The standard open $D_{+}(f)$ of a projective quotient is the affine chart $\mathrm{Spec}((S_f{)}_0)$

For a homogeneous prime $\mathfrak{p}\subseteq S$ with $S_{+}⊈\mathfrak{p}$ one has $D_{+}(f)\cap D_{+}(x_i)=D_{+}(f{x}_i)$: a point of $D_{+}(x_i)$ is such a prime with $x_i\mathrm{\notin }\mathfrak{p}$, and $f\mathrm{\notin }\mathfrak{p}$ and $x_i\mathrm{\notin }\mathfrak{p}$ hold together exactly when $f{x}_i\mathrm{\notin }\mathfrak{p}$. Under the chart correspondence of [L1] and [L2], this condition is $f\mathrm{/}{x}_i^d\mathrm{\notin }\mathfrak{q}$ for the corresponding prime $\mathfrak{q}\subseteq A_i$, so the intersection is a distinguished open in $D_{+}(x_i)$. Since the standard charts cover $X$, $D_{+}(f)$ is open in $X$.

Inside the chart $D_{+}(x_i)=\mathrm{Spec}(A_i)$ the element $f$ has degree-zero dehomogenization $f\mathrm{/}{x}_i^d\in A_i$, and a prime $\mathfrak{q}\subseteq A_i$ corresponds to a point of $D_{+}(f{x}_i)$ exactly when $f\mathrm{/}{x}_i^d\mathrm{\notin }\mathfrak{q}$ by [L2], so $D_{+}(f{x}_i)=\mathrm{Spec}((A_i{)}_{f\mathrm{/}{x}_i^d})$ is the distinguished open subscheme of $\mathrm{Spec}(A_i)$ determined by $f\mathrm{/}{x}_i^d$.

Put $B=(S_f{)}_0$ and $u_i=x_i^d\mathrm{/}f\in B$. Inside $S_{f{x}_i}$ the degree-zero subrings $(S_{f{x}_i}{)}_0$, $B_{u_i}$ and $(A_i{)}_{f\mathrm{/}{x}_i^d}$ coincide. Indeed, write a degree-zero fraction as $s\mathrm{/}(f^a{x}_i^b)$ with $s$ homogeneous of degree $ad+b$. Choose $c$ with $dc\ge b$. Then $\frac{s}{f^a{x}_i^b}=\left(\frac{s{x}_i^{dc-b}}{f^{a+c}}\right)u_i^{-c},$ and the parenthesized fraction has degree zero, proving membership in $B_{u_i}$. The same fraction equals $\left(\frac{s}{x_i^{ad+b}}\right){\left(\frac{f}{x_i^d}\right)}^{-a},$ with a degree-zero parenthesized fraction in $A_i$, proving membership in $(A_i{)}_{f\mathrm{/}{x}_i^d}$. The reverse inclusions into $(S_{f{x}_i}{)}_0$ follow from degree preservation of localization. These equalities include the zero-ring case.

The elements $u_0,\dots ,u_n$ generate the unit ideal of $B=(S_f{)}_0$. To see this, put $N=n+1$. Every monomial of degree $Nd$ in the variables $x_0,\dots ,x_n$ is divisible by some $x_i^d$: otherwise all exponents are at most $d-1$, and the total degree is at most $(n+1)(d-1)<Nd$. Write the homogeneous representative of $f^N$ in the polynomial ring as ${\sum }_i{x}_i^d{g}_i$, with $g_i$ homogeneous of degree $(N-1)d$, and pass to $S_f$. Dividing by $f^N$ gives $1={\sum }_i(x_i^d\mathrm{/}f)(g_i\mathrm{/}{f}^{N-1})={\sum }_i{u}_i(g_i\mathrm{/}{f}^{N-1})$ in $B$. Therefore the distinguished opens $D_B(u_i)$ cover $\mathrm{Spec}B$.

The ring maps $A_i\to (S_{f{x}_i}{)}_0=B_{u_i}$ are localizations of $A_i$ and are compatible on overlaps: in $S_{f{x}_i{x}_j}$ all comparisons become the identity of the canonical localization of $B$, so the cocycle condition holds.

By step 1.4 the opens $D_B(u_i)=\mathrm{Spec}(B_{u_i})$ cover $\mathrm{Spec}B$. The maps from these pieces induced by step 2.1 agree on their overlaps and hence glue to a morphism $\mathrm{Spec}B\to X$. By steps 1.2 and 1.3 each piece maps isomorphically onto $D_{+}(f{x}_i)=D_{+}(f)\cap D_{+}(x_i)$, and those opens cover $D_{+}(f)$ because the $D_{+}(x_i)$ cover $X$. The local inverses agree on overlaps by the same localization identity, so the glued morphism is an isomorphism onto $D_{+}(f)$.

The isomorphism of step 3.1 identifies global sections of the structure sheaf on $D_{+}(f)$ with the global sections of $\mathrm{Spec}((S_f{)}_0)$, namely $(S_f{)}_0$, and the restriction maps to the pieces $D_{+}(f{x}_i)$ are the localizations displayed in step 2.1; taking $f=x_i$ gives $d=1$ and $(S_{x_i}{)}_0=A_i$, so the standard chart is recovered. AC is used only through the construction of the affine structure sheaves in [L1]; the finite localization and gluing calculations themselves make no choice.