1 Proper morphisms

The functorial point of view

(1.1) Schemes as functors

References: [ GW1 ] Sections (4.1), (4.2); [ Mu ] II.6.

As we have discussed in Algebraic Geometry 1, to a scheme $X$ we can attach its functor of $T$-valued points:

\[ h_X\colon ({\rm Sch})^{{\rm opp}} \to {\rm (Sets)},\quad T\mapsto X(T):=\operatorname{Hom}(T, X), \]

which on morphisms is just given by composition: $f\colon T'\to T$ is mapped under $h_X$ to the map $X(T)\to X(T')$, $\alpha \mapsto \alpha \circ f$.

Example 1.1

For any affine scheme $X=\operatorname{Spec}A$, we have $X(T) = \operatorname{Hom}(A, \Gamma (T, {\mathscr O}_T))$. For example, this gives ${\mathbb A}^n_R(T) = \Gamma (T, {\mathscr O}_T)^n$ for any ring $R$ and any $R$-scheme $T$ (where we understand ${\mathbb A}^n_R(T)$ as the set of morphisms $T\to {\mathbb A}^n_R$ of $R$-schemes).
It is more difficult to describe $\mathbb {P}^n(T)$ for general $T$ (for $T$ the spectrum of a field, we have the description by homogeneous coordinates). We will come back to this later.

Using the notion of a morphism of functors, we can speak of the category $\widehat{{\mathscr C}} := \operatorname{Func}(({\rm Sch})^{{\rm opp}}, {\rm (Sets)})$ of all such functors, and we obtain a functor $h\colon ({\rm Sch}) \to \widehat{{\mathscr C}}$, $X\mapsto h_X$, which on morphisms is – once again – defined by composition: $\alpha \colon X'\to X$ is mapped by $h$ to the morphism $h_{X'}\to h_X$ of functors given by $X'(T)\to X(T)$, $\beta \mapsto \beta \circ \alpha $.

Even though at first sight this may look complicated, this is an entirely “formal” (i.e., category-theoretic) procedure which has nothing to do with schemes. In fact, if ${\mathscr C}$ is any category (as always, assumed to be locally small, i.e., each $\operatorname{Hom}(X, Y)$ is a set), for an object $X$ of ${\mathscr C}$ we can define the functor

\[ h_X\colon {\mathscr C}^{{\rm opp}} \to {\rm (Sets)},\quad T\mapsto X(T):=\operatorname{Hom}_{{\mathscr C}}(T, X), \]

and now setting $\widehat{{\mathscr C}} := \operatorname{Func}({\mathscr C}^{{\rm opp}}, {\rm (Sets)})$, we obtain a functor $h\colon {\mathscr C}\to \widehat{{\mathscr C}}$.

Theorem 1.2

(Yoneda Lemma) For any category ${\mathscr C}$, the functor $h$ constructed above is fully faithful.

Proof

Given $X$ and $Y$ and a morphism $\Phi \colon h_X\to h_Y$, we obtain a morphism $X\to Y$ by applying $\Phi $ to $\operatorname{id}_X\in h_X(X)$. One checks that this is an inverse of the map $\operatorname{Hom}(X, Y)\to \operatorname{Hom}(h_X, h_Y)$ given by $h$.

Remark 1.3

The Yoneda lemma says that the map $\operatorname{Hom}_{{\mathscr C}}(X, Y)\to \operatorname{Hom}(h_X, h_Y)$ is a bijection for all $X$, $Y$. We may rewrite the left hand side as $h_Y(X)$. More generally, with basically the same argument one can show that for every functor $F\colon {\mathscr C}^{{\rm opp}}\to {\rm (Sets)}$ there are identifications $F(X) = \operatorname{Hom}_{\operatorname{Func}({\mathscr C}^{{\rm opp}}, {\rm (Sets)})}(h_X, F)$ that are functorial in $X$.

We will mostly apply the Yoneda Lemma to the category of schemes, or the category of $S$-schemes for some fixed scheme $S$. Let us list some of its consequences (as before, these facts are not specific to the category of schemes):

Let $X$, $Y$ be schemes. The following are equivalent:
1. $X\cong Y$,
2. $h_X \cong h_Y$ (isomorphism of functors)
3. there exists a family $f_T\colon X(T)\to Y(T)$ of bijections of sets that is functorial in $T$, i.e., for every scheme morphism $T'\to T$, the diagram
  
  $\begin{tikzcd} X(T) \arrow[r, "f_T"]\arrow[d] & Y(T)\arrow[d] \\ X(T') \arrow[r, "f_T"]& Y(T') \end{tikzcd}$
  
  is commutative.
Let $X$, $Y$ be schemes. Giving a scheme morphism $X\to Y$ is equivalent to giving a family of maps $f_T\colon X(T)\to Y(T)$ of sets for each scheme $T$, that is functorial in $T$ (same condition as in (1) (iii)). Example. The determinant of a matrix is a scheme morphism $\mathbb A^{n^2}\to \mathbb A^1$.
A diagram of scheme morphisms is commutative if and only if for every scheme $T$ the diagram (in the category of sets) obtained by replacing each scheme by its set of $T$-valued points, and replacing the scheme morphisms by the induced maps of sets, is commutative.

Given a functor $F\colon {\mathscr C}^{\rm opp}\to {\rm (Sets)}$, it “usually” will not be isomorphic to a functor of the form $h_X$. If it is, it is called representable (by $X$), and this is a very special property. On the other hand, especially in algebraic geometry, i.e., when ${\mathscr C}$ is the category of schemes (or of $S$-schemes for some fixed scheme $S$), it turns out that many naturally appearing functors are in fact representable. Proving such representability results is often quite difficult, but at the same time extremely useful, and this approach yields some of the most interesting examples of schemes.

Fiber products and base change

(1.2) Recap: fiber products of schemes

References: [ GW1 ] Sections (4.4)–(4.6); [ H ] II.3; [ Mu ] II.2.

April 15,
2026

The universal property of a fiber product generalizes the universal property of a product (of two objects, in any category). It is defined as follows. (See the lecture notes for Algebraic Geometry 1 for a bit more background, e.g., an explanation of the term fiber product.)

Definition 1.4

Let ${\mathscr C}$ be a category. We fix morphisms $f\colon X\to S$ and $g\colon Y\to S$. Then an object $P$ in ${\mathscr C}$ together with morphisms $p\colon P\to X$ and $q\colon P\to Y$ such that $f\circ p = g\circ q$ is called a fiber product of $X$ and $Y$ over $S$, if for every object $T$ of ${\mathscr C}$ together with morphisms $\alpha \colon T\to X$ and $\beta \colon T\to Y$ with $f\circ \alpha = g\circ \beta $, there exists a unique morphism $\xi \colon T\to P$ with $p\circ \xi = \alpha $, $q\circ \xi = \beta $.

We say that a commutative square

$\begin{tikzcd} A \arrow[r]\arrow[d] & B\arrow[d] \\ C \arrow[r] & D \end{tikzcd}$

is cartesian, if it is a fiber product diagram, i.e., if $A$ satisfies the universal property defining the fiber product of $B$ and $C$ over $D$.

Theorem 1.5

If $f\colon X\to S$, $g\colon Y\to S$ are morphisms of schemes, then the fiber product of $X$ and $Y$ over $S$ exists.
If in (1) $X=\operatorname{Spec}A$, $Y=\operatorname{Spec}B$, $S=\operatorname{Spec}R$ are affine schemes (so that $f$ and $g$ are given by ring homomorphisms $R\to A$, $R\to B$, then $\operatorname{Spec}A\otimes _RB$ together with the morphisms induced by the natural maps $A\to A\otimes _RB$, $B\to A\otimes _RB$, is the fiber product of $X$ and $Y$ over $S$.
For $f$, $g$ as in (1), and open covers $S = \bigcup _i U_i$, $f^{-1}(U_i) = \bigcup _j V_{ij}$, $g^{-1}(U_i) = \bigcup _k W_{ik}$, for all $i$, $j$, $k$, the natural morphism $V_{ij}\times _{U_i}W_{ik}\to X\times _SY$ induced by the universal property of the fiber product is an open immersion, and taken together the open subschemes of the above form cover $X\times _SY$.

Example 1.6

If $f\colon X\to S$ is any morphism and $V\to S$ is an open immersion, then we can identify $X\times _SV = f^{-1}(V)$ as $X$-schemes, i.e., the projection $X\times _SV\to X$ is an open immersion which induces an isomorphism $X\times _SV \cong f^{-1}(V)$ (where we view $f^{-1}(V)$ as an open subscheme of $X$).

Example 1.7

(Fibers of a morphism) If $f\colon X\to S$ is a morphism of schemes and $s\in S$, then we call the fiber product $X_s := f^{-1}(s) := X\times _S\operatorname{Spec}\kappa (s)$ (with respect to $f$ and the natural morphism $\operatorname{Spec}\kappa (s)\to S$) the scheme-theoretic fiber of $f$ over $s$. The projection $X_s\to X$ induces a homeomorphism between the topological space of the scheme $X_s$ and the fiber of the continuous map $f$ over $s$. Cf. the Algebraic Geometry 1 class, or see [ GW1 ] Section (4.8).

Lemma 1.8

In the list below, the “obvious” maps between fiber products are isomorphisms:

$X\times _SS\cong X$,
$X\times _SY \cong Y\times _SX$,
$(X\times _SY)\times _TZ \cong X\times _S(Y\times _TZ)$ (and this allows us to omit the parentheses in expressions like these).
Let $f\colon X\to Y$ be a morphism and let $\Gamma _f\colon X\xrightarrow {(\operatorname{id}, f)} X\times _SY$ be the graph of $f$. Then the following diagram is a fiber product diagram:

$\begin{tikzcd} X \ar[r, "f"]\ar[d, "\Gamma_f"] & Y \ar[d, "\Delta"] \\ X\times_SY \ar[r, "f\times \id"] & Y\times_SY. \end{tikzcd}$

(These claims have to be read in the “natural way”, e.g., in (1) the fiber product on the left is formed with respect to $\operatorname{id}\colon S\to S$, and in (3) morphisms $X\to S$, $Y\to S$, $Y\to T$, $Z\to T$ are given and used to form the fiber products, and the morphism $X\times _S Y\to T$ is the composition $X\times _S Y\to Y\to T$ of the projection to the second factor with the given morphism $Y\to T$, and similarly on the right hand side.)

Proof

These properties can easily be checked using the universal property (or, what more or less amounts to the same, by the Yoneda lemma). In any case, this reduces to checking the above claims for fiber products of sets, where they follow immediately from the explicit description of fiber products of sets.

Example 1.9

(Group schemes) Let $S$ be a scheme. A group scheme over $S$ is an $S$-scheme $G$ together with a functor $h\colon ({\rm (Sch)}/S)^{{\rm opp}}\to {\rm (Grp)}$, such that $h_G$ is the composition of $h$ and the forgetful functor ${\rm (Grp)}\to {\rm (Sets)}$. In other words, for every $S$-scheme $T$, we are given a group structure on $G(T)$, and for every morphism $T'\to T$, the induced map $G(T)\to G(T')$ is a group homomorphism.

In view of the above discussion, we can express this structure equivalently by giving a multiplication morphism $m\colon G\times _SG\to G$, a morphism $i\colon G\to G$ (“inverse element”) that induces the map $g\mapsto g^{-1}$ on each $G(T)$, and a morphism $S\to G$ (“neutral element”) that induces the neutral element in each $G(T)$ (note that for every $S$-scheme $T$, the set $S(T)$ is a singleton). The morphisms $m$, $i$, $e$ have to satisfy certain conditions reflecting the group axioms; the conditions can be expressed by requiring that certain diagrams be commutative. See [ GW1 ] Section (4.15).

Some examples: Fix an affine scheme $S=\operatorname{Spec}(R)$. We will consider group schemes over $S$.

The trivial group scheme $G=S$. Then $G(T)$ is a singleton set for every $S$-scheme $T$, and we give it the structure of the trivial group.
The additive group $\mathbb {G}_a$ (over $S$). As a scheme, we define it as $\mathbb {G}_a = \operatorname{Spec}R[X]$. Then for every $S$-scheme $T$ we have, since $\mathbb {G}_a$ is affine,

\[ \mathbb {G}_a(T) = \operatorname{Hom}_S(T, \mathbb {G}_a) = \operatorname{Hom}_R(R[X], \Gamma (T, {\mathscr O}_T)) = \Gamma (T, {\mathscr O}_T) \]

and we equip it with the group structure given by the addition of the ring $\Gamma (T, {\mathscr O}_T)$.
The multiplicative group $\mathbb {G}_m$ (over $S$). As a scheme, we define it as $\mathbb {G}_m = \operatorname{Spec}R[X, X^{-1}]$. Then for every $S$-scheme $T$ we have, since $\mathbb {G}_m$ is affine,

\[ \mathbb {G}_m(T) = \operatorname{Hom}_S(T, \mathbb {G}_m) = \operatorname{Hom}_R(R[X, X^{-1}], \Gamma (T, {\mathscr O}_T)) = \Gamma (T, {\mathscr O}_T)^\times \]

and we equip it with the multiplicative group structure inherited from the multiplication of the ring $\Gamma (T, {\mathscr O}_T)$.
The general linear group $GL_n$ (as a group scheme over $S$). This should be a group scheme $GL_n$ such that for every $S$-scheme $T$ we have

\[ GL_n(T) = GL_n(\Gamma (T, {\mathscr O}_T)), \]

where the right hand side is the “usual” general linear group of invertible matrices with entries in the ring $\Gamma (T, {\mathscr O}_T)$. Clearly this is a functorial group structure on all these sets, but we still have to justify that there exists an $S$-scheme that has these sets as its $T$-valued points. But this is not difficult: We view the affine space $\mathbb {A}^{n^2}_S = \operatorname{Spec}(R[X_{i,j};\ i, j = 1, \dots , n])$ of dimension $n^2$ as the space of $n\times n$-matrices, i.e.,

\[ \mathbb {A}^{n^2}_S(T) = \Gamma (T, {\mathscr O}_T)^{n^2} = {\rm Mat}_n(\Gamma (T, {\mathscr O}_T)), \]

denote by $\det \in R[X_{i,j};\ i, j = 1, \dots , n]$ the determinant of the matrix $(X_{i,j})_{i,j}$ given by the indeterminates, and define $GL_n = D(\det )$, the principal open subscheme defined by $\det $, as a scheme. This has the desired $T$-valued points.
Elliptic curves also are an (interesting!) example, but what we have done in Part 1 of the lecture is not a full proof that they carry a group scheme structure (but could be made into one without too much work: one would have to express the group law that we have constructed on $E(k')$ for every extension field $k'$ of the base field $k$ in terms of explicit formulas and use those to show that the multiplication and the rule mapping each element to its inverse (=negative) are in fact given by scheme morphisms).

(1.3) Base change

References: [ GW1 ] Chapter 4, in particular Sections (4.7)–(4.10).

April 21,
2026

Given scheme morphisms $f\colon X\to S$ and $g\colon S'\to S$, we call the projection $X\times _SS'\to S'$ the morphism obtained from $f$ by base change along $g$. This defines a functor from the category of $S$-schemes to the category of $S'$-schemes.

A particularly simple example is the case where $g\colon V\to S$ is an open immersion. In that case the base change of $f$ is just the restriction of $f$ to $f^{-1}(V)\to V$.

Many properties of scheme morphisms are “stable under base change” in the following sense: A property $\mathbf P$ of scheme morphisms is called stable under base change if for every morphism $f\colon X\to Y$ of $S$-schemes that has property $\mathbf P$ and every scheme morphism $S'\to S$, the induced morphism $X\times _SS'\to Y\times _SS'$ also has property $\mathbf P$.

Given a property $\mathbf P$, to check that it is stable under base change, it is enough to check that whenever $f\colon X\to S$ has the property, and $g\colon S'\to S$ is a scheme morphism, then $X\times _SS'\to S'$ also has the property. In fact, this is clearly a special case of the above definition (namely the case where $Y=S$). On the other hand, suppose this special case is true and $f\colon X\to Y$ is any morphism of $S$-schemes. Identifying $X\times _SS' = X\times _Y(Y\times _SS')$ using the rules of “computations with fiber products” (Lemma 1.8), the base change $X\times _SS'\to Y\times _SS'$ is identified with the projection $X\times _Y(Y\times _SS')\to Y\times _SS'$. Applying the special case to $X\to Y$ and the base change $Y\times _SS'\to Y$, we obtain that $X\times _SS' = X\times _Y(Y\times _SS')\to Y\times _SS'$ has property $\mathbf P$.

Proposition 1.10

The following properties of scheme morphisms are stable under base change: Being …

an isomorphism,
an open immersion,
a closed immersion,
an immersion,
surjective,
…most of the properties of scheme morphisms that we will get to know later in the course …

A notable exception is the property of being injective: Can you find an example of an injective morphism $X\to S$ of schemes and a morphism $S'\to S$ such that the base change $X\times _SS'\to S'$ is not injective?

All the properties in the above list, and also being injective, are stable under composition, i.e., if two composable morphisms both have the property, then so does the composition.

Example 1.11

If $R\to R'$ is a ring homomorphism, then $\mathbb {A}^n_R\otimes _RR' := \mathbb {A}^n_R\times _{\operatorname{Spec}R}\operatorname{Spec}R' = \mathbb {A}^n_{R'}$. In view of this we define, for an arbitrary scheme $S$, $\mathbb {A}^n_S := \mathbb {A}^n_{\mathbb {Z}}\times _{\operatorname{Spec}\mathbb {Z}} S$.
If $R\to R'$ is a ring homomorphism, then $\mathbb {P}^n_R\otimes _RR' := \mathbb {P}^n_R\times _{\operatorname{Spec}R}\operatorname{Spec}R' = \mathbb {P}^n_{R'}$. In view of this we define, for an arbitrary scheme $S$, $\mathbb {P}^n_S := \mathbb {P}^n_{\mathbb {Z}}\times _{\operatorname{Spec}\mathbb {Z}} S$.

Remark 1.12

Recall the notion of separated morphism: $Y\to S$ is called separated, if the diagonal morphism $\Delta \colon Y\to Y\times _SY$ is a closed immersion (it is always an immersion, and the question really is whether the topological image is a closed subspace). This is a scheme-theoretic analogue of the Hausdorff property of topolgical spaces.

As an application of the above we obtain the following. Let $Y\to S$ be separated and let $f\colon X\to Y$ be any morphism. Then the graph $\Gamma _f \colon X\to X\times _SY$ is a closed immersion, because being a closed immersion is stable under base change and because of the cartesian diagram in Lemma 1.8 (4).

Proper morphisms

(1.4) Proper maps between topological spaces

References: [ Bou-TG ] Ch. I §10, [ Stacks ] Section 005M.

Most schemes that we have encountered so far (in particular, all affine schemes, projective space over any ring, subschemes $V_+(I)$ of projective space over a ring, …) are quasi-compact. On the other hand, from a geometric point of view, e.g., the affine line (or higher-dimensional affine space) “should not be viewed” as a compact space. The notion of properness is a suitable replacement in algebraic geometry for the notion of compactness in topology/differential geometry.

Similarly as separatedness, we will define properness in terms of fiber products of schemes, starting from a characterization of quasi-compact topological spaces, given by the notion of proper map between continuous spaces, which we discuss below as a motivation for the definition of proper scheme morphisms. The purpose of motivation aside, the rest of this section plays no role in the course.

Note that fiber products in the category of topological spaces exist. In fact, for continous maps $X\to S$, $Y\to S$, the set-theoretic fiber product $X\times _SY$, equipped with the subspace topology for the inclusion $X\times _SY\subseteq X\times Y$ (where the right hand side carries the product topology) is easily seen to satisfy the required universal property.

Recall that a continous map $f\colon X\times Y$ is closed, if for every closed subset $C\subseteq X$, the image $f(C)\subseteq Y$ is closed.

Definition 1.13

We call a continuous map $f\colon X\to Y$ universally closed, if for every continuous map $Z\to Y$, the “base change” of $f$ along $Z\to Y$, i.e., the induced map $X\times _YZ\to Z$, is closed.
We call a continuous map $f\colon X\to Y$ Bourbaki-proper, if for every topological space $Z$, the induced map $f\times \operatorname{id}_Z\colon X\times Z\to Y\times Z$ is closed.

Proposition 1.14

Let $f\colon X\to Y$ be a continuous map. The following properties are equivalent.

$f$ is universally closed,
$f$ is Bourbaki-proper,
$f$ is closed and for every $y\in Y$ the fiber $f^{-1}(y)$ is quasi-compact.
$f$ is closed and for every quasi-compact subset $K\subseteq Y$, the inverse image $f^{-1}(K)$ is quasi-compact.

See [ Stacks ] Theorem 005R.

(1.5) Morphisms of finite type

References: [ GW1 ] Sections (10.1), (10.2).

To define proper morphisms of schemes in the following section, we also need the following ingredients.

Definition 1.15

A morphism $f\colon X\to Y$ is called quasi-compact, if for every quasi-compact open $V\subseteq Y$ the inverse image $f^{-1}(V)$ is quasi-compact.

Lemma 1.16

Let $f\colon X\to Y$ be a morphism of schemes. The following are equivalent.

The morphism $f$ is quasi-compact.
For every affine open subscheme $V\subseteq Y$, the inverse image $f^{-1}(V)$ is quasi-compact.
There exists a cover $Y=\bigcup _i V_i$ by affine open subschemes such that for every $i$ the inverse image $f^{-1}(V_i)$ is quasi-compact.

Note that in part (iii) of the lemma it is important to consider a cover by affine open subschemes.

Recall that an algebra $B$ over a ring $A$ is called of finite type (or equivalently, finitely generated) if there exists $n\ge 0$ and a surjective $A$-algebra homomorphism $A[X_1,\dots , X_n]\to B$. We then also say that the ring homomorphism $A\to B$ is of finite type. For example, if $R$ is any ring, $f\in R$, then the homomorphism $R\to R_f$ is of finite type, since $R_f \cong R[X]/(fX-1)$. (On the other hand, if ${\mathfrak p}\subset R$ is a prime ideal, then the homomorphism $R\to R_{{\mathfrak p}}$ typically is not of finite type.) If $A\to B$ and $B\to C$ are ring homomorphisms of finite type, then the composition $A\to C$ is also of finite type.

Definition 1.17

A morphism $f\colon X\to Y$ of schemes is called locally of finite type (or: $X$ is called a $Y$-scheme locally of finite type, of locally of finite type over $Y$), if for every affine open subscheme $V\subseteq Y$ and every open subscheme $U\subseteq f^{-1}(V)$, the ring homomorphism $\Gamma (V, {\mathscr O}_Y)\to \Gamma (U, {\mathscr O}_X)$ induced by the restriction $U\to V$ of $f$ makes $\Gamma (U, {\mathscr O}_X)$ a $\Gamma (V, {\mathscr O}_Y)$-algebra of finite type.
A morphism $f\colon X\to Y$ of schemes is called of finite type (or: $X$ is called a $Y$-scheme of finite type, or of finite type over $Y$), if $f$ is locally of finite type and quasi-compact.

Lemma 1.18

Let $f\colon X\to Y$ be a morphism of schemes. The following are equivalent.

The morphism $f$ is locally of finite type.
There exist a cover $Y=\bigcup _i V_i$ by affine open subschemes, and for each $i$ a cover $f^{-1}(V_i) = \bigcup _j U_{ij}$ by affine open subschemes such that for all $i$, $j$ the $\Gamma (V_i, {\mathscr O}_Y)$-algebra $\Gamma (U_{ij}, {\mathscr O}_X)$ is of finite type.

Proof

This kind of statement holds for many interesting properties of morphisms of schemes. While the proof does not use any difficult input, it is quite long; but it nicely illustrates how to work with schemes and to eventually reduce scheme-theoretic statements to commutative algebra.

Step 1. We first show that without loss of generality, we may assume $V=Y$. In fact, if $D_{V_i}(s) \subset V_i$ is a principal open subset, then $f^{-1}(D_{V_i}(s)) = \bigcup _j D_{U_{ij}}(s_j)$, where $s_j\in \Gamma (U_{ij}, {\mathscr O}_X)$ is the image of $s$, and the $\Gamma (D_{V_i}(s), {\mathscr O}_Y)$-algebra $\Gamma (D_{U_{ij}}(s_j), {\mathscr O}_X)$ is of finite type. Thus replacing the $V_i$ by suitable principal open subsets, we may assume that $V$ is a union of some of the $V_i$, and can thus replace $Y$ by $V$. Moreover, we may choose these principal opens in $V_{i}$ so that at the same time they are principal open in $V$ (Lemma 1.19 below), so we may assume in addition that each $V_i$ is principal open in $Y$.

Step 2. We may replace $U_{ij}$ by a cover by principal opens, and thus assume that $U$ is covered by some of the $U_{ij}$. Moreover, as in Step 1 we may choose these principal opens in $U_{ij}$ so that at the same time they are principal open in $U$. We may therefore assume, without loss of generality, that $U=X$ and that every $U_{ij}$ is principal open in $X$.

Step 3. We now have the following situation: $X\to Y$ is a morphism of affine schemes, say corresponding to a ring homomorphism $A\to B$. We want to show that $A\to B$ is of finite type. We have a covering $Y = \bigcup V_i$ by principal open subsets (which we may assume to be finite, since $Y$ is quasi-compact), and coverings $f^{-1}(V_i) = \bigcup _j U_{ij}$ (which likewise we may assume to be finite) by principal opens of $X$, such that the homomorphism $\Gamma (V_i, {\mathscr O}_Y)\to \Gamma (U_{ij}, {\mathscr O}_X)$ is of finite type. Since $\Gamma (V, {\mathscr O}_Y)\to \Gamma (V_i, {\mathscr O}_Y)$ is also of finite type and this property is stable under composition, we also have that $\Gamma (V, {\mathscr O}_Y)\to \Gamma (U_{ij}, {\mathscr O}_X)$ is of finite type. We may therefore “forget about the $V_i$” and simply apply Lemma 1.20 to $A\to B$ and the cover $\operatorname{Spec}(B) = \bigcup _{i,j} U_{ij}$.

In the proof we have used the following lemmas.

Lemma 1.19

Let $X$ be a scheme, $x\in X$, and $U$, $V$ affine open subschemes of $X$ which both contain $x$. Then there exists an open neighborhood $W \subseteq U\cap V$ of $x$ which is principal open both in $U$ and in $V$.

Proof

One easily checks that a principal open in a principal open of some affine scheme $U$ is principal open in $U$. Thus after replacing $U$ by a suitable principal open, we may assume $U \subseteq V$. Now let $s \in \Gamma (V, {\mathscr O}_X)$ such that $D_V(s) \subseteq U\cap V$. This has the desired property since $D_V(s) = D_U(s_{|U})$. (By assumption, $D_V(s) \subseteq U$, and we can check whether a point of $U$ lies in this set by checking whether the image of $s$ in the residue class field is $\ne 0$, and the residue class field does not depend on whether we compute it inside $U$ or $V$.)

Lemma 1.20

Let $A\to B$ be a ring homomorphism, and let $b_1, \dots , b_r\in B$ be a family of elements which generates the unit ideal and such that for every $i$ the homomorphism $A\to B_{b_i}$ is of finite type. Then $A\to B$ is of finite type.

Proof

For each $i$ let $\left(\dfrac {c_{ij}}{b_i^{\nu _j}}\right)_j$ be a finite system of generators of $B_{b_i}$ as an $A$-algebra, where $c_{ij}\in B$. Write $1 = \sum a_i b_i$. Let $B' \subseteq B$ be the $A$-subalgebra generated by all $b_i$, all $a_i$ and all $c_{ij}$. We will show that $B' = B$, thus proving the lemma. So let $b\in B$. Its image in each $B_{b_i}$ can be written as a polynomial expression in the $\frac{c_{ij}}{b_i^{\nu _j}}$ with coefficients in $A$, and clearing denominators, we find $N\ge 0$ such that $b_i^Nb\in B'$. We may assume that the same $N$ works for every $i$. Since $B'$ contains the elements $a_i$, the $b_i$ generate the unit ideal in $B'$, so the same is true for the $b_i^N$. Therefore the above implies that $b\in B'$, as desired.

Each of the properties of being locally of finite type, quasi-compact, and of finite type is stable under composition and under base change.

(1.6) Proper morphisms

References: [ GW1 ] Section (12.13); [ H ] II.4; [ Mu ] II.7.

Definition 1.21

Let $f\colon X\to Y$ be a morphism of schemes.

The morphism $f$ is called closed, if for every closed subset $Z\subseteq X$, the image $f(Z)$ is a closed subset of $Y$.
The morphism $f$ is called universally closed, if for every morphism $Y'\to Y$ the base change $X\times _YY'\to Y'$ of $f$ along $Y'$ is a closed morphism.
The morphism $f$ is called proper, if it is separated, of finite type, and universally closed.

Example 1.22

The affine line is not proper. More precisely, let $k$ be a field, let $Y=\operatorname{Spec}(k)$, and let $X = \mathbb {A}^1_k$. Let $f\colon X\to Y$ be the natural morphism. Then $f$ is separated, of finite type and closed, but (why?) not universally closed.
Every closed immersion is proper.
The property of being proper is stable under composition and under base change.

(1.7) Projective schemes are proper

Definition 1.23

Let $S$ be a scheme. An $S$-scheme $X$ is called projective (we also say that $X$ is projective over $S$, or that the morphism $X\to S$ is projective), if there exist $N\ge 0$ and a closed immersion $X\hookrightarrow \mathbb {P}^N_S$ of $S$-schemes.

This definition of projective schemes differs slightly from the one in [ GW1 ] (Definition 13.68, which requires only that the above property holds locally on $S$). If $S$ is affine, they coincide, however, and the difference will not be of any concern for us in this course. See [ GW1 ] , Summary 13.71 for a discussion. The definition given here is the one used in [ H ] and in [ Stacks ] .

Suppose that $S= \operatorname{Spec}R$ is an affine scheme. For any homogeneous ideal $I\subseteq R[X_0,\dots , X_N]$, $V_+(I)$ is a closed subscheme of $\mathbb {P}^N_S$, and hence in particular a projective $S$-scheme. One can show that for $S$ affine every projective scheme is isomorphic to a scheme of this form.

We will study this notion in more detail later (see Chapter 5).

Before we come to the main theorem of this section (Theorem 1.27), recall that for a homogeneous ideal $I\subseteq R[X_0,\dots , X_n]$ (where $R$ is some ring) we have defined a closed subscheme $V_+(I)$ of $\mathbb {P}^n_R$. We need the following two results on closed subschemes of projective space.

Lemma 1.24

Let $R$ be a ring, $n\ge 0$, and let $I\subseteq (X_0, \dots , X_n) \subseteq R[X_0,\dots , X_n]$ be a homogeneous ideal. Then $V_+(I) = \emptyset $ if and only if $\operatorname{rad}(I) \supseteq (X_0, \dots , X_n)$.

Proposition 1.25

( [ GW1 ] Proposition 13.24) Let $R$ be a ring, $n\ge 0$, and let $Z\subseteq \mathbb {P}^n_R$ be a closed subscheme. Then there exists a homogeneous ideal $I\subseteq R[X_0,\dots , X_n]$ such that $Z\cong V_+(I)$.

In addition, we will use the following commutative algebra lemma which is easily proved using the definitions of the localizations appearing in the lemma. (We will later generalize the lemma when we prove that given an ${\mathscr O}_X$-module ${\mathscr F}$ of finite type on a locally ringed space $X$, the support of ${\mathscr F}$, i.e., the set of all points $x$ such that the stalk ${\mathscr F}_x$ does not vanish, is closed. See Proposition 2.18.)

Lemma 1.26

Let $R$ be a ring, let $\mathfrak p\subset R$ be a prime ideal and let $M$ be a finitely generated $R$-module. If the localization $M_{\mathfrak p}$ vanishes, then there exists $s\in R\setminus \mathfrak p$ such that already the localization $M_s$ is zero.

Theorem 1.27

( [ GW1 ] Theorem 13.40) Every projective morphism of schemes is proper.

Sketch of proof

Since closed immersions are proper, it is enough to prove that projective space is proper, i.e., that for every scheme $S$ the morphism $\mathbb {P}^n_S\to S$ is closed. Since this property can be checked locally on $S$, we may assume that $S=\operatorname{Spec}R$ is affine.

If $Z\subseteq \mathbb {P}^n_S$ is a closed subset, there exists a closed subscheme with underlying topological space $Z$, and hence (Proposition 1.25) a homogeneous ideal $I\subseteq R[X_0,\dots , X_n]$ such that $V_+(I)$ has underlying topological space $Z$. We need to show that the image of $V_+(I)$ in $S$ is closed, or equivalently, that its complement $U\subseteq S$ is open.

Denote by $f$ the composition $V_+(I)\hookrightarrow \mathbb {P}^n_S\to S$, and let $x\in U$. Then the scheme-theoretic fiber $f^{-1}(x) = V_+(I) \times _U\operatorname{Spec}\kappa (x)$ is empty. We want to show that there exists $s\in R$ such that $x\in D(s)\subseteq U$. The inclusion $D(s)\subseteq U$ amounts to saying that $f^{-1}(D(s)) = \emptyset $. To translate the problem into a commutative algebra statement, let $\overline{I}$ be the image of $I$ in $\kappa (x)[X_0,\dots , X_n]$. It follows from the assumption $f^{-1}(x) = \emptyset $ and Lemma 1.24 that $\operatorname{rad}(\overline{I})$ contains the ideal $(X_0, \dots , X_n)$ ($\subset \kappa (x)[X_0, \dots , X_n]$). Thus for $d$ sufficiently large, for the degree $d$ components we have $\overline{I}_d = (X_0,\dots , X_n)_d$. By the lemma of Nakayama, we obtain $I_d\otimes {\mathscr O}_{S, x} = (X_0, \dots , X_n)_d \subseteq {\mathscr O}_{S,x}[X_0, \dots , X_n]$. It then follows that the analogous equality holds already over the localization of $R$ with respect to a suitable element $s$ not contained in the prime ideal $x$.

Example 1.28

(The resultant of polynomials) Let $k$ be an algebraically closed field (with some “obvious” adaptations, the results below hold over an arbitrary field). Let $m,n\in \mathbb {N}$. We identify the set $A$ of pairs $(f,g)$ of monic polynomials with the set of $k$-valued points of the affine space $\mathbb {A}^{m+n}_k = \operatorname{Spec}k[S_0,\dots , S_{m-1}, T_0,\dots , T_{n-1}]$, where a tuple $(s_i, t_j)\in k^{m+n} = \mathbb {A}^{m+n}_k(k)$ corresponds to $\left(X^m+s_{m-1}X^{m-1}+\cdots + s_0, X^n+t_{n-1}X^{n-1}+\cdots + t_0 \right)$.

Viewing $A$ as the set of closed points of $\mathbb {A}^{m+n}_k$, $A$ is equipped with a topology, namely the topology induced by the Zariski topology.

Let $Z\subset A$ be the subset consisting of those pairs $(f,g)$ such that $f$ and $g$ have a common zero in $k$. Write $R=k[S_0,\dots , S_{m-1}, T_0,\dots , T_{n-1}]$.

Claim. The set $Z$ is a closed subset.

Proof of claim. Let

\begin{align*} F =& X^m + S_{m-1}X^{m-1} + \cdots + S_1X + S_0,\\ G =& X^n + T_{n-1}X^{n-1} + \cdots + T_1X + T_0 \in R[X] \end{align*}

be the “universal” monic polynomials, and let

\begin{align*} \tilde{F} = & X^m + S_{m-1}X^{m-1}Y + \cdots + S_1XY^{m-1} + S_0Y^m,\\ \tilde{G} = & X^n + T_{n-1}X^{n-1}Y + \cdots + T_1XY^{n-1} + T_0Y^n \in R[X, Y] \end{align*}

be their homogenizations with respect to a second variable $Y$.

Let $p\colon \mathbb {P}^1_R\to \operatorname{Spec}R$ be the projection. Then $Z = p(V_+(\tilde{F}, \tilde{G}))\cap A$. By the above theorem, $p(V_+(\tilde{F}, \tilde{G}))$ is closed in $\mathbb {A}^{m+n}_k$, hence the claim follows. To see the equality, fix $x = (f,g)\in A$ and let $\tilde{f}, \tilde{g}$ be their homogenizations. Then $f$ and $g$ have a common zero in $k$ if and only if $V_+(\tilde{f}, \tilde{g}) \ne \emptyset $ (inside $\mathbb {P}^1_{\kappa (x)}$). Note that the point $(1:0)$, the “point at infinity” in $\mathbb {P}^1$ is never a zero of $\tilde{f}$ or $\tilde{g}$. Since $V_+(\tilde{f}, \tilde{g}) = V_+(\tilde{F}, \tilde{G})\times _{\operatorname{Spec}R}\operatorname{Spec}\kappa (x)$ can be identified with the (scheme-theoretic) fiber of $p$ over the point $x$, this proves the desired description of $Z$.

More precisely one can show (using other methods) that $Z$ is the zero locus of a single polynomial in $R$, the so-called resultant of a pair of monic polynomials. See [ GW1 ] Section (B.20) for a sketch and further references, or [ Bo ] Abschnitt 4.4 for a detailed account in German.

Algebraic Geometry 2, Summer term 2026.
Lecture course notes.

Content

1 Proper morphisms

The functorial point of view

Fiber products and base change

Proper morphisms