Relative Proj commutes with arbitrary base change

The affine chart comparison. Let $B$ be a graded $R$-algebra, $R^{\mathrm{\prime }}$ an $R$-algebra and $B^{\mathrm{\prime }}=B{\otimes }_R{R}^{\mathrm{\prime }}$ with its induced grading. For homogeneous $f\in B_{+}$ of positive degree and its image $f^{\mathrm{\prime }}\in B^{\mathrm{\prime }}$, [F4] gives $(B_{(f)}){\otimes }_R{R}^{\mathrm{\prime }}\cong (B^{\mathrm{\prime }}{)}_{(f^{\mathrm{\prime }})}$ naturally; the transition identifications on $D_{+}(fg)$ are the localisations $B_{(f)}\to B_{(fg)}$, which base change to the corresponding localisations of $B^{\mathrm{\prime }}$ by [F4], so the chart data of $\mathrm{Proj}B$ base changed along $R\to R^{\mathrm{\prime }}$ agree with the chart data of $\mathrm{Proj}{B}^{\mathrm{\prime }}$, and the shifted modules base change by the same computation: $(B(n{)}_{(f)}){\otimes }_R{R}^{\mathrm{\prime }}\cong (B^{\mathrm{\prime }}(n){)}_{(f^{\mathrm{\prime }})}$ for every $n\in \mathbb{Z}$.

The affine quotient comparison. If $I\subseteq B$ is a homogeneous ideal and $I^{\mathrm{\prime }}=I{B}^{\mathrm{\prime }}$, then for homogeneous $f$ one has $(B_{(f)}\mathrm{/}{I}_{(f)}){\otimes }_R{R}^{\mathrm{\prime }}\cong B_{(f^{\mathrm{\prime }})}^{\mathrm{\prime }}\mathrm{/}{I}_{(f^{\mathrm{\prime }})}^{\mathrm{\prime }}$, because localisation at $f$ commutes with base change and tensor product is right exact by [F4]; hence the chart ideals of $\mathrm{Proj}(B\mathrm{/}I)$ base change to those of $\mathrm{Proj}(B^{\mathrm{\prime }}\mathrm{/}{I}^{\mathrm{\prime }})$.

The affine case of the isomorphism. For every ring map $R\to R^{\mathrm{\prime }}$ the chart comparison of step 1.1 produces a canonical isomorphism $\mathrm{Proj}B{\times }_{\mathrm{Spec}R}\mathrm{Spec}{R}^{\mathrm{\prime }}\cong \mathrm{Proj}(B{\otimes }_R{R}^{\mathrm{\prime }}),$ since on the $f$-chart the left side is $\mathrm{Spec}(B_{(f)}){\times }_{\mathrm{Spec}R}\mathrm{Spec}{R}^{\mathrm{\prime }}\cong \mathrm{Spec}\left(B_{(f)}{\otimes }_R{R}^{\mathrm{\prime }}\right)$ by [F2], the right side is $\mathrm{Spec}(B^{\mathrm{\prime }}{)}_{(f^{\mathrm{\prime }})}$, and step 1.1 identifies these rings and their transition maps. The identification carries $\widetilde{B(n)}$ base changed to $\widetilde{B^{\mathrm{\prime }}(n)}$ on each chart.

The affine quotient case. By step 1.2 the isomorphism of step 2.1 restricts on charts to the closed subschemes cut out by $I_{(f)}$ and $I_{(f^{\mathrm{\prime }})}^{\mathrm{\prime }}$; passing to the definitions of the chart ideals, this gives a canonical isomorphism $\mathrm{Proj}(B\mathrm{/}I){\times }_{\mathrm{Spec}R}\mathrm{Spec}{R}^{\mathrm{\prime }}\cong \mathrm{Proj}(B^{\mathrm{\prime }}\mathrm{/}{I}^{\mathrm{\prime }})$ compatible with the closed immersions into $\mathrm{Proj}B{\times }_{\mathrm{Spec}R}\mathrm{Spec}{R}^{\mathrm{\prime }}$ and $\mathrm{Proj}{B}^{\mathrm{\prime }}$.

Reduction to affine charts. Let $U=\mathrm{Spec}R\subseteq S$ be affine and let $V=\mathrm{Spec}{R}^{\mathrm{\prime }}\subseteq S^{\mathrm{\prime }}$ be affine with $g(V)\subseteq U$. By [F3] the graded pieces of the pulled-back algebra satisfy $\mathrm{\Gamma }(V,{\mathcal{A}}^{\mathrm{\prime }}{)}_d=\mathrm{\Gamma }(V,g^{\ast }{\mathcal{A}}_d)=\mathrm{\Gamma }(U,{\mathcal{A}}_d){\otimes }_R{R}^{\mathrm{\prime }}$, so the graded $R^{\mathrm{\prime }}$-algebra $\mathrm{\Gamma }(V,{\mathcal{A}}^{\mathrm{\prime }})$ is $\mathrm{\Gamma }(U,\mathcal{A}){\otimes }_R{R}^{\mathrm{\prime }}$; hence step 2.1 applies with $B=\mathrm{\Gamma }(U,\mathcal{A})$ and gives a canonical isomorphism $\mathrm{Proj}\mathrm{\Gamma }(U,\mathcal{A}){\times }_{U}V\cong \mathrm{Proj}\mathrm{\Gamma }(V,{\mathcal{A}}^{\mathrm{\prime }})\mathrm{.}$ By [F1] the left side is $({\mathrm{Proj}}_S\mathcal{A}{\times }_S{S}^{\mathrm{\prime }}){\mathrm{\mid }}_V$ and the right side is $({\mathrm{Proj}}_{S^{\mathrm{\prime }}}{\mathcal{A}}^{\mathrm{\prime }}){\mathrm{\mid }}_V$, so the theorem holds over every such pair $U,V$.

Gluing the local isomorphisms. The affine opens $V$ of $S^{\mathrm{\prime }}$ mapping into some affine open $U$ of $S$ cover $S^{\mathrm{\prime }}$: for $x\in S^{\mathrm{\prime }}$ take an affine neighbourhood $U$ of $g(x)$ and then an affine neighbourhood $V$ of $x$ inside $g^{-1}(U)$. For two such pairs the local isomorphisms of step 3.2 agree on the intersection, because restricted to an affine open of $V\cap V^{\mathrm{\prime }}$ both are given by the same affine computation of step 2.1; hence they glue to a canonical isomorphism ${\mathrm{Proj}}_S\mathcal{A}{\times }_S{S}^{\mathrm{\prime }}\to {\mathrm{Proj}}_{S^{\mathrm{\prime }}}{\mathcal{A}}^{\mathrm{\prime }}$ over $S^{\mathrm{\prime }}$, natural in the map $S^{\mathrm{\prime }}\to S$.

Twists and quotients. On each pair $U,V$ as in step 3.2 the isomorphism of step 2.1 identifies the base change of $\widetilde{B(n)}$ with $\widetilde{B^{\mathrm{\prime }}(n)}$ on charts by step 1.1, so the glued twists correspond: $p^{\ast }{\mathcal{O}}_{{\mathrm{Proj}}_S\mathcal{A}}(n)\cong {\mathcal{O}}_{{\mathrm{Proj}}_{S^{\mathrm{\prime }}}{\mathcal{A}}^{\mathrm{\prime }}}(n)$. Similarly, for a quasi-coherent homogeneous ideal $\mathcal{J}\subseteq \mathcal{A}$ with ${\mathcal{J}}^{\mathrm{\prime }}=\mathrm{Im}(g^{\ast }\mathcal{J}\to {\mathcal{A}}^{\mathrm{\prime }})$, the affine comparison of step 3.1 applies on each chart with $I=\mathrm{\Gamma }(U,\mathcal{J})\subseteq \mathrm{\Gamma }(U,\mathcal{A})$, and the local closed subschemes glue to the restriction isomorphism ${\mathrm{Proj}}_S(\mathcal{A}\mathrm{/}\mathcal{J}){\times }_S{S}^{\mathrm{\prime }}\cong {\mathrm{Proj}}_{S^{\mathrm{\prime }}}({\mathcal{A}}^{\mathrm{\prime }}\mathrm{/}{\mathcal{J}}^{\mathrm{\prime }})$.

Conclusion. Step 4.1 gives the canonical base-change isomorphism over $S^{\mathrm{\prime }}$, step 5.1 its compatibility with relative twists and with graded algebra quotients. No flatness or finite generation of $\mathcal{A}$ was used: only the affine case of tensor products and localisations, as displayed in steps 1.1, 1.2 and 2.1, and the affine-local description of relative Proj. the Axiom of Choice [A1] is inherited through the Proj and pullback constructions. If $S^{\mathrm{\prime }}=\mathrm{\varnothing }$ or the affine pieces are empty, the identifications are between empty schemes and the construction is vacuous. [A1, step 4.1, step 5.1, cases: empty base change] \qed