Schemes Subschemes and Morphisms Locally of Finite Type

For $x\in U\cap V$, regard $U\cap V$ as an open neighbourhood of $x$ in the affine scheme $U$. Distinguished opens form a basis there, so choose $f$ with $x\in D_U(f)\subseteq U\cap V$.

The same construction in an affine neighbourhood in $V$ refines $D_U(f)$ around each of its points by a distinguished open in that chart. These refinements cover $U\cap V$, and each is principal in the indicated affine charts.

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.

By Stacks Project, Tag 01IN, there is an ideal $I\subseteq A$ whose associated quasi-coherent sheaf is the kernel of ${\mathcal{O}}_{\mathrm{Spec}A}\to i_{\ast }{\mathcal{O}}_Z$, and for every distinguished open $D(f)$ one has $i^{-1}(D(f))\cong \mathrm{Spec}(A_f\mathrm{/}{I}_f)\mathrm{.}$ In particular, for $f=1$ this gives $Z\cong \mathrm{Spec}(A\mathrm{/}I)$ over $\mathrm{Spec}A$. This avoids the false inference that an arbitrary surjection of sheaves must be surjective on global sections.

Conversely, a quotient $A\twoheadrightarrow A\mathrm{/}I$ induces a morphism $\mathrm{Spec}(A\mathrm{/}I)\to \mathrm{Spec}A$ with closed image $V(I)$; on every distinguished open it is the quotient $A_f\twoheadrightarrow A_f\mathrm{/}{I}_f$, so the structure-sheaf map is surjective. Thus it is a closed immersion.

The kernel ideal and quotient construction recover each other, giving the claimed classification up to unique isomorphism.

Let $\mathcal{I}\subseteq {\mathcal{O}}_X$ be quasi-coherent and let $U=\mathrm{Spec}A$ be an affine open. By its defining local form, $\mathcal{I}{\mathrm{\mid }}_U$ is associated to an ideal $I_U\subseteq A$, and [F1] gives the closed immersion $\mathrm{Spec}(A\mathrm{/}{I}_U)\to U$.

Conversely, let $i:Z\hookrightarrow X$ be a closed immersion. Restricting to $U=\mathrm{Spec}A$, [F1] identifies $Z\cap U\to U$ with $\mathrm{Spec}(A\mathrm{/}{I}_U)\to \mathrm{Spec}A$, and identifies the kernel of ${\mathcal{O}}_U\to i_{\ast }{\mathcal{O}}_{Z\cap U}$ with the ideal sheaf associated to $I_U$. Thus the global kernel is quasi-coherent.

On an overlap of two affine opens, both local quotient immersions have kernel the restricted ideal sheaf $\mathcal{I}$; hence their quotient rings and their closed immersions agree after restriction. They therefore glue to a closed subscheme, denoted $V(\mathcal{I})$, of $X$.

On every affine open the two constructions are the inverse quotient-ring constructions of [F1]. Since both the ideal sheaves and closed immersions agree on the affine open cover, the constructions are mutually inverse on $X$.

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. If a source cover already verifies local finite type, quasi-compactness of $U$ and the principal-open refinement lemma give a finite distinguished cover $U={\bigcup }_{i=1}^{n}D(a_i)$ for which every $A_{a_i}$ is a finite-type $R$-algebra. Choose finitely many localized generators on each member and clear their finitely many denominators. Since $(a_1,\dots ,a_n)=A$, the standard finite-localization criterion then shows that $A$ is finite type over $R$. This is the ring argument in Stacks Project, Tag 01T2.

Refining target overlaps by distinguished opens gives the same argument on the target side. 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.

Let $V=\mathrm{Spec}A$ be an affine open of $Y$. By [F1], $f^{-1}(V)$ is quasi-compact, so its affine-open cover has a finite subcover $U_1,\dots ,U_n$; when $f^{-1}(V)$ is empty, take the finite empty cover.

Write $U_i=\mathrm{Spec}{B}_i$. The sheaf condition makes $\mathrm{\Gamma }(f^{-1}(V),{\mathcal{O}}_X)$ inject into ${\prod }_i{B}_i$, so the kernel of $A\to \mathrm{\Gamma }(f^{-1}(V),{\mathcal{O}}_X)$ equals the kernel $I_V$ of $A\to {\prod }_i{B}_i$. The latter is an ideal of $A$, and its localizations give the corresponding kernels on principal opens of $V$.

Thus $\mathcal{I}{\mathrm{\mid }}_V$ is the ideal sheaf associated to $I_V$ on every affine $V$, so $\mathcal{I}$ is quasi-coherent.

By [F2], $\mathcal{I}$ defines a closed subscheme $V(\mathcal{I})$. On each affine $V$, the map $A\to {\prod }_i{B}_i$ factors through $A\mathrm{/}{I}_V$, so $f$ factors through it. If a closed subscheme defined on $V$ by $J\subseteq A$ also receives $f$, then $J\subseteq I_V$; hence $V(I_V)$ factors through that competing closed subscheme. This proves minimality.

Repeating the kernel computation on an open $W\subseteq Y$ gives the restriction of $\mathcal{I}$ to $W$. The correspondence in [F2] therefore identifies $V(\mathcal{I})\cap W$ with the scheme-theoretic image of $f^{-1}(W)\to W$.