Glue relative spectra of affine-local algebras

For every affine open $U\subseteq S$, let $R_U=\mathrm{\Gamma }(U,{\mathcal{O}}_S)$, $B_U=\mathrm{\Gamma }(U,\mathcal{A})$, and let ${\phi }_U:R_U\to B_U$ be the algebra structure map. The affine-local presentation in [F1] identifies these global sections with its chart algebra. Thus [F2] gives a morphism $p_U:Y_U=\mathrm{Spec}{B}_U\to U=\mathrm{Spec}{R}_U$. For an inclusion of affine opens $U\subseteq V$, restriction of sections gives $B_V\to B_U$ and hence a morphism ${\rho }_U^V:Y_U\to Y_V$ over $U\to V$. Identity and composition of these chart maps follow from identity and composition of sheaf restrictions.

Fix affine opens $U\subseteq V$. The distinguished opens $D_V(f)\subseteq U$, for $f\in R_V$, cover $U$ because distinguished opens form a basis in the affine scheme $V$. Write $\bar{f}$ for the restriction of $f$ to $U$; the same open is $D_U(\bar{f})$. By [F1], the restriction map identifies $B_V[{\phi }_V(f{)}^{-1}]$ and $B_U[{\phi }_U(\bar{f}{)}^{-1}]$ with the same ring $\mathrm{\Gamma }(D_V(f),\mathcal{A})$. Therefore [F2] and [F3] identify the restriction of ${\rho }_U^V$ on the corresponding distinguished spectrum opens with an isomorphism. These opens cover both $Y_U$ and $p_V^{-1}(U)$, so ${\rho }_U^V$ identifies $Y_U$ with the full open subscheme $p_V^{-1}(U)$. If $U=\mathrm{\varnothing }$, both sides are empty; localization at $0$ gives the empty distinguished open and localization at $1$ is the identity.

For affine opens $U,V\subseteq S$, the affine opens $W\subseteq U\cap V$ cover their intersection. By step 2.1, each $Y_W$ is identified with $p_U^{-1}(W)$ and with $p_V^{-1}(W)$. These identifications define isomorphisms between the two inverse-image opens over $U\cap V$. They agree on overlaps: cover any $W\cap W^{\mathrm{\prime }}$ by affine opens $T$ contained in it, and on each $Y_T$ both composites are induced by the same restriction maps of $\mathcal{A}$. The sheaf restriction maps compose, so the local isomorphisms glue uniquely.

The overlap isomorphisms of step 3.1 are the identity when $U=V$ and inverse to one another when the indices are reversed. On a triple overlap, affine opens $T\subseteq U\cap V\cap W$ cover the base overlap; over each $T$, all three chart identifications are the maps from the same chart $Y_T$. The restriction-map composition from step 1.1 therefore gives the cocycle condition. Apply [F4] to glue the affine charts to a scheme $Y$. The maps $p_U:Y_U\to U\to S$ agree on overlaps. Their continuous maps glue; for each open $Q\subseteq S$, the local pullbacks of sections of ${\mathcal{O}}_S(Q)$ agree on chart overlaps and glue by the sheaf axiom to the structure-sheaf map. Stalk locality is checked on each chart. This defines $\pi :Y\to S$, and [F4] gives uniqueness up to unique isomorphism over $S$.

The chart $Y_U$ maps into ${\pi }^{-1}(U)$. Conversely, a point of ${\pi }^{-1}(U)$ lies in some chart $Y_V$ and maps to a point of $V\cap U$; choose an affine open $W$ around that base point with $W\subseteq V\cap U$. The overlap identification from step 3.1 moves the point into $Y_U$, proving ${\pi }^{-1}(U)=Y_U$. Taking $D(r)\subseteq U$ in step 2.1 and using [F1] and [F3] gives the asserted principal-open localization formula.

Let $T\subseteq S$ be any open subscheme. Its affine opens are exactly the affine opens of $S$ contained in $T$. If a point of ${\pi }^{-1}(T)$ lies in a chart $Y_V$, an affine open $U$ around its base point with $U\subseteq V\cap T$ moves it into $Y_U$ by step 3.1. Thus the charts over affines contained in $T$ are precisely an open cover of ${\pi }^{-1}(T)$ with the restriction maps from the original atlas. The glued construction for $\mathcal{A}{\mathrm{\mid }}_T$ is canonically this open subscheme, including when $T=\mathrm{\varnothing }$. By [F6] it represents $Y{\times }_{S}T$, proving the restriction claim. No choice axiom is used: all affine opens and distinguished opens in the proof are considered as sets, and the local cover arguments select no simultaneous family of points or charts.