## 26.18 Base change in algebraic geometry

One motivation for the introduction of the language of schemes is that it gives a very precise notion of what it means to define a variety over a particular field. For example a variety $X$ over $\mathbf{Q}$ is synonymous (Varieties, Definition 33.3.1) with $X \to \mathop{\mathrm{Spec}}(\mathbf{Q})$ which is of finite type, separated, irreducible and reduced1. In any case, the idea is more generally to work with schemes over a given base scheme, often denoted $S$. We use the language: “let $X$ be a scheme over $S$” to mean simply that $X$ comes equipped with a morphism $X \to S$. In diagrams we will try to picture the structure morphism $X \to S$ as a downward arrow from $X$ to $S$. We are often more interested in the properties of $X$ relative to $S$ rather than the internal geometry of $X$. For example, we would like to know things about the fibres of $X \to S$, what happens to $X$ after base change, and so on.

We introduce some of the language that is customarily used. Of course this language is just a special case of thinking about the category of objects over a given object in a category, see Categories, Example 4.2.13.

Definition 26.18.1. Let $S$ be a scheme.

1. We say $X$ is a scheme over $S$ to mean that $X$ comes equipped with a morphism of schemes $X \to S$. The morphism $X \to S$ is sometimes called the structure morphism.

2. If $R$ is a ring we say $X$ is a scheme over $R$ instead of $X$ is a scheme over $\mathop{\mathrm{Spec}}(R)$.

3. A morphism $f : X \to Y$ of schemes over $S$ is a morphism of schemes such that the composition $X \to Y \to S$ of $f$ with the structure morphism of $Y$ is equal to the structure morphism of $X$.

4. We denote $\mathop{\mathrm{Mor}}\nolimits _ S(X, Y)$ the set of all morphisms from $X$ to $Y$ over $S$.

5. Let $X$ be a scheme over $S$. Let $S' \to S$ be a morphism of schemes. The base change of $X$ is the scheme $X_{S'} = S' \times _ S X$ over $S'$.

6. Let $f : X \to Y$ be a morphism of schemes over $S$. Let $S' \to S$ be a morphism of schemes. The base change of $f$ is the induced morphism $f' : X_{S'} \to Y_{S'}$ (namely the morphism $\text{id}_{S'} \times _{\text{id}_ S} f$).

7. Let $R$ be a ring. Let $X$ be a scheme over $R$. Let $R \to R'$ be a ring map. The base change $X_{R'}$ is the scheme $\mathop{\mathrm{Spec}}(R') \times _{\mathop{\mathrm{Spec}}(R)} X$ over $R'$.

Here is a typical result.

Lemma 26.18.2. Let $S$ be a scheme. Let $f : X \to Y$ be an immersion (resp. closed immersion, resp. open immersion) of schemes over $S$. Then any base change of $f$ is an immersion (resp. closed immersion, resp. open immersion).

Proof. We can think of the base change of $f$ via the morphism $S' \to S$ as the top left vertical arrow in the following commutative diagram:

$\xymatrix{ X_{S'} \ar[r] \ar[d] & X \ar[d] \ar@/^4ex/[dd] \\ Y_{S'} \ar[r] \ar[d] & Y \ar[d] \\ S' \ar[r] & S }$

The diagram implies $X_{S'} \cong Y_{S'} \times _ Y X$, and the lemma follows from Lemma 26.17.6. $\square$

In fact this type of result is so typical that there is a piece of language to express it. Here it is.

Definition 26.18.3. Properties and base change.

1. Let $\mathcal{P}$ be a property of schemes over a base. We say that $\mathcal{P}$ is preserved under arbitrary base change, or simply that $\mathcal{P}$ is preserved under base change if whenever $X/S$ has $\mathcal{P}$, any base change $X_{S'}/S'$ has $\mathcal{P}$.

2. Let $\mathcal{P}$ be a property of morphisms of schemes over a base. We say that $\mathcal{P}$ is preserved under arbitrary base change, or simply that preserved under base change if whenever $f : X \to Y$ over $S$ has $\mathcal{P}$, any base change $f' : X_{S'} \to Y_{S'}$ over $S'$ has $\mathcal{P}$.

At this point we can say that “being a closed immersion” is preserved under arbitrary base change.

Definition 26.18.4. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$ be a point. The scheme theoretic fibre $X_ s$ of $f$ over $s$, or simply the fibre of $f$ over $s$, is the scheme fitting in the following fibre product diagram

$\xymatrix{ X_ s = \mathop{\mathrm{Spec}}(\kappa (s)) \times _ S X \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(\kappa (s)) \ar[r] & S }$

We think of the fibre $X_ s$ always as a scheme over $\kappa (s)$.

Lemma 26.18.5. Let $f : X \to S$ be a morphism of schemes. Consider the diagrams

$\xymatrix{ X_ s \ar[r] \ar[d] & X \ar[d] & \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \times _ S X \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(\kappa (s)) \ar[r] & S & \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \ar[r] & S }$

In both cases the top horizontal arrow is a homeomorphism onto its image.

Proof. Choose an open affine $U \subset S$ that contains $s$. The bottom horizontal morphisms factor through $U$, see Lemma 26.13.1 for example. Thus we may assume that $S$ is affine. If $X$ is also affine, then the result follows from Algebra, Remark 10.17.8. In the general case the result follows by covering $X$ by open affines. $\square$

[1] Of course algebraic geometers still quibble over whether one should require $X$ to be geometrically irreducible over $\mathbf{Q}$.

Comment #655 by Anfang Zhou on

Typo? There are two 'etc' at the end of the first paragraph.

Comment #665 by on

Sometimes people use 'etc, etc' to say 'and so on'. I have changed it here.

Is "Anfang Zhou" your full name? I'm asking because if so I can add it to the list of contributors. I won't add people who only leave a first or a last name to the list.

Comment #668 by Anfang Zhou on

Yeah, it's my full name. This is the first time I have seen double 'etc' in a sentence :) I am not a native English speaker. In fact, my English is very poor.

Comment #671 by on

No problem. I am from the Netherlands and English isn't my first language either. Added your name to the contributors list here.

Comment #2342 by on

In parts (1) and (2) of Definition 25.18.3, I think that instead of "or simply that preserved under base change", it should either read "or simply that $\mathcal{P}$ is preserved under base change" or "or simply that it is preserved under base change".

Also, in Definition 25.18.4, maybe there should be a comma right after the sentence "or simply the fibre of f over". s

Comment #3039 by Samir Canning on

I think the future reference you're looking for in the first paragraph is Tag 020D

Comment #5489 by Kyle Broder on

In Definition 01k0 concerning the scheme-theoretic fiber, $\kappa(s)$ is not defined or referenced anywhere. I assume it is the residue field of $s$, but this should be pointed out to the reader, in my opinion.

Comment #5490 by on

@Kyle: The residue field of $s$ is defined and the notation $\kappa(s)$ is introduced for points of locally ringed spaces in Definition 26.2.1. Moreover, the correspondence between points and morphisms from the spectrum of the residue field is discussed in Lemma 26.13.3. OK?

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