Schemes Subschemes and Morphisms Locally of Finite Type

Fix $x\in U\cap V$, and write $U=\mathrm{Spec}A$ and $V=\mathrm{Spec}B$. The basis property in [L1] gives $f\in A$ and $g\in B$ such that $x\in D_U(f)\subseteq V$ and $x\in D_V(g)\subseteq U$. Put $W=D_U(f)\cap D_V(g)$; it is an open neighbourhood of $x$.

The function $g$ on $V$ restricts to a section on $D_U(f)$. By [L1], write that restriction as $a\mathrm{/}{f}^n$ for some $a\in A$ and $n\ge 0$. On $D_U(f)$, its nonvanishing locus is $D_U(a)\cap D_U(f)=D_U(fa)$. Hence $W=D_U(fa)$ is principal in $U$.

Symmetrically, the restriction of $f$ to $D_V(g)$ is a section $b\mathrm{/}{g}^m$ for some $b\in B$ and $m\ge 0$. Its nonvanishing locus there is $D_V(gb)$, so the same open $W$ is principal in $V$.

Every $x\in U\cap V$ has such a common principal neighbourhood $W$. These neighbourhoods cover the intersection and prove the claim.

The data are gluing data for locally ringed spaces, so the gluing theorem produces a locally ringed space $X$ covered by open subspaces identified with the given affine schemes.

Since every point of $X$ lies in one of those affine open subspaces, $X$ is a scheme by definition.

Let $X^{\mathrm{\prime }}$ be another gluing with the same chart identifications. On every affine chart, compose the identification into $X$ with the inverse of the corresponding identification into $X^{\mathrm{\prime }}$. The cocycle condition says these chartwise locally ringed-space isomorphisms agree on overlaps, so their underlying maps and sheaf maps glue to an isomorphism $X\to X^{\mathrm{\prime }}$. The inverse is obtained by reversing the chart maps. Any isomorphism respecting all chart identifications has these restrictions and is therefore equal to this one. Thus the gluing is unique up to a unique chart-compatible isomorphism.

A morphism $f:X\to \mathrm{Spec}A$ induces the ring map on global sections $A=\mathrm{\Gamma }(\mathrm{Spec}A,\mathcal{O})\to \mathrm{\Gamma }(X,{\mathcal{O}}_X)$.

Conversely, a ring map $A\to \mathrm{\Gamma }(X,{\mathcal{O}}_X)$ gives the canonical morphism from $X$ to the affine scheme $\mathrm{Spec}A$, checked on affine opens by the affine anti-equivalence and glued over an affine cover of $X$.

On each affine open the two constructions are inverse by the affine anti-equivalence; equality of scheme morphisms is local on the source, hence they are mutually inverse and natural.

The underlying continuous maps agree on overlaps and therefore glue to a unique continuous map from $X$.

Compatibility of the maps of structure sheaves on overlaps glues their sheaf maps, and locality on stalks is local on $X$; thus the glued map is a scheme morphism.

Restricting a hypothetical equality to the cover proves necessity, while uniqueness in the gluing construction proves sufficiency; refining by affine opens preserves the same argument.

Replace $U$ by its isomorphic open subscheme of $X$, so that $j$ is the inclusion of an open subset with restricted sheaf.

If $a,b:T\to U$ have $ja=jb$, their underlying maps and their sheaf maps agree after restriction to that open subscheme, hence $a=b$; thus $j$ is a monomorphism.

A composite of two such identifications identifies the source with an open subscheme of the final target, so it is again an open immersion.

Write $X=\mathrm{Spec}A$. Since $U$ and $V$ are quasi-compact open subsets of $X$, each is a finite union of distinguished opens, say $U={\bigcup }_{i}D(f_i)$ and $V={\bigcup }_{j}D(g_j)$.

Hence $U\cap V={\bigcup }_{i,j}D(f_i{g}_j)$ is a finite union of quasi-compact distinguished opens, and is therefore quasi-compact.

This is exactly the affine-intersection criterion in the definition of quasi-separatedness.

Write $E=i(Z)=V(J)\subseteq X$. By [F1], $E$ and hence $Z$ are quasi-compact. Choose a finite affine open cover $W_j=\mathrm{Spec}{C}_j$ of $Z$. Since $Z$ has the subspace topology on $E$, the principal-open basis in [F2] and quasi-compactness supply finitely many labelled functions $f_i\in A$ whose $E\cap D(f_i)$ cover $E$, each lying in one $W_j$. The inverse image of $D(f_i)$ is the principal open defined by $f_i$ in that affine chart, so is affine. Empty covers are allowed.

The cover implies $V(J+(f_i))=\mathrm{\varnothing }$, hence $J+(f_i)=A$ by [F2]: a proper ideal in a nonzero ring would lie in a maximal ideal. For the zero ring the equality is immediate. Thus $1={\sum }_i{a}_i{f}_i+{\sum }_{\mathrm{\ell }}{b}_{\mathrm{\ell }}{g}_{\mathrm{\ell }}$ for finitely many $g_{\mathrm{\ell }}\in J$. On each affine $W_j$, every image of $g_{\mathrm{\ell }}$ lies in all primes, since every point maps into $V(J)$; it is nilpotent by [F2]. Finiteness of the cover supplies a common exponent, so each restriction $g_{\mathrm{\ell }}{\mathrm{\mid }}_Z$ is nilpotent globally by the sheaf axiom. The finite sum $h={\sum }_{\mathrm{\ell }}{b}_{\mathrm{\ell }}{g}_{\mathrm{\ell }}{\mathrm{\mid }}_Z$ is nilpotent by the multinomial expansion. Consequently ${\sum }_i{a}_i{f}_i{\mathrm{\mid }}_Z=1-h$ is a unit, with inverse a finite geometric sum. The $f_i{\mathrm{\mid }}_Z$ generate the unit ideal.

Put $B=\mathrm{\Gamma }(Z,{\mathcal{O}}_Z)$ and write $f_i$ also for the pulled-back global sections. The affine opens $U_i=Z_{f_i}$ cover $Z$ by step 1.1. If this list is empty, $Z$ is empty and $B=0$, giving $Z=\mathrm{Spec}0$. Otherwise put $C_i=\mathrm{\Gamma }(U_i,{\mathcal{O}}_Z)$ and $C_{ij}=\mathrm{\Gamma }(U_i\cap U_j,{\mathcal{O}}_Z)$. Each intersection is the principal open defined by $f_j$ in the affine $U_i$, so is affine by [F2]. By [F5], $B$ is the kernel of the difference of restriction maps ${\prod }_i{C}_i\to {\prod }_{i,j}{C}_{ij}$, a map of $B$-modules. Fix $k$. Exact localization in [F4] preserves this kernel, and localization commutes with finite products by taking a common denominator in any finite tuple. Basic-open sections in [F4] identify $(C_i{)}_{f_k}$ and $(C_{ij}{)}_{f_k}$ with sections on $U_i\cap U_k$ and $U_i\cap U_j\cap U_k$, respectively, respecting restrictions. Their kernel is therefore $\mathrm{\Gamma }(U_k,{\mathcal{O}}_Z)=C_k$ by [F5], since the $U_i\cap U_k$ cover $U_k$. Thus restriction gives a ring isomorphism $B_{f_k}\cong C_k$: all maps come from restriction and fraction multiplication. By [F3] the identity of $B$ defines a canonical map $j:Z\to \mathrm{Spec}B$. Its inverse image of $D_B(f_k)$ is $U_k$, and on that open its comorphism is precisely this ring isomorphism. The sections $f_i$ generate the unit ideal by step 2.1, so the $D_B(f_i)$ cover $\mathrm{Spec}B$. These local inverse morphisms agree on overlaps because each inverts $j$, and glue to an inverse of $j$. Hence $Z=\mathrm{Spec}B$ in every case.

Let $\phi :A\to B$ correspond to $i$ by [F3]. For any prime $\mathfrak{p}\subset A$, sections of $i_{\ast }{\mathcal{O}}_Z$ on $D(f)$ are $B_{\phi (f)}$. Taking the colimit over $f\mathrm{\notin }\mathfrak{p}$ identifies its stalk with $(A\setminus \mathfrak{p}{)}^{-1}B$. The surjective sheaf map in [F1] therefore gives a surjection $A_{\mathfrak{p}}\to (A\setminus \mathfrak{p}{)}^{-1}B$. Exact localization in [F4] implies $M_{\mathfrak{p}}=0$ for $M=\mathrm{coker}\phi$. If $m\in M$ were nonzero, [F2] would give a maximal ideal $\mathfrak{p}$ containing its proper annihilator. Then $m\mathrm{/}1\mathrm{\ne }0$ in $M_{\mathfrak{p}}$: vanishing would require a denominator outside $\mathfrak{p}$ annihilating $m$, contrary to that containment. Thus $M=0$ and $\phi$ is onto. For $A=0$, unitality already forces $B=0$. Setting $I=\ker \phi$ gives $B\cong A\mathrm{/}I$ and the required isomorphism over $X$.

Conversely, the spectrum map of $A\twoheadrightarrow A\mathrm{/}I$ is a homeomorphism onto $V(I)$ by [F4]: contraction identifies its primes and all their closed vanishing sets. At a prime containing $I$, its direct-image stalk map is the surjection $A_{\mathfrak{p}}\to (A\mathrm{/}I{)}_{\mathfrak{p}\mathrm{/}I}$; at a prime not containing $I$, the target stalk is zero, since an element of $I$ is inverted. Hence its structure-sheaf map is surjective by [F4], proving it is a closed immersion. The kernel of $A\to \mathrm{\Gamma }(Z,{\mathcal{O}}_Z)$ recovers $I$ uniquely. Any isomorphism of quotient spectra over $X$ is induced by the unique compatible quotient-ring map, so is unique. In particular $I=A$ gives the empty immersion, also when $A=0$.

On affine opens, the map on coordinate rings sends every nilpotent of the target ring to a nilpotent of the reduced source ring, hence to zero.

It therefore factors uniquely through the quotient by the nilradical on each affine chart, producing compatible local maps to $X_{\mathrm{r}\mathrm{e}\mathrm{d}}$.

The local maps glue uniquely, and their uniqueness on charts proves uniqueness globally.

Restricting a finite-type ring map to distinguished affine opens localizes the map and retains a finite generating set, so the condition survives affine refinement.

Conversely, let $V=\mathrm{Spec}R$ be an affine target and $U=\mathrm{Spec}A\subseteq f^{-1}(V)$ an affine source open. Around each point of $U$, choose an affine chart $W=\mathrm{Spec}C$ from the given finite-type source cover. First choose a principal open in $U$ contained in $W$, then a smaller principal open in $W$ contained in the first one. The latter is principal in $U$ as well: its defining function on the first open belongs to a localization of $A$, so clearing a power of the first defining function gives one element of $A$. Quasi-compactness of $U$ now supplies a finite cover $U={\bigcup }_{i=1}^n{D}_U(a_i)$ by such common principal opens. Each $A_{a_i}$ is a localization of a finite-type $R$-algebra $C$, hence is finite type over $R$.

Here is the needed finite-localization descent, rather than just an appeal to it. Choose $h_i\in A$ with ${\sum }_i{h}_i{a}_i=1$. For every $i$, choose finitely many $R$-algebra generators of $A_{a_i}$ and write each as $y_{ij}\mathrm{/}{a}_i^{k_{ij}}$ with $y_{ij}\in A$. Let $B\subseteq A$ be the $R$-subalgebra generated by all $a_i,h_i,y_{ij}$. Then $B_{a_i}=A_{a_i}$ for every $i$, and the $a_i$ generate the unit ideal already in $B$. For any $x\in A\mathrm{/}B$, localization gives $a_i^{N_i}x=0$ for some $N_i$ for each $i$. The powers $a_i^{N_i}$ also generate the unit ideal in $B$, so $x=0$. Thus $A=B$ is finitely generated over $R$. This is the finite-type part of Stacks Project, Tag 00EP.

On an affine target $V$, refine any target cover to a finite cover by opens $D_V(r_j)$ that are principal also in the chosen affine target charts, using the common-principal construction of step 2.1. Base change to each $D_V(r_j)$ preserves finite type of the source charts. For an affine $U\subseteq f^{-1}(V)$, the inverse images are the principal opens $D_U(f^{\mathrm{\#}}{r}_j)$; steps 2.1 and 3.1 make their coordinate rings finite type over ${\mathcal{O}}_S(D_V(r_j))$, hence over ${\mathcal{O}}_S(V)$. Applying step 3.1 again gives finite type on $U$. Finally, quasi-compactness of $f$ supplies a finite affine source subcover over each affine target open, so locally finite type plus quasi-compactness is exactly finite type.

For a classical space $V$, define $V^{\ast }$ to have one point ${\eta }_Z$ for each nonempty irreducible closed subset $Z\subseteq V$, identifying an existing point $v$ with ${\eta }_{\{v\}}$. Thus only nonsingleton subsets supply new points. For each open $U\subseteq V$, put $U^{\ast }=\{{\eta }_Z:Z\cap U\mathrm{\ne }\mathrm{\varnothing }\}$. Irreducibility gives $(U\cap W{)}^{\ast }=U^{\ast }\cap W^{\ast }$; unions are also preserved, and singleton points show injectivity. Give $V^{\ast }$ this open-set lattice and set ${\mathcal{O}}_{V^{\ast }}(U^{\ast })={\mathcal{O}}_V(U)$ with the same restrictions. On a classical affine chart with coordinate ring $A$, nonempty irreducible closed subsets correspond to prime ideals of $A$ by the affine Nullstellensatz; $D(a{)}^{\ast }$ corresponds to the prime-spectrum open $D(a)$, and both sheaves have ring $A_a$ on this basis. Thus this ringed chart is $\mathrm{Spec}A$, so $V^{\ast }$ is a reduced scheme. Conversely the closed-point construction recovers these classical charts by lem-classical-points-inside-affine-scheme.

For a classical regular map $f:V\to W$, define $\tilde{f}({\eta }_Z)$ as the point indexed by $\overline{f(Z)}$. This is a nonempty irreducible closed subset. For any open $O\subseteq W$, its closure meets $O$ exactly when $f(Z)$ meets $O$, so the inverse image of $O^{\ast }$ is $(f^{-1}O{)}^{\ast }$. The classical sheaf map therefore defines a sheaf map on the new open lattices. On affine source/target charts a regular map induces a $k$-algebra map $B\to A$; the extension is the spectrum map on primes, and its stalk homomorphisms $B_{\mathfrak{q}}\to A_{\mathfrak{p}}$, with $\mathfrak{q}$ the contracted prime, are local. Hence it is a scheme morphism. These local maps agree on overlaps, being the same point and sheaf maps just defined. Conversely, a scheme $k$-morphism between finite-type $k$-schemes sends a closed point to a closed point: on affine neighbourhoods the composite to its residue field $k$ is a surjective $k$-algebra map, so its kernel is maximal. Restricting the sheaf map then gives a classical regular map. The affine ring maps show that the two operations on morphisms are inverse and respect composition.

The constructions are inverse on objects as locally ringed spaces, since on each affine chart prime ideals and their closed-point zero loci are inverse correspondences, and the sheaves agree on the principal-open basis. These chart identifications are compatible on overlaps. A finite classical affine cover becomes a finite affine cover by spectra of finite-type $k$-algebras, hence a finite-type scheme. Conversely a finite-type scheme has such a finite affine cover. The open-lattice correspondence preserves irreducibility and nonemptiness, and all classical coordinate rings are reduced. Thus it restricts to nonempty irreducible classical prevarieties and integral finite-type schemes.

It remains to compare separation. For two nonempty classical affine opens $U,V$ in an irreducible classical prevariety, their affine product exists with ring $k[U]{\otimes }_{k}k[V]$ by thm-affine-variety-product-coordinate-ring. The equalizer of its two maps into the prevariety is the locus of pairs representing the same point, naturally isomorphic to $U\cap V$ by either projection; locally the inverse is the pair of the two open inclusions. Classical separatedness makes this locus closed, hence affine, and restriction from the product coordinate ring onto its coordinate ring is surjective. Under the affine identifications of step 1.1, these are exactly the requirements of def-affine-overlap-separation-condition. Conversely that condition makes each such overlap a closed locus in the affine product. For any pair of classical maps into the space, cover the source by opens on which their images lie in such $U,V$; their equalizer is the inverse image of this closed locus and is therefore locally, hence globally, closed. Thus separation is preserved in both directions. Combining this with step 3.1 proves the claimed equivalence.