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$\textbf{Liu Qing, Proposition 5.1.31.}$

Let $Y=\operatorname{Proj}A[T_0,\cdots,T_d]$ be a projective space over a ring $A$, and let $X$ be a scheme over $A$.

(b) Conversely, for any invertible sheaf $\mathscr L$ on $X$ generated by $d+1$ global sections $s_0,\cdots,s_d$, there exists a morphism $f:X\to Y$ such that $\mathscr L\simeq f^*\mathscr O_Y(1)$ and $f^*T_i=s_i$ via this isomorphism.

$\textbf{Proof}$

(b) The open subsets $X_{s_i}$ cover $X$. For every $i\le d$, let us consider the morphism $f_i:X_{s_i}\to D_+(T_i)$ corresponding to the ring homomorphism $\mathscr O_Y(D_+(T_i))\to \mathscr O_X(X_{s_i}),$

$T_j/T_i\mapsto s_j/s_i\in \mathscr O_X(X_{s_i})$.

It is clear that the morphisms $f_i$ glue to a morphism $f:X\to Y$, and that $\mathscr L\simeq f^*\mathscr O_Y(1)$ by Proposition 1.14(b) and the description in Example 1.19.

Now I'll explain the above proof.

For every $i\le d$, let us consider the morphism $f_{ij}:X_{s_i}\cap X_{s_j}\to X_{s_i}\to D_+(T_i)$ corresponding to the ring homomorphism $\mathscr O_Y(D_+(T_i))\to \mathscr O_X(X_{s_i})\to\mathscr O_X(X_{s_i}\cap X_{s_j}),$

$T_k/T_i\mapsto s_k/s_i\mapsto s_k/s_i\in \mathscr O_X(X_{s_i}\cap X_{s_j})$.

It can be factored as follows: $\mathscr O_Y(D_+(T_i))\to \mathscr O_Y(D_+(T_iT_j))\to \mathscr O_X(X_{s_i}\cap X_{s_j})$

$T_k/T_i\mapsto (T_kT_j)/(T_iT_j)\mapsto s_k/s_i\in \mathscr O_X(X_{s_i}\cap X_{s_j})$.

The second ring homomorphism

$\mathscr O_Y(D_+(T_iT_j))\to \mathscr O_X(X_{s_i}\cap X_{s_j})$

is defined as follows: $(T_mT_n)/(T_iT_j)\mapsto (s_m/s_i)(s_n/s_j)\in \mathscr O_X(X_{s_i}\cap X_{s_j})$.

Let $s_j/s_i=a, s_i/s_j=b$.

$\because s_j=as_i=(ab)s_j$,

$\therefore ab=1$,

$\therefore (s_m/s_i)(s_n/s_j)=(s_n/s_i)(s_m/s_j)$.

$\therefore$The ring homomorphism is defined well.

$\therefore f_{ij}(X_{s_i}\cap X_{s_j})\subset D_+(T_iT_j)$ and $f_{ij}=f_{ji}.$

It is clear that the morphisms $f_i$ glue to a morphism $f:X\to Y$.

Is my argument correct?

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