I have a question concerning the construction of $\mathbf{Proj}$.
We take a graded quasicoherent sheaf $\mathscr{R} = \bigoplus \limits_{n=0}^\infty \mathscr{R}_n$.
Next for all affine subsets $U = \text{Spec}(A)$ we have $\Gamma(U, \mathscr{R}) = \bigoplus \limits_{n=0}^\infty \Gamma(U,\mathscr{R}_n)$ and
therefore $\Gamma(U, \mathscr{R})$ has a structure of a graded ring over $\Gamma(U,\mathscr{O}_X)$.
Next we claim the existence of a morphism
$$\pi: \operatorname{Proj}(\Gamma(U,\mathscr{R})) \longrightarrow U$$
But how does this map work?
My first idea was to use the fact that for any scheme $X$ and for any affine scheme $Y = \operatorname{Spec}(A)$
the set $\operatorname{Hom}(X,Y)$ is in one-to-one correspondence with the set
$\operatorname{Hom}(A,\Gamma(X,\mathscr{O}_X))$. But I want a concrete mapping, and the latter fact gives (as far as I can tell) no information.
I also find a lemma on a Stacksproject: https://stacks.math.columbia.edu/tag/01M7, but I can't figure out the way
it could be applied here.
Thanks in advance?
Best Answer
If $R$ is a ring and $A$ is a graded $R$-algebra, then one can construct a morphism of schemes $\text{Proj}(A) \to \text{Spec}(R)$. This just follows from the Proj construction. To see exactly why, go through the first two pages of chapter $13$, section $2$ of Algebraic Geometry 1 by Görtz and Wedhorn.
In your case, $\Gamma(U, \mathscr{R})$ has the structure of a graded ring over $\Gamma(U,\mathscr{O}_X)$. So there is a morphism $$ \operatorname{Proj}(\Gamma(U,\mathscr{R})) \to \text{Spec}(\Gamma(U,\mathscr{O}_X)) \cong U.$$