The Stacks project

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 10.18.5. 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 $\mathbf{Q}$.

Comments (12)

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 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, is not defined or referenced anywhere. I assume it is the residue field of , but this should be pointed out to the reader, in my opinion.

Comment #5490 by on

@Kyle: The residue field of is defined and the notation 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?

Comment #8464 by on

I think the following could be an interesting remark to add after 26.18.3: Let be a property of morphisms of schemes. Then is preserved under arbitrary base change as in 26.18.3 (1) if and only if it is preserved under arbitrary base change as in 26.18.3 (2). To see (), use the diagram of the proof of 26.18.2, where the top square is cartesian. To see (), use again the same diagram but with , so .

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