Every scheme is glued from affine open subschemes along open subschemes of their pairwise intersections.
Gluing
Gluing schemes along open subschemes (Tag D-SCHGLUE)
Let \(\ts{X_i}\) be schemes, and for each pair \(i,j\) let \(U_{ij} \subseteq X_i\) be open with isomorphisms \(\varphi_{ij}: U_{ij} \to U_{ji}\) satisfying \(\varphi_{ii} = \id\), \(\varphi_{ji} = \inverseof{\varphi_{ij}}\), and the cocycle condition \(\varphi_{ik} = \varphi_{jk} \circ \varphi_{ij}\) on \(U_{ij} \intersect U_{ik}\). Then there is a scheme \(X\) with an open cover by copies of the \(X_i\) inducing the \(\varphi_{ij}\), unique up to isomorphism.
(Projective space by charts)
Take \(n+1\) copies of \(\AA^n\slice k = \Spec k[x_0/x_i, \dots, x_n/x_i]\) and glue \(U_{ij} = D(x_j/x_i)\) to \(U_{ji} = D(x_i/x_j)\) by inverting the ratio. The result is \(\PP^n\slice k\), and its global sections are \(k\), so it is not affine for \(n\geq1\).
The same charts with different transition maps produce different schemes: gluing two copies of \(\AA^1\) along \(\AA^1 \sm \ts{0}\) by the identity gives the line with a doubled origin, while gluing by \(t \mapsto \inverseof{t}\) gives \(\PP^1\). The line with a doubled origin is not separated: its two open embeddings of \(\AA^1\) agree on \(\AA^1\sm\ts{0}\) and differ at the origin. The scheme \(\PP^1\) is separated, and it is proper over the base field.
Gluing two copies of \(\AA^1_k=\Spec k[t]\) along \(\AA^1_k\sm\ts{0}\) by \(t\mapsto t^{-1}\) gives \(\PP^1_k\); gluing them by the identity gives the line with doubled origin, which is not separated.
The line with a doubled origin (Tag FE-SCHLINE)
Let \(k\) be a field. Glue \(U=\Spec k[t]\) and \(V=\Spec k[t]\) along \(\Spec k[t,t^{-1}]\) by the identity. The resulting scheme \(X\) has two origins \(0_U,0_V\). Every neighborhood of either origin contains the common generic point, so neighborhoods of the two origins always meet [@Har10a, Example II.4.0.1].
The diagonal \(\Delta_{X/k}:X\to X\times_kX\) is not closed. On the open subset \(U\times_kV\cong\Spec k[u,v]\), its image is \(V(u-v)\cap D(u)\). The closure is the whole line \(V(u-v)\), which contains \((0_U,0_V)\), a point not on the diagonal. Thus \(X\) is not separated over \(k\).
The two chart inclusions \(\AA_k^1\to X\) agree on the punctured affine line but send the origin to different points. For comparison, two \(S\)-morphisms from a reduced \(S\)-scheme \(Z\) to a separated \(S\)-scheme \(Y\) that agree on a dense open subset of \(Z\) are equal [@Har10a, Exercise II.4.2], as proved in P-AGH242AGREEDENSE. The reducedness of the source and separatedness of the target are separate hypotheses.
(A nonreduced source and a separated target)
Put \(R=k[t,\varepsilon]/(\varepsilon^2,t\varepsilon)\) and \(Z=\Spec R\). Its underlying space is \(\Spec k[t]\), since quotienting by the nilpotent ideal \((\varepsilon)\) does not change the prime spectrum. Thus \(D(t)\) is dense in \(Z\). The two ring homomorphisms \[ k[z]\rightrightarrows R,\qquad z\longmapsto0,\quad z\longmapsto\varepsilon \] define distinct morphisms \(Z\to\AA_k^1\): the class of \(\varepsilon\) is nonzero, as its image in \(R/(t)=k[\varepsilon]/(\varepsilon^2)\) is nonzero. They agree on \(D(t)\) because \(t\varepsilon=0\) makes \(\varepsilon\) zero after inverting \(t\). The target is affine and therefore separated; the failure of dense-open uniqueness here comes from the nonreduced source, not from the target.
The calculation of invertible sheaves on $X$ gives \(\Pic(X)\cong\ZZ\). Only the trivial class is globally generated, and no class is ample.
Proj
$\Proj$ of a graded ring (Tag D-SCHPROJ)
Let \(S = \bigoplus_{d \geq 0} S_d\) be a graded ring and \(S_+ \da \bigoplus_{d \geq 1} S_d\) the irrelevant ideal. Then \(\Proj S\) is the set of homogeneous primes \(\mfp\) with \(\mfp \not\supseteq S_+\), topologised by the closed sets \(V(\mfa) = \ts{\mfp \st \mfp \supseteq \mfa}\) for \(\mfa\) homogeneous. Its structure sheaf sends \(U\) to the functions \(s: U \to \coprod_{\mfp \in U} S_{(\mfp)}\) with \(s(\mfp) \in S_{(\mfp)}\) and locally of the form \(a/f\) with \(a, f\) homogeneous of the same degree and \(f \notin \mfp\), where \(S_{(\mfp)}\) is the degree-zero part of the homogeneous localisation.
Let \(k\) be an algebraically closed field and \(S=k[x_0,\ldots,x_n]\). The closed points of \(\Proj S\) are the homogeneous primes generated by the linear forms vanishing on a line through the origin of \(k^{n+1}\), so they correspond to the points of \(\PP^n(k)\). For homogeneous \(f\) of degree \(d\) and \(v\in k^{n+1}\sm\ts{0}\), \(f(\lambda v)=\lambda^df(v)\), so whether \(f\) vanishes on the line \(kv\) does not depend on \(v\). For \(a,f\) homogeneous of the same degree \(d\) and \(f(v)\ne0\), \(a(\lambda v)/f(\lambda v)=a(v)/f(v)\), so \(a/f\) has a well-defined value at the point \([v]\). The irrelevant ideal \(S_+\) is the ideal of the origin of \(k^{n+1}\) and the only homogeneous prime containing \(S_+\); it contains the ideal of every line through the origin and corresponds to no point of \(\PP^n(k)\).
\(\Proj S\) is covered by the affine opens \(D_+(f)=\Spec S_{(f)}\) for homogeneous \(f\) of positive degree, glued along \(D_+(fg)\). For \(S=k[x_0,\ldots,x_n]\), the charts \(D_+(x_i)\) with transition functions \(x_j/x_i\) give \(\PP^n_k\). A homogeneous ideal \(I\subseteq S\) gives the closed subscheme \(\Proj(S/I)\subseteq\Proj S\).
$\Proj S$ is a scheme, with affine charts $D_+(f) \cong \Spec S_{(f)}$ (Tag T-SCHPRJC)
For \(f \in S_d\) homogeneous of positive degree, let \(D_+(f) \da \ts{\mfp \in \Proj S \st f \notin \mfp}\). Then \[ (D_+(f), \ro{\OO_{\Proj S}}{D_+(f)}) \cong \Spec S_{(f)} , \] where \(S_{(f)}\) is the degree-zero part of \(S[\inverseof{f}]\). The \(D_+(f)\) for \(f \in S_+\) cover \(\Proj S\), so \(\Proj S\) is a scheme.
For \(S = \kxnz\) with the usual grading, \(D_+(x_i) \cong \Spec k[x_0/x_i, \dots, x_n/x_i] \cong \AA^n\slice k\), recovering the standard charts of \(\PP^n\slice k\).
The scheme structure is exhibited by the affine charts \(D_+(f)\cong\Spec S_{(f)}\): inverting a homogeneous element and passing to degree zero produces ordinary affine functions. Discarding \(V(S_+)\) is what makes the \(D_+(f)\) a cover, since a prime containing all of \(S_+\) lies in no chart.
Over a base, define \(\PP^n\slice \ZZ \da \Proj \ZZ[x_0, \dots, x_n]\) and then \[ \PP^n\slice S \da \fiberprod{\PP^n\slice \ZZ}{\Spec \ZZ}{S} . \] The same pattern gives \(\AA^n\slice S\) from \(\Spec \ZZ[x_1, \dots, x_n]\). Thus projective space over a base scheme is obtained by base change from the universal projective space over \(\Spec\ZZ\), rather than by choosing new gluing data for each base.
For a ring \(A\), \(\PP^n_A=\PP^n_\ZZ\times_{\Spec\ZZ}\Spec A\), and \(\OO_{\PP^n_A}(d)\) is the pullback of \(\OO_{\PP^n_\ZZ}(d)\). Since \(\PP^n_\ZZ\to\Spec\ZZ\) is proper and properness is stable under base change, \(\PP^n_A\to\Spec A\) is proper.
Quotients by group actions
GIT quotients (Tag D-GITQUOT)
Let \(G\) be a reductive group over an algebraically closed field \(k\) acting on an affine variety \(X = \Spec A\). The ring of invariants \(A^G\) is finitely generated, and the GIT quotient is \(X /\!\!/ G = \Spec A^G\), with the morphism \(\pi \colon X \to X/\!\!/G\) induced by \(A^G \subseteq A\).
\(\pi\) is surjective and constant on orbits, and \(\pi(x) = \pi(y)\) exactly when the orbit closures \(\overline{Gx}\) and \(\overline{Gy}\) meet. Each fibre contains a unique closed orbit, so the closed points of \(X/\!\!/G\) correspond to closed \(G\)-orbits in \(X\).
(Projective GIT quotient)
If \(G\) acts on a projective variety \(X\) with a \(G\)-linearized ample line bundle \(L\), a point \(x\) is semistable if some invariant section \(s \in H^0(X, L^m)^G\) has \(s(x) \neq 0\). The GIT quotient is \(X /\!\!/_L G = \Proj \bigoplus_{m \geq 0} H^0(X, L^m)^G\), a projective variety receiving a morphism from the open set \(X^{ss}\) of semistable points.
Let \(\GG_m\) act on \(\AA^2\). With weights \((1,1)\), \(k[x,y]^{\GG_m} = k\) and \(\AA^2/\!\!/\GG_m\) is a point, because every orbit closure contains the origin. With weights \((1,-1)\), \(k[x,y]^{\GG_m} = k[xy]\) and \(\AA^2/\!\!/\GG_m = \AA^1\); the fibre over \(0\) is the union of the two axes and contains three orbits, of which only the origin is closed. With weights \((1,1)\) and the linearization \(\OO\) twisted by the identity character, the semistable locus is \(\AA^2 \setminus \{0\}\) and the projective quotient is \(\PP^1\).
Relative Spec and relative Proj
Relative Spec and relative Proj (Tag D-SCHRELSPECPROJ)
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 \(\PP^n_X\).
Blowing up and blowing down
Blowing up a scheme along a closed subscheme, and blowing down (Tag D-SCHBLOWUP)
Let \(X\) be a Noetherian scheme and \(Z \subseteq X\) a closed subscheme with ideal sheaf \(\mathcal{I}\). The Rees algebra of \(\mathcal{I}\) is the graded quasicoherent \(\OO_X\)-algebra \(\bigoplus_{d \geq 0} \mathcal{I}^d\), with \(\mathcal{I}^0 = \OO_X\); it is generated in degree \(1\). The blowup of \(X\) along \(Z\) is the \(X\)-scheme \[\pi \colon \Bl_Z X \coloneqq \operatorname{\mathbf{Proj}}_X \bigoplus_{d \geq 0} \mathcal{I}^d \to X,\] using the relative Proj of D-SCHRELSPECPROJ. The exceptional divisor is the closed subscheme \(E = \pi^{-1}(Z)\) of \(\Bl_Z X\), with ideal sheaf \(\mathcal{I} \cdot \OO_{\Bl_Z X}\).
(Properties and universal property)
Let \(\pi \colon \Bl_Z X \to X\) be the blowup of a Noetherian scheme along a closed subscheme \(Z\).
\(\pi\) is projective, \(E\) is an effective Cartier divisor, and \(\pi\) restricts to an isomorphism \(\Bl_Z X \setminus E \to X \setminus Z\).
If \(X\) is integral and \(Z \neq X\), then \(\Bl_Z X\) is integral and \(\pi\) is birational.
For every morphism \(f \colon Y \to X\) such that \(f^{-1}(Z)\), the closed subscheme with ideal sheaf \(f^{-1}\mathcal{I} \cdot \OO_Y\), is an effective Cartier divisor on \(Y\), there is a unique morphism \(g \colon Y \to \Bl_Z X\) with \(f = \pi \circ g\).
Let \(X = \AA^2_k = \Spec k[x,y]\) and \(Z\) the origin, \(\mathcal{I} = (x, y)\). Since \(x, y\) is a regular sequence, the Rees algebra is \(k[x,y][s,t]/(xt - ys)\), with \(s \mapsto x\) and \(t \mapsto y\) in degree \(1\). So \(\Bl_0 \AA^2 = V(xt - ys) \subseteq \AA^2_k \times \PP^1_k\). On the chart \(s \neq 0\) with coordinate \(u = t/s\), the equation becomes \(y = xu\), the chart is \(\Spec k[x, u] \cong \AA^2\), and \(E\) is the line \(V(x)\) there; the two charts glue \(E\) to \(\PP^1_k\). This recovers the blowup of a variety at a point in D-VARBLOW.
(Blowdown)
A proper birational morphism \(\rho \colon Y \to X\) of integral Noetherian schemes is the blowdown of a closed subscheme \(F \subseteq Y\) if there are a closed subscheme \(Z \subseteq X\) and an isomorphism \(Y \cong \Bl_Z X\) over \(X\) carrying \(F\) to the exceptional divisor. Then \(X\) is obtained from \(Y\) by blowing down \(F\), and \(\rho\) contracts \(F\) to \(Z\).
If \(Y\) is a smooth projective surface over an algebraically closed field and \(F \subseteq Y\) is a curve with \(F \cong \PP^1\) and \(F^2 = -1\), then by Castelnuovo’s criterion T-SRFCAST there is a smooth projective surface \(X\) and a point \(p \in X\) with \(Y \cong \Bl_p X\) and \(F\) the exceptional curve, so \(F\) can be blown down.