Affine morphisms and the relative spectrum

Choose a witnessing affine cover \(Y=\bigcup_iV_i\) with every \(f^{-1}(V_i)\) affine, and fix an affine open \(V=\Spec A\subseteq Y\). For each \(y\in V\cap V_i\), there is a neighborhood \(W\) of \(y\) distinguished in both \(V\) and \(V_i\). To construct it, choose \(y\in D_V(a)\subseteq V\cap V_i\), then choose \(y\in D_{V_i}(c)\subseteq D_V(a)\). The restriction of \(c\) to \(D_V(a)\) is \(b/a^n\) for some \(b\in A\). Consequently \[ D_{V_i}(c)=D_V(a)\cap D_V(b)=D_V(ab). \] The inverse image of \(W\) is distinguished in the affine scheme \(f^{-1}(V_i)\), hence affine.

Quasi-compactness of \(V\) gives finitely many such neighborhoods \(W_j=D_V(a_j)\) covering \(V\). The elements \(a_j\) generate the unit ideal in \(A\). Put \(Z=f^{-1}(V)\) and let \(b_j\in\Gamma(Z,\OO_Z)\) be their pullbacks. They still generate the unit ideal, and \(Z_{b_j}=f^{-1}(W_j)\) is affine. The global-function affineness criterion makes \(Z\) affine. Conversely, the condition on every affine open holds in particular on an affine cover, giving the defining condition.

By step 1, the inverse image of every affine open in \(Y\) is affine and therefore quasi-compact, so \(f\) is quasi-compact. On \(V=\Spec A\subseteq Y\), write \(f^{-1}(V)=\Spec B\). The restriction of the diagonal of \(f\) over this open is induced by the surjection \[ B\otimes_A B\longrightarrow B,\qquad b\otimes b'\longmapsto bb'. \] Hence it is a closed immersion. These opens cover the target \(X\times_YX\) of the diagonal, so the diagonal is a closed immersion and \(f\) is separated. Finally, the defining affine cover for a finite morphism already has affine inverse images, so every finite morphism is affine.

For \(V=\Spec A\), put \(B=\mca(V)\) and let \(h_V:\Spec B\to V\) be induced by its \(A\)-algebra structure. Quasi-coherence gives canonical algebra isomorphisms \[ \mca(D_V(a))\cong B_a\qquad(a\in A), \] compatible with restriction [@Har10a, Proposition II.5.4]. Thus \(X_{D_V(a)}=\Spec B_a\) identifies with \(h_V^{-1}(D_V(a))\) as an open subscheme of \(X_V\).

On this distinguished-open basis, the sections of \((h_V)_*\OO_{X_V}\) are \(B_a\), with the same restrictions as \(\mca|_V\). Therefore \((h_V)_*\OO_{X_V}\cong\mca|_V\) as sheaves of algebras. The morphism \(h_V\) is affine by the one-member affine cover of \(V\). For any affine open \(U\subseteq V\), step 1 makes \(h_V^{-1}(U)\) affine, and its coordinate ring is \[ \Gamma(h_V^{-1}(U),\OO_{X_V})\cong\mca(U). \] This gives an open immersion \(X_U\hookrightarrow X_V\) induced by the restriction map \(\mca(V)\to\mca(U)\). For nested affine opens these maps compose as their restriction homomorphisms do.

Choose an affine open cover \((V_i)\) of \(Y\). For two members, cover \(V_i\cap V_j\) by affine opens \(U\). The schemes \(X_U\) in step 3 form an open cover of both \(h_{V_i}^{-1}(V_i\cap V_j)\) and \(h_{V_j}^{-1}(V_i\cap V_j)\) and identify them canonically. For two such opens \(U,U'\), refine \(U\cap U'\) by affine opens again. The identifications agree there because they are induced by the same restriction homomorphisms from \(\mca\). This proves consistency of the overlap maps; on triple intersections the same argument proves the cocycle identity.

Gluing schemes along these open identifications gives a scheme \(X\) and a morphism \(f:X\to Y\) whose pieces are the \(h_{V_i}\) [@Har10a, Exercise II.2.12]. For any affine open \(V\subseteq Y\), covering \(V\cap V_i\) by affine opens and using step 3 identifies \(f^{-1}(V)\) with \(X_V\). The restriction maps for all affine inclusions are exactly those stipulated in part (c). Any other scheme with these compatible identifications has the same local isomorphisms to \(X_V\). They agree on overlaps and glue to a unique \(Y\)-isomorphism preserving those identifications. This proves the required uniqueness and defines \(\Spec_Y\mca\).

For the relative spectrum just constructed, each \(f^{-1}(V)=\Spec\mca(V)\) is affine, so \(f\) is affine. On every affine open \(V\), \[ (f_*\OO_X)(V)=\Gamma(\Spec\mca(V),\OO_X)\cong\mca(V). \] The identifications respect restriction and algebra multiplication, giving \(f_*\OO_X\cong\mca\).

Conversely, let \(f\) be affine and put \(\mca=f_*\OO_X\). For an affine open \(V=\Spec A\), write \(f^{-1}(V)=\Spec B\). On \(D_V(a)\) its direct image has sections \(B_a\), so \(\mca|_V\) is the quasi-coherent algebra associated to the \(A\)-algebra \(B\). This holds on an affine cover, proving quasi-coherence on \(Y\). The relative spectrum of \(\mca\) is glued from the affine schemes \(\Spec B\) and the restriction maps already present in \(X\). The uniqueness in step 4 gives \(X\cong\Spec_Y\mca\) over \(Y\).

Let \(\mathcal N\) be a quasi-coherent \(\OO_X\)-module. For \(V=\Spec A\subseteq Y\) with \(f^{-1}(V)=\Spec B\), put \(N=\Gamma(f^{-1}(V),\mathcal N)\). Its \(B\)-module structure makes the direct image a module over \(\mca\). For \(a\in A\), the associated-sheaf description of \(\mathcal N\) gives \[ (f_*\mathcal N)(D_V(a))=N_a. \] Thus \((f_*\mathcal N)|_V\) is the sheaf associated to \(N\) viewed as an \(A\)-module, and is quasi-coherent over \(\OO_V\). Morphisms of \(\OO_X\)-modules induce \(B\)-linear maps on these section modules and hence \(\mca\)-linear sheaf morphisms after direct image.

On \(V=\Spec A\subseteq Y\), put \(B=\mca(V)\) and \(M_V=\mcm(V)\), a \(B\)-module. On \(f^{-1}(V)=\Spec B\), take the associated sheaf \(\widetilde{M_V}\). For a distinguished open \(D_V(a)\), quasi-coherence over \(\OO_Y\) gives \(\mcm(D_V(a))\cong(M_V)_a\). The associated sheaf of this localized module is exactly the restriction of \(\widetilde{M_V}\) to \(\Spec B_a\). These identifications commute with further localization.

Equivalently, the direct image of \(\widetilde{M_V}\) to \(V\) is canonically \(\mcm|_V\), by the distinguished-open calculation. For any smaller affine open \(U\subseteq V\), its restriction to the affine scheme \(f^{-1}(U)\) therefore has global module \(\mcm(U)\). The affine sheaf-module equivalence identifies it with \(\widetilde{M_U}\). Using common affine refinements as in step 4, these identifications glue the local sheaves to \(\widetilde{\mcm}\) on \(X\).

An \(\mca\)-linear map \(\mcm\to\mcm'\) gives \(B\)-linear maps \(M_V\to M'_V\) and thus sheaf morphisms on \(f^{-1}(V)\). They commute with the gluing identifications, defining this construction on morphisms and preserving composition and identities.

The same calculations give a natural isomorphism \(f_*\widetilde{\mcm}\cong\mcm\) on every affine open of \(Y\), hence on \(Y\). For a quasi-coherent \(\mathcal N\) on \(X\), the isomorphism \(\widetilde{f_*\mathcal N}\cong\mathcal N\) on \(f^{-1}(V)\) is the canonical affine identification \[ \widetilde{\Gamma(f^{-1}(V),\mathcal N)}\cong\mathcal N|_{f^{-1}(V)}. \] It is natural and compatible with restrictions, so it glues on \(X\). These two natural isomorphisms prove the equivalence of categories, including its morphisms.

Step 1 proves part (a), step 2 proves part (b), steps <1>3–<1>4 prove part (c), step 5 proves part (d), and steps <1>6–<1>7 prove part (e).