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.
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.
If R is a ring we say X is a scheme over R instead of X is a scheme over \mathop{\mathrm{Spec}}(R).
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.
We denote \mathop{\mathrm{Mor}}\nolimits _ S(X, Y) the set of all morphisms from X to Y over S.
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'.
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).
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.
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}.
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
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