Let \(V\) be a variety over the algebraically closed field \(k\). Show that a point \(P \in t(V)\) is a closed point if and only if its residue field is \(k\).
If \(f: X \to Y\) is a morphism of schemes over \(k\) and \(P \in X\) is a point with residue field \(k\), then \(f(P) \in Y\) also has residue field \(k\).
Now show that if \(V, W\) are any two varieties over \(k\), then the natural map \[ \Hom_{\mathsf{Var}_k}(V, W) \to \Hom_{\Sch_k}\qty{t(V), t(W)} \] is bijective. Injectivity follows because a morphism of varieties over \(k\) is determined by its action on closed points; surjectivity requires the residue-field preservation established in parts (a) and (b).
Solution
1. A point of \(t(V)\) corresponds to an irreducible closed subvariety\[ Z\subseteq V, \]and its residue field is the function field\[ \boxed{ \kappa(\eta_Z)=k(Z). } \]
Choose an affine open subset \[ U=\operatorname{Spec}A \] of the associated scheme meeting the point \(\eta_Z\). In the classical variety, this corresponds to an affine open subset of \(V\), and \[ Z\cap U \] is defined by a prime ideal \[ \mathfrak p\subseteq A. \] The point \(\eta_Z\) is the prime \(\mathfrak p\). Its residue field is \[ \kappa(\mathfrak p) =\operatorname{Frac}(A/\mathfrak p), \] which is precisely the rational function field of the irreducible variety \(Z\).
2. If \(P\in t(V)\) is a closed point, then\[ \boxed{\kappa(P)=k.} \]
A closed point of \(t(V)\) corresponds to an ordinary closed point \[ p\in V. \] Choose an affine neighborhood with coordinate ring \(A\). The point is a maximal ideal \[ \mathfrak m\subseteq A. \] Since \(A\) is a finitely generated \(k\)-algebra and \(k\) is algebraically closed, the weak Nullstellensatz gives \[ A/\mathfrak m\cong k. \] This quotient is the residue field at the closed point.
3. Conversely, if\[ P\in t(V) \]has residue field \(k\), then \(P\) is closed.
Let \(Z\) be the irreducible closed subvariety whose generic point is \(P\). By 1, \[ k(Z)=\kappa(P)=k. \] The transcendence degree of the function field of an irreducible variety equals its dimension, so \[ \dim Z =\operatorname{trdeg}_k k(Z) =0. \]
An irreducible zero-dimensional variety over an algebraically closed field is a single closed point. Therefore \[ Z=\{P\}, \] so \(P\) is closed in \(t(V)\).
4. Hence\[ \boxed{ P\in t(V)\text{ is closed} \iff \kappa(P)=k. } \]
Combine 2 and 3.
5. Let\[ f:X\longrightarrow Y \]be a morphism of schemes over \(k\), and let \(P\in X\) satisfy\[ \kappa(P)=k. \]Then\[ \boxed{\kappa(f(P))=k.} \]
Put \[ Q=f(P). \] The local homomorphism \[ f_P^\sharp: \mathcal O_{Y,Q} \longrightarrow \mathcal O_{X,P} \] induces an injective homomorphism of residue fields \[ \kappa(Q) \hookrightarrow \kappa(P)=k. \] Because \(f\) is a morphism over \(k\), the composite \[ k \longrightarrow \kappa(Q) \longrightarrow k \] is the identity. Thus the image of \(\kappa(Q)\) inside \(k\) contains the image of \(k\), which is all of \(k\). Since it is also a subfield of \(k\), the image is exactly \(k\).
The injection is therefore a \(k\)-algebra isomorphism \[ \kappa(Q)\cong k. \]
6. Let\[ F:t(V)\longrightarrow t(W) \]be a morphism of schemes over \(k\). Then \(F\) sends closed points to closed points and therefore determines a set map\[ \boxed{ f:V\longrightarrow W. } \]
By 4, the closed points of \(t(V)\) are exactly the points with residue field \(k\). By 5, \(F\) sends each such point to a point of \(t(W)\) whose residue field is \(k\), which by 4 is closed.
The closed points of \(t(V)\) and \(t(W)\) identify respectively with the ordinary points of the classical varieties \(V\) and \(W\). Restricting the underlying map of \(F\) to those closed points gives the stated set map \(f\).
7. The map \(f:V\to W\) is continuous in the Zariski topology.
The classical variety \(V\) identifies with the subspace of closed points of \(t(V)\), and similarly for \(W\). Let \[ C\subseteq W \] be closed. The corresponding subset \[ t(C)\subseteq t(W) \] is closed, and its closed points are exactly \(C\).
Since \(F\) is continuous, \[ F^{-1}(t(C)) \] is closed in \(t(V)\). Intersecting with the closed-point subspace \(V\) gives \[ f^{-1}(C), \] which is therefore closed in \(V\). Hence \(f\) is continuous.
8. Fix \(P\in V\). There are affine open neighborhoods\[ P\in V_0\subseteq V, \qquad f(P)\in W_0\subseteq W, \]such that\[ f(V_0)\subseteq W_0. \]
Choose an affine open neighborhood \[ W_0\subseteq W \] of \(f(P)\). The corresponding open subscheme \[ t(W_0)\subseteq t(W) \] has open inverse image \[ F^{-1}(t(W_0)) \subseteq t(V). \] Its closed points form the open subset \[ f^{-1}(W_0)\subseteq V, \] which contains \(P\).
Affine opens form a basis for a variety, so choose an affine open \[ P\in V_0\subseteq f^{-1}(W_0). \] Then \[ f(V_0)\subseteq W_0. \]
9. Write\[ V_0=\operatorname{Spec}_{\mathrm{var}}A, \qquad W_0=\operatorname{Spec}_{\mathrm{var}}B \]for their affine coordinate rings. The restriction\[ F|_{t(V_0)}: t(V_0)=\operatorname{Spec}A \longrightarrow t(W_0)=\operatorname{Spec}B \]is induced by a \(k\)-algebra homomorphism\[ \varphi:B\longrightarrow A. \]
By 8, the scheme morphism \(F\) maps the open subscheme \(t(V_0)\) into \(t(W_0)\). Both are affine schemes associated to the affine varieties. The affine anti-equivalence therefore identifies the restricted morphism with a unique ring homomorphism \[ B\to A. \] Because \(F\) is a morphism over \(k\), this homomorphism is a \(k\)-algebra homomorphism.
10. The \(k\)-algebra map \(\varphi:B\to A\) defines a morphism of affine varieties\[ f_0:V_0\longrightarrow W_0, \]and on closed points\[ f_0=f|_{V_0}. \]
The classical affine-coordinate correspondence sends a \(k\)-algebra homomorphism \[ B\to A \] to a morphism of affine varieties \[ \operatorname{Spec}_{\mathrm{var}}A \to \operatorname{Spec}_{\mathrm{var}}B. \]
The associated scheme morphism is exactly the affine scheme morphism induced by the same ring map, namely \[ F|_{t(V_0)}. \] Hence their maps on closed points agree. By definition of \(f\) in 6, this says \[ f_0=f|_{V_0}. \]
11. The set map \(f:V\to W\) is a morphism of varieties.
Every point \(P\in V\) has an affine neighborhood \(V_0\) on which 10 identifies \(f\) with a morphism of affine varieties. Regularity of a map of varieties is local on the source and target. Therefore \(f\) is a morphism globally.
12. The associated scheme morphism\[ t(f):t(V)\longrightarrow t(W) \]is the original morphism \(F\).
9. Thus they agree on an open cover of \(t(V)\), hence agree globally.
13. Therefore the natural map\[ \operatorname{Hom}_{\mathsf{Var}_k}(V,W) \longrightarrow \operatorname{Hom}_{\mathsf{Sch}_k}(t(V),t(W)) \]is surjective.
Given any scheme morphism \(F\), steps <1>6–<1>12 construct a variety morphism \(f\) satisfying \[ t(f)=F. \]
14. The same natural map is injective.
If \[ t(f)=t(g), \] then the two associated scheme morphisms have the same map on closed points. Under the identification of the closed points of \(t(V)\) and \(t(W)\) with the ordinary points of \(V\) and \(W\), this means \[ f(P)=g(P) \] for every \(P\in V\).
A morphism of varieties is in particular a function on the underlying point sets, so equality pointwise gives \[ f=g. \]
15. Hence the functor\[ t:\mathsf{Var}_k\longrightarrow\mathsf{Sch}_k \]is fully faithful:\[ \boxed{ \operatorname{Hom}_{\mathsf{Var}_k}(V,W) \xrightarrow{\sim} \operatorname{Hom}_{\mathsf{Sch}_k}(t(V),t(W)). } \]
Surjectivity is 13 and injectivity is 14.
16. Q.E.D.
Steps <1>1–<1>4 prove part (a), 5 proves part (b), and <1>6–<1>15 prove part (c).