, G)$ I mean homotopy classes of continuous maps $X \to BG$, where $BG$ is the delooping of $G$. Here $H^1(X, \mathbb{C}^{\times}) \cong \text{Pic}(X) \cong H^2(X, \mathbb{Z})$ is the Picard group of continuous line bundles on $X$, so the exact sequence tells us that the obstruction to lifting an element in $\text{PGL}_n(C(X))$ to an element of $\text{GL}_n(C(X))$, up to homotopy, is given by a line bundle on $X$; roughly speaking this is because there is a $\mathbb{C}^{\times}$s worth of lifts locally but no guarantee that it is possible to patch them up globally into a consistent global lift.