Algebraic Geometry 1, Winter term 2025/26.
Lecture notes.

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6 Fiber products of schemes

References: [ GW1 ] Sections (4.4) – (4.8)

In this section we will discuss the notion of (fiber) product of schemes, which turns out to be useful in many respects. One motivation is this: It is a natural question whether we can define products in the setting of schemes, or \(k\)-schemes, say, where \(k\) is a field. For example, the bijection

\begin{align*} \mathbb {P}^1(k)\times \mathbb {P}^1(k) & \to V_+(X_0 X_3-X_1X_2) \quad \subset \mathbb {P}^3(k),\\ ((x:y), (x’:y’)) & \mapsto (xx’ : xy’ : yx’ : yy’), \end{align*}

leads us to expect that there is an isomorphism \(\mathbb {P}^1_k\times \mathbb {P}^1_k\to V_+(X0 X_3-X_1X_2)\) of \(k\)-schemes — only that the symbol \(\mathbb {P}^1_k\times \mathbb {P}^1_k\) is undefined at this point! Note that one has to be a bit careful here: It is not hard to check that the above map is bijective, but it is not a homeomorphism if we equip \(\mathbb {P}^1(k)\times \mathbb {P}^1(k)\) with the product topology; rather, the suitable topology is a different one (cf. also Problem 4).

(6.1) Fiber products in a category

We take a categorical approach and define the product (and the more general notion of fiber product) by a universal category as follows.

Definition 6.1

Let \({\mathcal C}\) be a category, and let \(f\colon X\to S\), \(g\colon Y\to S\) be morphisms in \({\mathcal C}\). An object \(P\) of \({\mathcal C}\) together with morphisms \(p\colon P\to X\) and \(q\colon P\to Y\) with \(f\circ p = g\circ q\) is called a fiber product of \(X\) and \(Y\) over \(S\), if the following universal property holds:

For every scheme \(T\) together with morphisms \(\varphi \colon T\to X\) and \(\psi \colon T\to Y\) with \(f\circ \varphi = g\circ \psi \) there exists a unique morphism \(\xi \colon T\to P\) such that the following diagram commutes:

\begin{tikzcd} T \ar[drr, bend left, "\varphi"]\ar[ddr, bend right, "\psi"]\ar[dr, dashed, "\xi"]\\ & P \ar[r, "p"]\ar[d, "q"] & X\ar[d, "f"]\\ & Y \ar[r, "g"] & S \end{tikzcd}

In view of the universal property, a fiber product (i.e., the object \(P\) together with the morphisms \(P\to X\), \(P\to Y\)) is uniquely determined up to unique isomorphism, if it exists. We therefore speak of “the” fiber product of \(X\) and \(Y\) over \(S\) and denote it by \(X\times _SY\). The morphisms \(X\times _SY\to X\), \(X\times _SY\to Y\) are called the projections from the fiber product to its factors.

Terminology: Saying that the (commutative) diagram

\begin{tikzcd} P \ar[r]\ar[d] & X\ar[d]\\ Y \ar[r] & S \end{tikzcd}

is a fiber product diagram or is cartesian is equivalent to saying that \(P\) together with the given morphisms to \(X\) and \(Y\) is a fiber product of \(X\) and \(Y\) over \(S\). One also speaks of a pullback diagram, or calls \(P\) the pullback of \(X\) along \(Y\to S\) (of of \(Y\) along \(X\to S\)).

Remarks and Examples 6.2

  1. Let \({\mathcal C}\) be the category of sets and let \(f\colon X\to S\), \(g\colon Y\to S\) be maps of sets. Then the fiber product \(X\times _SY\) exists and is given by (as always, up to unique isomorphism)

    \[ X\times _SY = \left\{ (x,y)\in X\times Y; f(x)=g(y) \right\} , \]

    where the maps \(X\times _SY\to X\) and \(X\times _SY\to Y\) are the restrictions of the usual projections from the cartesian product \(X\times Y\) to its factors.

    As a subset of \(X\times Y\), we can rewrite this as

    \[ X\times _S Y = \bigsqcup _{s\in S} f^{-1}(s)\times g^{-1}(s). \]

    This explains the terminology fiber product.

  2. Let \({\mathcal C}\) be the category of topological spaces and let \(f\colon X\to S\), \(g\colon Y\to S\) be continuous maps. Then the fiber product \(X\times _SY\) exists and is given by (as always, up to unique isomorphism)

    \[ X\times _SY = \left\{ (x,y)\in X\times Y; f(x)=g(y) \right\} , \]

    where the maps \(X\times _SY\to X\) and \(X\times _SY\to Y\) are the restrictions of the usual projections from the cartesian product \(X\times Y\) to its factors. The topology is defined as the coarsest topology for which both projections are continuous. Equivalently, if we equip the product \(X\times Y\) with the product topology, then \(X\times _SY\) carries the subspace topology. More explicitly, the open subsets of \(X\times _SY\) are those that can be written as unions of sets of the form \(U\times _SV\), where \(U \subseteq X\) and \(V \subseteq Y\) are open.

  3. If \({\rm pt}\) is a terminal object in \({\mathcal C}\) (i.e., every object of \({\mathcal C}\) admits a unique morphism to \({\rm pt}\)), then the universal property of \(X\times _{\rm pt} Y\) (with respect to the unique maps \(X\to {\rm pt}\), \(Y\to {\rm pt}\)) is precisely the universal property of the product of \(X\) and \(Y\).

    This applies, for example, the the category of sets (with \({\rm pt}\) a set with one element), to the category of schemes (with terminal object \(\operatorname{Spec}(\mathbb {Z})\)) and to the category of \(S\)-schemes (with terminal object \(S\), or more precisely \(\operatorname{id}_S\colon S\to S\)).

  4. If \({\mathcal C}\) is the category of sets or the category of topological spaces, \(f\colon X\to S\) is a morphism and \(s\in S\), then the inclusion \(\left\{ s \right\} \to S\) is a morphism and there is a natural identification of \(X\times _S \left\{ s \right\} \) with the fiber \(f^{-1}(s)\) (in the case of topological spaces, we equip the fiber \(f^{-1}(s) \subseteq X\) with the subspace topology.

Dually to the notion of fiber product we have the notion of algamated sum or pushout. The corresponding commutative diagrams are called cocartesian. An example is the tensor product of rings.

(6.2) Fiber products of schemes

Jan. 21, 2026

Theorem 6.3

All fiber products in the category of schemes exist.

Moreover, for scheme morphisms \(X\to S\), \(Y\to S\), the fiber product \(X\times _SY\) also satisfies the universal property of the fiber product in the category of locally ringed spaces.

With more work, one can show that also in the category of locally ringed spaces all fiber products exist; we will not need to use this.

Proof

We first consider the case that \(X=\operatorname{Spec}(A)\), \(Y=\operatorname{Spec}(B)\) and \(S=\operatorname{Spec}(R)\) are all affine. In this case, we claim that the (affine) scheme \(\operatorname{Spec}(A\otimes _R B)\) satisfies the universal property of the fiber product in the category of locally ringed spaces. (Note that it follows formally from the universal property of the tensor product (it is the pushout of \(A\) and \(B\) with respect to the homomorphisms \(R\to A\) and \(R\to B\)) and the antiequivalence of the categories of rings and of affine schemes, that \(\operatorname{Spec}(A\otimes _R B)\) is the fiber product in the category of affine schemes.) This claim follows easily from Theorem 4.11, in fact for every locally ringed space \(T\) we have

\begin{align*} & \operatorname{Hom}_{\rm locRS}(T, \operatorname{Spec}(A\otimes _RB))\\ & = \operatorname{Hom}_{\rm Ring}(A\otimes _RB, \Gamma (T, {\mathscr O}_T))\\ & = \operatorname{Hom}_{\rm Ring}(A, \Gamma (T, {\mathscr O}_T))\times _{\operatorname{Hom}_{\rm Ring}(R, \Gamma (T, {\mathscr O}_T)} \operatorname{Hom}_{\rm Ring}(B, \Gamma (T, {\mathscr O}_T)) \\ & = \operatorname{Hom}_{\rm locRS}(T, X)\times _{\operatorname{Hom}_{\rm locRS}(T, S)}\operatorname{Hom}_{\rm locRS}(T, Y). \end{align*}

The composition is the map \(\xi \mapsto (p\circ \xi , q\circ \xi )\), and the statement that this map is a bijection is precisely the universal property of the fiber product.

In the general case, the fiber product may be constructed by gluing of schemes. We give a rough sketch and refer to  [ GW1 ] Theorem 4.18 and the references given there for more details.

Choose affine open covers

\[ S = \bigcup _i W_i,\quad f^{-1}(W_i) = \bigcup _j U_{ij},\quad g^{-1}(W_i) = \bigcup _k V_{ik}. \]

Then one constructs a gluing datum for the family

\[ U_{ij} \times _{W_i} V_{ik}. \]

For a pair of indices \((i,j,k)\), \((i', j', k')\), one shows that, denoting by \(p_1\), \(p_2\) the projections from \(U_{ij} \times _{W_i} V_{ik}\) to its two factors,

\[ p_1^{-1}(U_{ij}\cap U_{i'j'}) \cap p_2^{-1}(V_{ik}\cap V_{i'k'}) \]

is an open subscheme of \(U_{ij} \times _{W_i} V_{ik}\) which satisfies the universal property of the fiber product

\[ (U_{ij}\cap U_{i'j'})\times _{W_i\cap W_{i'}} (V_{ik}\cap V_{i'k'}). \]

This open subschemes will be, in the scheme obtained by gluing, the intersection of (the images of) \(U_{ij}\times _{W_i}V_{ik}\) and \(U_{i'j'}\times _{W_{i'}}V_{i'k'}\). The identification morphisms are chosen as the unique isomorphisms obtained from the characterization by the universal property of a fiber product. This also ensures that the cocycle condition is satisfied.

Notation: For schemes \(X\), \(Y\), \(S\), and rings \(R\), \(A\), we sometimes write \(X\otimes _R A := X\times _{\operatorname{Spec}(R)}\operatorname{Spec}(A)\), \(X\otimes _S A := X\times _S\operatorname{Spec}(A)\) and \(X\times _R Y := X\times _{\operatorname{Spec}(R)} Y\) (provided that we are given the scheme morphisms required to define these fiber products).

Example 6.4
As the following example shows, the topological space of a fiber product of schemes is different, in general, from the fiber product (in the category of topological spaces) of the underlying spaces. Since
\[ \mathbb {C}\otimes _{\mathbb {R}}\mathbb {C}= \mathbb {R}[X]/(X^2+1)\otimes _{\mathbb {R}}\mathbb {C}\cong \mathbb {C}[X]/(X^2+1) = \mathbb {C}[X]/(X-i)(X+i) \cong \mathbb {C}\times \mathbb {C}, \]
where the final isomorphism uses the Chinese Remainder Theorem and the natural isomorphisms \(\mathbb {C}[X]/(X-a)\cong \mathbb {C}\), \(a\in \mathbb {C}\), we have that
\[ \operatorname{Spec}(\mathbb {C})\times _{\operatorname{Spec}(\mathbb {R})}\operatorname{Spec}(\mathbb {C}) \cong \operatorname{Spec}(\mathbb {C}\times \mathbb {C}) = \operatorname{Spec}(\mathbb {C})\sqcup \operatorname{Spec}(\mathbb {C}) \]
consists of two points (both are closed and have residue class field \(\mathbb {C}\); this scheme is the disjoint union of two copies of \(\operatorname{Spec}(\mathbb {C})\)).

Remarks 6.5

  1. For any ring homomorphism \(R\to R'\), we have \(\mathbb {A}^n_R\otimes _RR' = \mathbb {A}^n_{R'}\). In particular, for any ring \(R\), we have \(\mathbb {A}^n_R = \mathbb {A}_{\mathbb {Z}}\otimes _{\mathbb {Z}}R\). In view of this, we define, for an arbitrary scheme \(S\), \(\mathbb {A}^n_S = \mathbb {A}^n_{\mathbb {Z}}\times _{\operatorname{Spec}(\mathbb {Z})} S\). Via the second projection, this is an \(S\)-scheme, called the affine space of relative dimension \(n\) over \(S\).

(6.3) Base change

Let \(S'\to S\) be a morphism of schemes. We obtain a functor

\[ \text{(Sch/$S$)}\longrightarrow \text{(Sch/$S'$)},\quad X\to X\times _SS', \]

where \(X\times _SS'\) is an \(S'\)-scheme via the second projection.

On morphisms, the functor is defined as follows. Given a morphism \(X\to Y\) of \(S\)-schemes, we need to define a morphism \(X\times _SS'\to Y\times _SS'\). To do so, by the universal property of \(Y\times _SS'\), it is enough to specify morphisms \(X\times _SS'\to Y\) and \(X\times _SS'\to S'\) (compatible with the morphisms to \(S\)). For the first one, we take the composition \(X\times _SS'\to X\to Y\), for the second one the projection to the second factor.

(6.4) Fibers of morphisms

In view of Example 6.2 (4), we define fibers of morphisms of schemes as follows:

Definition 6.6
Let \(f\colon X\to S\) be a morphism of schemes, and let \(s\in S\) (i.e., \(s\) is a point of the underlying topological space of \(S\)). Denote by \(i_s\colon \operatorname{Spec}(\kappa (s))\to S\) the canonical map (Section 4.4). Then the scheme
\[ f^{-1}(s) := X\times _S \operatorname{Spec}(\kappa (s)) \]
is called the (schematic) fiber of \(f\) over \(s\).

Jan. 27, 2026

In the case of the schematic fiber, the fiber product behaves well regarding the underlying topological space.

Proposition 6.7
Let \(f\colon X\to S\) be a morphism of schemes, \(s\in S\), and let \(f^{-1}(s) = X\otimes _S \kappa (s)\) be the schematic fiber. Then the projection
\[ f^{-1}(s)\to X \]
on topological spaces is a homeomorphism onto the fiber of the underlying continuous map \(f\colon X\to S\).

Proof

It is clear that we may assume that \(S\) is affine. Furthermore, covering \(X\) by affine open subschemes, it is easy to reduce to the case that \(X\) is affine. So assume \(S = \operatorname{Spec}(R)\), \(X=\operatorname{Spec}(A)\), and \(f\) corresponds to the ring homomorphism \(\varphi \colon R\to A\). Let \({\mathfrak p}\subset R\) be the prime ideal corresponding to \(s\), and denote by \(\kappa ({\mathfrak p})\) the residue class field of \({\mathfrak p}\). We may describe \(\kappa ({\mathfrak p})\) equivalently as \(R_{{\mathfrak p}}/{\mathfrak p}R_{\mathfrak p}\) or as \(\operatorname{Frac}(R/{\mathfrak p})\).

Then the definition of the schematic fiber amounts to

\[ f^{-1}(s) = \operatorname{Spec}(A\otimes _R \kappa ({\mathfrak p})). \]

We may compute the tensor product as

\[ A\otimes _R \kappa ({\mathfrak p}) = A\otimes _R S^{-1} R \otimes _{S^{-1}R} S^{-1}R/{\mathfrak p}\, S^{-1} R = S^{-1} A / {\mathfrak p}\, S^{-1}A, \]

where \(S = R\setminus {\mathfrak p}\). (In the term on the right, by abuse of notation we write \(S\) instead of \(\varphi (S)\) and \({\mathfrak p}\) instead of \(\varphi ({\mathfrak p})\).) This gives us a bijection between \(\operatorname{Spec}(A\otimes _R\kappa ({\mathfrak p}))\) and the set of prime ideals \({\mathfrak q}\) in \(A\) which satisfy

\[ \varphi ({\mathfrak p}) \subseteq {\mathfrak q},\quad \varphi (S)\cap {\mathfrak q}= \emptyset , \]

or equivalently,

\[ \varphi ^{-1}({\mathfrak q}) = {\mathfrak p}. \]

This is the desired statement on sets. To finish the proof, one checks that the resulting bijective map is a homeomorphism (similarly as in Proposition 2.8, Proposition 2.13).

Remark 6.8
Let \(S\) be a scheme. Let \(X\) be an \(S\)-scheme (i.e., we are given a scheme morphism \(f\colon X\to S\). With the construction of the schematic fiber, the morphism \(f\) gives rise to a family \((X_s)_{s\in S}\), where each \(X_s := X\otimes _S\kappa (s)\) is a scheme over a field (namely over the residue class field \(\kappa (s)\)), and hence an object that is closer to classical algebraic geometry. Properties of the morphism \(f\) may often be viewed as properties of this family.