This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.
By definition, \(\Sch\) is a full subcategory of \(\LRS\). (§Schemes, ⁋Definition 1) That is, given two schemes \(X,Y\), a scheme morphism from \(X\) to \(Y\) is given by a continuous map \(\varphi: X \rightarrow Y\) and a morphism of structure sheaves \(\varphi^\sharp: \mathscr{O}_Y \rightarrow \varphi_\ast \mathscr{O}_X\), where \(\varphi^\sharp\) must become a local homomorphism when restricted to each stalk. (§Affine Schemes, ⁋Definition 2)
As above, a scheme morphism \(f:X \rightarrow Y\) is essentially an object we have already defined. In the next post we will examine the properties of scheme morphisms, but before that we present four ways to understand scheme morphisms.
Gluing Ring Homomorphisms
The first perspective is a fairly natural one. A scheme is essentially constructed by gluing affine schemes, and by the categorical equivalence \(\AffSch\cong\cRing^\op\), a morphism between affine schemes is essentially a ring homomorphism. Therefore, one should be able to understand a scheme morphism as gluing morphisms between affine schemes. That is, it is reasonable to expect the following proposition.
Proposition 1 Let a scheme morphism \(\varphi: X \rightarrow Y\) be given. Then if \(X\) has an affine open subset \(U\cong\Spec A\) and \(Y\) has an affine open subset \(V\cong\Spec B\) satisfying \(\varphi(U)\subseteq V\), the restriction \(\varphi\vert_U: U \rightarrow V\) is a morphism of affine schemes.
Conversely, let two affine open coverings \(\{U_i\}\) of \(X\) and \(\{V_j\}\) of \(Y\) be given, and let morphisms of affine schemes \(\varphi_{ij}: U_i \rightarrow V_j\) be given. If these satisfy the gluing condition on each intersection and are thereby well defined, then the \(\varphi_{ij}\) define a scheme morphism \(\varphi: X \rightarrow Y\).
One direction is merely a new version of the assertion that \(\AffSch\) is a full subcategory of \(\LRS\), by §Affine Schemes, ⁋Proposition 11, and the gluing for the converse is also obtained in an obvious manner.
Example 2 As an example of a scheme morphism that is not a morphism between affine schemes, consider the map
\[\varphi:\mathbb{A}_\mathbb{K}^{n+1}\setminus \{0\} \rightarrow \mathbb{P}^n_\mathbb{K}\]that first appeared for motivation in §Projective Schemes, §§Projective Space. This formula was traditionally used to construct projective space, but it did not appear in §Projective Schemes, ⁋Example 12 when translating the classical projective space into the language of algebraic geometry. Of course this morphism satisfies the formula \((x_0,\ldots, x_n)\mapsto [x_0:\cdots:x_n]\), but the points of \(\mathbb{A}^{n+1}_\mathbb{K}\) are not only of this form, and moreover this formula carries no information about the structure sheaf, so it would be inappropriate to call it a scheme morphism.
Now, to define \(\varphi\) as a scheme morphism, consider the affine open subscheme of \(\mathbb{P}^n_{\mathbb{K}}\)
\[D_+(\x_i)\cong \Spec \mathbb{K}[\x_0,\ldots, \x_n]_{(\x_i)}\cong \Spec \mathbb{K}[\x_{0/i},\ldots, \x_{n/i}]/(\x_{i/i}-1)\](§Projective Schemes, ⁋Example 12). Also consider the affine space
\[\mathbb{A}^{n+1}_\mathbb{K}=\Spec \mathbb{K}[\x_0,\ldots, \x_n]\]Then
\[\mathbb{A}^{n+1}_\mathbb{K}\setminus \{0\}=\bigcup_{i=0}^n D(\x_i)\]and \(D(\x_i)\cong \Spec \mathbb{K}[\x_0,\ldots, \x_n]_{\x_i}\). Now for each \(i\), the map \(\varphi_i: D(\x_i) \rightarrow D_+(\x_i)\) is a morphism of affine schemes, hence corresponds to a ring homomorphism. Then the affine scheme morphism \(\varphi_i\) obtained by applying the first isomorphism theorem to the formula
\[\phi_i:\mathbb{K}[\x_{0/i},\ldots, \x_{n/i}]\rightarrow\mathbb{K}[\x_0,\ldots, \x_n]_{\x_i};\qquad \x_{k/i}\mapsto \frac{\x_k}{\x_i}\]becomes the desired morphism. That these satisfy the conditions of Proposition 1 can also be checked by a short calculation. Now, borrowing the notation from §Projective Schemes, §§Projective Space, on each \(D(\x_i)\) they are given by the formula
\[(x_0,\ldots, x_n) \rightarrow \left[\frac{x_0}{x_i}:\cdots:\frac{x_{i-1}}{x_i}:1:\frac{x_{i+1}}{x_i}:\cdots:\frac{x_n}{x_i} \right]\]so it is appropriate to denote this as
\[(x_0,\ldots, x_n)\rightarrow [x_0:\cdots:x_n]\]We shall regard this perspective almost as the definition, and the remaining three perspectives introduced below are better understood as ways of interpreting it.
Schemes over a Scheme
First we define the following.
Definition 3 For any scheme \(S\), we call the slice category \(\Sch_{/S}\) over \(S\) the category of \(S\)-schemes. ([Category Theory] §Categories, ⁋Example 13)
That is, an \(S\)-scheme is another name for a scheme morphism \(X \rightarrow S\) to \(S\), which is also called the structure morphism.
This becomes a little more intuitive when we look at the following example.
Example 4 Consider affine \(n\)-space \(\mathbb{A}^n_\mathbb{K}=\Spec \mathbb{K}[\x_1,\ldots, \x_n]\). Then \(\mathbb{K}[\x_1,\ldots, \x_n]\) is a \(\mathbb{K}\)-algebra, and this means that a \(\mathbb{K}\)-algebra structure is given via the structure morphism
\[\mathbb{K}\hookrightarrow \mathbb{K}[\x_1,\ldots, \x_n]\]([Algebraic Structures] §Algebras, ⁋Definition 1 and the argument following it.)
Then through this structure morphism we may view \(\mathbb{A}^n_\mathbb{K}\) as a \(\Spec\mathbb{K}\)-scheme
\[\mathbb{A}^n_\mathbb{K}=\Spec \mathbb{K}[\x_1,\ldots, \x_n] \rightarrow \Spec \mathbb{K}\]As above, when \(S\) is an affine scheme \(S=\Spec A\), it is common by a slight abuse of language to call an \(S\)-scheme \(X\) an \(A\)-scheme. Then by §Affine Schemes, ⁋Theorem 13, fixing an arbitrary ring \(A\) and giving an \(A\)-scheme structure on a scheme \(X\) is precisely the same as
\[\Hom_\Sch(X, \Spec A)=\Hom_\LRS(X, \Spec A)\cong \Hom_\cRing(A, \Gamma(X, \mathscr{O}_X))\]That is, giving an \(A\)-scheme structure on a scheme \(X\) is algebraically equivalent to giving an \(A\)-algebra structure on \(\Gamma(X, \mathscr{O}_X)\). In particular, when \(A=\mathbb{Z}\), since \(\mathbb{Z}\) is the initial object of \(\cRing\), every scheme can be regarded as a \(\mathbb{Z}\)-scheme in a unique way.
Now let us look at the following example, which generalizes Example 2 further.
Example 5 Consider a ring \(A\) and an \(A\)-scheme \(X\), let functions \(f_0,\ldots, f_n\in \Gamma(X, \mathscr{O}_X)\) defined on \(X\) be given, and let \(X=\bigcup U_j\) be an affine open covering of \(X\). Then
\[U_{ij}:=D(f_i)\cap U_j=D(f_i\vert_{U_j})\subseteq U_j\]form an affine open covering of \(X\). On the other hand, consider the projective space over \(A\)
\[\mathbb{P}^n_A=\Proj A[\x_0,\ldots, \x_n]\]and its open covering \(D_+(\x_i)\). Now for each pair \(i,j\), define the map \(\varphi_{ij}: U_{ij} \rightarrow D_+(\x_i)\) via the ring homomorphism
\[A[\x_{0/i},\ldots, \x_{n/i}]\rightarrow \Gamma(U_{ij});\qquad \x_{k/i}\mapsto \frac{f_k\vert_{U_{ij}}}{f_i\vert_{U_{ij}}}\]Then by definition it is obvious that these satisfy the gluing condition of Proposition 1, and hence they define a scheme morphism
\[X \rightarrow \mathbb{P}^n_A\]Explicitly, this scheme morphism is given, in the same way as in Example 2, by
\[x\mapsto [f_0(x):\cdots: f_n(x)]\]Points
Next, we define the following.
Definition 6 We call a scheme morphism \(f: X \rightarrow Y\) an \(X\)-point of \(Y\).
Again, it helps intuitively to examine the case where \(X\) is an affine scheme.
Example 7 Consider an algebraically closed field \(\mathbb{K}\), the affine \(n\)-space \(Y=\mathbb{A}^n_\mathbb{K}=\Spec \mathbb{K}[\x_1,\ldots, \x_n]\) defined over it, and \(X=\Spec \mathbb{K}\). Then a scheme morphism \(X \rightarrow Y\) is a morphism of affine schemes
\[\Spec \mathbb{K} \rightarrow \Spec \mathbb{K}[\x_1,\ldots, \x_n]\]and thus corresponds to a ring homomorphism
\[\phi:\mathbb{K}[\x_1,\ldots, \x_n] \rightarrow \mathbb{K}\]This ring homomorphism must be surjective, because by the definition of a ring homomorphism \(\phi(1)=1\) and therefore \(\phi(x)=x\) for any \(x\in \mathbb{K}\). Hence by the first isomorphism theorem
\[\mathbb{K}[\x_1,\ldots, \x_n]/\ker\phi\cong \mathbb{K}\]Then by the fourth result of [Algebraic Structures] §Quotient Rings, Ring Homomorphisms, ⁋Theorem 3, \(\ker\phi\) must be a maximal ideal of \(\mathbb{K}[\x_1,\ldots, \x_n]\), and therefore by [Commutative Algebra] §Nullstellensatz, ⁋Lemma 5
\[\ker\phi=(\x_1-x_1,\ldots, \x_n-x_n)\]for some \(x=(x_1,\ldots, x_n)\), and \(\phi\) becomes the evaluation homomorphism \(\ev_x\) at the point \(x\). Moreover, considering the proof of that lemma, we also see that \(x_i=\phi(\x_i)\). Thus there exist the following two mutually inverse bijections
\[\begin{aligned}\{\text{$\mathbb{K}$-point $\Spec \phi:\Spec\mathbb{K}\rightarrow \mathbb{A}^n_\mathbb{K}$}\}&\rightarrow \{\text{points $(x_1,\ldots, x_n)\in \mathbb{A}^n_\mathbb{K}$}\}\\\Spec\phi&\mapsto (\phi(\x_1),\ldots,\phi(\x_n))\end{aligned}\]and
\[\begin{aligned}\{\text{points $(x_1,\ldots, x_n)\in \mathbb{A}^n_\mathbb{K}$}\}&\rightarrow \{\text{$\mathbb{K}$-point $\Spec \phi:\Spec\mathbb{K}\rightarrow \mathbb{A}^n_\mathbb{K}$}\}\\a=(a_1,\ldots, a_n)&\mapsto \Spec \ev_a\end{aligned}\]As above, if \(X\) is of the form \(\Spec A\), we simply call it an \(A\)-point. The usefulness of this concept can also be seen in the following example.
Example 8 Let a \(\mathbb{C}\)-scheme \(X=\Spec\frac{\mathbb{C}[\x_1,\ldots, \x_n]}{(f_1,\ldots, f_r)}\) be given, and consider the \(\mathbb{Q}\)-points of this scheme. Then from [Commutative Algebra] §Nullstellensatz, ⁋Lemma 5 and the calculation of Example 6, we know that there exists a one-to-one correspondence between the \(\mathbb{Q}\)-points \(\Spec\phi: \Spec \mathbb{Q}\rightarrow X\) of \(X\) and the rational solutions of
\[f_1(x_1,\ldots, x_n)=\cdots=f_r(x_1,\ldots, x_n)=0\]Similarly, the integer solutions of the above equations correspond exactly to the \(\mathbb{Z}\)-points of \(X\).
Based on this perspective we define the following.
Definition 9 We call the functor \(\Hom_\Sch(-,X): \Sch^\op \rightarrow \Set\) the functor of points of \(X\).
Then \(\Hom_\Sch(-,X)\) is the functor that takes a scheme \(S\) and returns the set of \(S\)-valued points of \(X\).
Families of Schemes
The final perspective is one for which our present language is still insufficient to give a rigorous definition, so we shall only explain the geometric intuition. We call a scheme morphism \(f:X \rightarrow S\) a family parametrized by \(S\), or simply an \(S\)-family. Therefore by definition \(\Sch_{/S}\) can be thought of as the category of families parametrized by \(S\).
For geometric intuition, it basically suffices to consider the following (non-scheme) situation.
Example 10 Consider the sphere \(S:x^2+y^2+z^2=1\) defined in the coordinate space \(\mathbb{R}^3\), and the projection \(\pi: S \rightarrow \mathbb{R}_x\) onto the \(x\)-axis. Then for any \(x_0\in \mathbb{R}_x\),
\[\pi^{-1}(x_0)=\{(x_0,y,z)\in \mathbb{R}: y^2+z^2=1-x_0^2\}\]Geometrically, this can be viewed as a situation in which to each \(x_0\in \mathbb{R}_x\) there corresponds a circle \(y^2+z^2=1-x_0^2\), and therefore we may think of \(\pi\) as
Among the reasons why this example cannot be immediately represented in the language of schemes, the less essential one is that \(S\) is a closed subset of \(\mathbb{R}^3\) and we do not yet know how to put a scheme structure on a closed subset. This will be resolved in §Closed Subschemes. The more subtle and essential point is that there is no way to represent the fiber \(\pi^{-1}(x_0)\) of the function \(\pi\) at a point \(x_0\). Of course a scheme morphism is basically a continuous map, so one could regard this as the fiber of a continuous map; but even if we do so, there is no way to give \(\pi^{-1}(x_0)\) a scheme structure (even assuming the contents of §Closed Subschemes). We will have to wait a little longer to explain this.
References
[Har] R. Hartshorne, Algebraic geometry. Graduate texts in mathematics. Springer, 1977.
[Vak] R. Vakil, The rising sea: Foundation of algebraic geometry. Available online.
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