Morphisms of Schemes

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — 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 \(\varphi^\sharp: \mathcal{O}_Y \rightarrow \varphi_\ast \mathcal{O}_X\) of structure sheaves, where \(\varphi^\sharp\) must become a local homomorphism upon restriction to each stalk. (§Affine Scheme, ⁋Definition 2)

Thus a scheme morphism \(\varphi:X \rightarrow Y\) is fundamentally an object we have already defined. In the next post we will examine properties of scheme morphisms; before that, we present four ways to understand them.

Gluing Ring Homomorphisms

The first perspective is quite natural. A scheme is essentially built by gluing affine schemes, and by the categorical equivalence \(\AffSch\cong\cRing^\op\), a morphism between affine schemes is essentially a ring homomorphism. Therefore a scheme morphism should also be understood as gluing morphisms between affine schemes. That is, the following proposition is to be expected.

Proposition 1 Let a scheme morphism \(\varphi: X \rightarrow Y\) be given. If an affine open subset \(U\cong\Spec A\) of \(X\) and an affine open subset \(V\cong\Spec B\) of \(Y\) satisfy \(\varphi(U)\subseteq V\), then the map

\[(\varphi\vert_U)\vert^V: U \rightarrow V\]

obtained by restricting the domain of \(\varphi\) to \(U\) and viewing the codomain as \(V\) is a morphism between affine schemes, i.e. a ring homomorphism \(B \rightarrow A\).

Conversely, let an affine open covering \(\{U_i\}\) of \(X\) be given, and suppose for each \(i\) we are given an affine open subset \(V_i\) of \(Y\) and a morphism \(\varphi_i: U_i \rightarrow V_i\) between affine schemes. If these satisfy the gluing condition

\[\varphi_i\vert_{U_i\cap U_k}=\varphi_k\vert_{U_i\cap U_k}\qquad\text{(as morphisms $U_i\cap U_k \rightarrow Y$)}\]

for all \(i,k\), then the \(\varphi_i\) glue to a unique scheme morphism \(\varphi: X \rightarrow Y\).

Proof

For the first claim, for any open subset \(W\subseteq V\), the maps

\[\mathcal{O}_Y(W) \rightarrow \varphi_\ast \mathcal{O}_X(W)=\mathcal{O}_X(\varphi^{-1}(W)) \rightarrow \mathcal{O}_X(\varphi^{-1}(W)\cap U)\]

obtained by composing \(\varphi^\sharp(W):\mathcal{O}_Y(W)\rightarrow\varphi_\ast \mathcal{O}_X(W)\) with the restriction to \(U\) define the sheaf morphism \(\mathcal{O}_Y\vert_V \rightarrow ((\varphi\vert_U)\vert^V)_\ast(\mathcal{O}_X\vert_U)\) that we must examine. The map this induces on the stalk at any \(x\in U\) is determined by germs alone, so it coincides with \(\mathcal{O}_{Y,\varphi(x)} \rightarrow \mathcal{O}_{X,x}\) induced by the original \(\varphi\), and hence is a local homomorphism. Thus \((\varphi\vert_U)\vert^V\) is a morphism of \(\LRS\), and since \(U\) and \(V\) are affine schemes, by §Affine Scheme, ⁋Proposition 11 it is induced from a unique ring homomorphism \(B \rightarrow A\).

For the second claim, by the gluing condition the continuous maps \(\varphi_i: U_i \rightarrow V_i\hookrightarrow Y\) agree on overlaps, so by [Topology] §Presheaves, ⁋Lemma 1 they glue to a continuous map \(\varphi: X \rightarrow Y\). We now define the sheaf morphism \(\varphi^\sharp\). Given an open subset \(W\subseteq Y\) and \(s\in \mathcal{O}_Y(W)\), consider the sections

\[s_i:=\varphi_i^\sharp(W)(s)\in \mathcal{O}_X(\varphi^{-1}(W)\cap U_i).\]

These also agree on overlaps for the same reason, and since \(\{\varphi^{-1}(W)\cap U_i\}\) is an open covering of \(\varphi^{-1}(W)\), the two conditions of [Topology] §Sheaves, ⁋Definition 1 yield a unique glued section \(s'\in \mathcal{O}_X(\varphi^{-1}(W))\). Defining \(\varphi^\sharp(W): s\mapsto s'\), compatibility with restriction maps can be checked on each \(U_i\), and the map induced by \(\varphi^\sharp\) on the stalk at \(x\in U_i\) coincides with that induced by \(\varphi_i^\sharp\), so it is a local homomorphism. Hence \(\varphi\) is a scheme morphism, and since the condition \((\varphi\vert_{U_i})\vert^{V_i}=\varphi_i\) completely determines \(\varphi\), such a morphism is unique.

The first claim is nothing more than applying the fact from §Affine Scheme, ⁋Proposition 11 that \(\AffSch\) is a full subcategory of \(\LRS\) to the local picture of a scheme morphism. However, one must be careful: the gluing condition in the second claim is not expressed as a condition between ring homomorphisms. When \(V_i\neq V_k\), we cannot compare \(\varphi_i\) and \(\varphi_k\) within a single affine scheme, so we must compare them inside \(Y\); moreover \(U_i\cap U_k\) is generally not an affine scheme. Thus the data given for gluing scheme morphisms are ring homomorphisms, but the condition determining whether they glue is not.

Example 2 As an example of a scheme morphism that is not a morphism between affine schemes, for \(n\geq 1\) 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 the traditional projective space was translated into the language of schemes. This morphism of course 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 merely of this form, and moreover this formula encodes no information about the structure sheaf, so it would be inappropriate to call it a scheme morphism.

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 \(i\), since \(\varphi_i: D(\x_i) \rightarrow D_+(\x_i)\) is a morphism between affine schemes, it is the same as a ring homomorphism. Then the formula

\[\phi_i:\mathbb{K}[\x_{0/i},\ldots, \x_{n/i}]\rightarrow\mathbb{K}[\x_0,\ldots, \x_n]_{\x_i};\qquad \x_{k/i}\mapsto \frac{\x_k}{\x_i}\]

defines, by the first isomorphism theorem, a morphism \(\varphi_i\) between affine schemes that gives the desired morphism. That these satisfy the conditions of Proposition 1 can also be checked by a brief computation. Now borrowing the notation from §Projective Schemes, §§Projective Space again, on each \(D(\x_i)\) these 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 essentially take this perspective as the definition; the three perspectives to be introduced in the remainder are closer to ways of interpreting it.

Schemes over a Scheme

First we define the following.

Definition 3 For an arbitrary 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 simply another name for a scheme morphism \(X \rightarrow S\), also called the structure morphism. This becomes slightly more intuitive upon examining the following example.

Example 4 Consider the 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, which is to say 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 via this structure morphism we may regard \(\mathbb{A}^n_\mathbb{K}\) 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 to call an \(S\)-scheme an \(A\)-scheme by a slight abuse of language. Then by §Affine Scheme, ⁋Theorem 13, fixing an arbitrary ring \(A\) and giving a scheme \(X\) an \(A\)-scheme structure is precisely the same as

\[\Hom_\Sch(X, \Spec A)=\Hom_\LRS(X, \Spec A)\cong \Hom_\cRing(A, \Gamma(X, \mathcal{O}_X)).\]

That is, giving an \(A\)-scheme structure to a scheme \(X\) is algebraically equivalent to giving an \(A\)-algebra structure to \(\Gamma(X, \mathcal{O}_X)\). 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 see the following example, which generalizes Example 2 further.

Example 5 Consider a ring \(A\) and an \(A\)-scheme \(X\), and suppose functions \(f_0,\ldots, f_n\in \Gamma(X, \mathcal{O}_X)\) defined on \(X\) are given. Assume that these generate the unit ideal, i.e. \((f_0,\ldots, f_n)=\mathcal{O}_X\). Also consider an affine open covering \(X=\bigcup U_j\) of \(X\). Then

\[U_{ij}:=D(f_i)\cap U_j=D(f_i\vert_{U_j})\subseteq U_j\]

is 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 given a pair \(i,j\), define the function \(\varphi_{ij}: U_{ij} \rightarrow D_+(\x_i)\) via the ring homomorphism

\[A[\x_0,\ldots, \x_n]_{(\x_i)}\rightarrow \Gamma(U_{ij});\qquad \x_{k/i}\mapsto \frac{f_k\vert_{U_{ij}}}{f_i\vert_{U_{ij}}}.\]

Then by definition it is obvious that this morphism satisfies the gluing condition of Proposition 1, and hence these define a scheme morphism

\[X \rightarrow \mathbb{P}^n_A.\]

Explicitly, this scheme morphism is given, in the same manner as Example 2, by

\[x\mapsto [f_0(x):\cdots: f_n(x)].\]

Points

We also define the following.

Definition 6 We call a scheme morphism \(\varphi: X \rightarrow Y\) an \(X\)-point of \(Y\).

Similarly, examining the case where \(X\) is an affine scheme is intuitively helpful.

Example 7 Consider a field \(\mathbb{K}\) and the affine \(n\)-space \(Y=\mathbb{A}^n_\mathbb{K}=\Spec \mathbb{K}[\x_1,\ldots, \x_n]\) defined over it. As we saw in Example 4, \(Y\) is a \(\Spec\mathbb{K}\)-scheme. According to Definition 6, a \(\mathbb{K}\)-point of \(Y\) is an arbitrary scheme morphism \(\Spec\mathbb{K}\rightarrow Y\); but since \(Y\) is a \(\Spec\mathbb{K}\)-scheme, among these we are interested in sections of the structure morphism \(Y\rightarrow\Spec\mathbb{K}\), that is, morphisms \(X=\Spec\mathbb{K}\rightarrow Y\) over \(\Spec\mathbb{K}\). This is a \(\mathbb{K}\)-morphism between affine schemes

\[\Spec \mathbb{K} \rightarrow \Spec \mathbb{K}[\x_1,\ldots, \x_n]\]

and hence corresponds to a \(\mathbb{K}\)-algebra homomorphism

\[\phi:\mathbb{K}[\x_1,\ldots, \x_n] \rightarrow \mathbb{K}.\]

Now \(\mathbb{K}[\x_1,\ldots, \x_n]\) is a polynomial algebra over \(\mathbb{K}\), so by the universal property of [Algebraic Structures] §Algebras, ⁋Proposition 8 such a \(\phi\) is uniquely determined by the images \(x_i=\phi(\x_i)\in \mathbb{K}\) of each variable, and conversely any \(x=(x_1,\ldots, x_n)\in \mathbb{K}^n\) yields the evaluation homomorphism \(\ev_x\). That is, \(\phi=\ev_x\), and in particular

\[\ker\phi=(\x_1-x_1,\ldots, \x_n-x_n)\]

since the quotient by the right-hand side is already \(\mathbb{K}\). Thus the following two mutually inverse bijections exist:

\[\begin{aligned}\{\text{$\mathbb{K}$-point $\Spec \phi:\Spec\mathbb{K}\rightarrow \mathbb{A}^n_\mathbb{K}$}\}&\rightarrow \{\text{points $(x_1,\ldots, x_n)\in \mathbb{K}^n$}\}\\\Spec\phi&\mapsto (\phi(\x_1),\ldots,\phi(\x_n))\end{aligned}\]

and

\[\begin{aligned}\{\text{points $(x_1,\ldots, x_n)\in \mathbb{K}^n$}\}&\rightarrow \{\text{$\mathbb{K}$-point $\Spec \phi:\Spec\mathbb{K}\rightarrow \mathbb{A}^n_\mathbb{K}$}\}\\a=(a_1,\ldots, a_n)&\mapsto \Spec \ev_a\end{aligned}\]

As above, if \(X\) is of the form \(\Spec A\), we simply call this an \(A\)-point. The usefulness of this concept can also be seen in the following example.

Example 8 Consider the \(\mathbb{Z}\)-scheme \(X=\Spec\mathbb{Z}[\x_1,\ldots, \x_n]/(f_1,\ldots, f_r)\) defined by integer-coefficient polynomials \(f_1,\ldots, f_r\in\mathbb{Z}[\x_1,\ldots, \x_n]\). Then by §Affine Scheme, ⁋Theorem 13, a \(\mathbb{Q}\)-point \(\Spec\phi: \Spec \mathbb{Q}\rightarrow X\) of \(X\) corresponds to a ring homomorphism \(\phi:\mathbb{Z}[\x_1,\ldots, \x_n]/(f_1,\ldots, f_r)\rightarrow\mathbb{Q}\), and since \(\phi\) is canonically given over \(\mathbb{Z}\), this is again in bijection with the rational solutions \((x_1,\ldots, x_n)\in\mathbb{Q}^n\) of

\[f_1(x_1,\ldots, x_n)=\cdots=f_r(x_1,\ldots, x_n)=0.\]

Similarly, integer solutions of the above equation correspond exactly to \(\mathbb{Z}\)-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 cannot yet be defined rigorously because our language is insufficient, so we shall only explain the geometric intuition. We call a scheme morphism \(\varphi:X \rightarrow S\) a family parametrized by \(S\), or simply an \(S\)-family. Thus by definition \(\Sch_{/S}\) can be regarded as the category of families parametrized by \(S\).

For geometric intuition, one should basically think of the following (non-scheme) situation.

Example 10 Consider the sphere \(S:x^2+y^2+z^2=1\) defined in coordinate space \(\mathbb{R}^3\), and the projection \(\pi: S \rightarrow \mathbb{R}_x\) onto the \(x\)-axis. Then for any \(x_0\in \mathbb{R}_x\),

\[\pi^{-1}(x_0)=\{(x_0,y,z)\in \mathbb{R}^3\mid y^2+z^2=1-x_0^2\}.\]

Geometrically, this can be viewed as a situation in which, for each \(x_0\in \mathbb{R}_x\), a circle \(y^2+z^2=1-x_0^2\) is assigned; hence we may regard \(\pi\) as a family of circles parametrized by the \(x\)-axis. Of course if \(\lvert x_0\rvert>1\) this fiber is empty, and if \(x_0=\pm 1\) it is a single point, so it is a circle only when \(\lvert x_0\rvert<1\). The manner in which members of a family degenerate in this way will become an issue again when we treat flatness later.

Among the reasons we cannot directly represent this example as a scheme, the less essential one is that \(S\) is a closed subset of \(\mathbb{R}^3\) and we do not yet know how to endow a closed subset with a scheme structure. This will be resolved in §Closed Subschemes. The more subtle and essential difficulty is that there is no way to represent the fiber \(\pi^{-1}(x_0)\) of the function \(\pi\) at a point \(x_0\). A scheme morphism is of course a continuous map, so we could view this as the fiber of a continuous map; but even doing so (even assuming the content of §Closed Subschemes) there is no way to give \(\pi^{-1}(x_0)\) a scheme structure. To explain this we must wait a little longer.

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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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