(Relative Spec)
Let \(X\) be a scheme and \(\mathcal{A}\) a quasicoherent sheaf of \(\OO_X\)-algebras. The relative Spec of \(\mathcal{A}\) is an \(X\)-scheme \(\pi \colon \operatorname{\mathbf{Spec}}_X \mathcal{A} \to X\) representing the functor on \(X\)-schemes \[(t \colon T \to X) \longmapsto \Hom_{\OO_X\text{-alg}}(\mathcal{A}, t_* \OO_T),\] so that for every \(X\)-scheme \(t \colon T \to X\) there is a bijection, natural in \(T\), \[\Hom_X(T, \operatorname{\mathbf{Spec}}_X \mathcal{A}) \cong \Hom_{\OO_X\text{-alg}}(\mathcal{A}, t_* \OO_T).\]
The relative Spec exists and is unique up to unique isomorphism. For every affine open \(V = \Spec B \subseteq X\) with \(\mathcal{A}|_V \cong \widetilde{A}\) for a \(B\)-algebra \(A\), \(\pi^{-1}(V) \cong \Spec A\) over \(V\). In particular \(\pi\) is affine, \(\pi_* \OO_{\operatorname{\mathbf{Spec}}_X \mathcal{A}} \cong \mathcal{A}\), and \(\mathcal{A} \mapsto \operatorname{\mathbf{Spec}}_X \mathcal{A}\) is an anti-equivalence from quasicoherent \(\OO_X\)-algebras to affine morphisms with target \(X\) (P-AGH2517AFFMOR).
(Relative Proj)
Let \(X\) be a scheme and \(\mathcal{S} = \bigoplus_{d \geq 0} \mathcal{S}_d\) a quasicoherent sheaf of graded \(\OO_X\)-algebras with \(\mathcal{S}_0 = \OO_X\), generated in degree \(1\): the map \(\operatorname{Sym}^\bullet_{\OO_X} \mathcal{S}_1 \to \mathcal{S}\) is surjective. The relative Proj \(\operatorname{\mathbf{Proj}}_X \mathcal{S} \to X\) is the \(X\)-scheme obtained by gluing the schemes \(\Proj \mathcal{S}(V) \to V\), for affine opens \(V \subseteq X\), along the isomorphisms \[\Proj \mathcal{S}(V) \times_V V' \cong \Proj \mathcal{S}(V')\] for affine opens \(V' \subseteq V\), which come from \(\mathcal{S}(V') \cong \mathcal{S}(V) \otimes_{\OO_X(V)} \OO_X(V')\). It carries the invertible sheaf \(\OO(1)\) obtained by gluing the sheaves \(\OO_{\Proj \mathcal{S}(V)}(1)\).
For \(X = \Spec A\) affine, \(\operatorname{\mathbf{Proj}}_X \mathcal{S} = \Proj \mathcal{S}(X)\). For \(\mathcal{S} = \operatorname{Sym}^\bullet \OO_X^{\oplus(n+1)}\) it is \(\PP^n_X\). A morphism \(Y \to X\) isomorphic over \(X\) to \(\operatorname{\mathbf{Proj}}_X \mathcal{S}\), for such an \(\mathcal{S}\) with \(\mathcal{S}_1\) of finite type, is projective in the sense of EGA; when \(X\) is affine this agrees with the definition in D-MORPROJ by a closed immersion into \(\PP^n_X\).
Dependencies
D-QNTZYQuasicoherent and coherent sheavesuses
Backlinks
D-SCHBLOWUPBlowing up a scheme along a closed subscheme, and blowing downuses