math 216: foundations of algebraic geometry - Stanford University

200 Math 216: Foundations of Algebraic Geometry
is in the family, and (iii) the complement of a family member is also in the family.
For example the image of (x, y) ↦→ (x, xy) is constructible. (An extension of
the notion of constructibility to more general topological spaces is mentioned in
Exercise 10.3.H.)
8.4.A. EXERCISE: CONSTRUCTIBLE SUBSETS ARE FINITE UNIONS OF LOCALLY CLOSED
SUBSETS. Recall that a subset of a topological space X is locally closed if it is the
intersection of an open subset and a closed subset. (Equivalently, it is an open
subset of a closed subset, or a closed subset of an open subset. We will later have
trouble extending this to open and closed and locally closed subschemes, see Exercise
9.1.M.) Show that a subset of a Noetherian topological space X is constructible
if and only if it is the finite disjoint union of locally closed subsets. As a consequence,
if X → Y is a continuous map of Noetherian topological spaces, then the
preimage of a constructible set is a constructible set.
8.4.B. EXERCISE (USED IN EXERCISE 25.5.F).
(a) Show that a constructible subset of a Noetherian scheme is closed if and only if
it is “stable under specialization”. More precisely, if Z is a constructible subset of a
Noetherian scheme X, then Z is closed if and only if for every pair of points y1 and
y2 with y1 ∈ y2, if y2 ∈ Z, then y1 ∈ Z. Hint for the “if” implication: show that Z
can be written as n
i=1 Ui ∩ Zi where Ui ⊂ X is open and Zi ⊂ X is closed. Show
that Z can be written as n
i=1 Ui ∩ Zi (with possibly different n, Ui, Zi) where
each Zi is irreducible and meets Ui. Now use “stability under specialization” and
the generic point of Zi to show that Zi ⊂ Z for all i, so Z = ∪Zi.
(b) Show that a constructible subset of a Noetherian scheme is open if and only if
it is “stable under generization”. (Hint: this follows in one line from (a).)
The image of a morphism of schemes can be stranger than a constructible set.
Indeed if S is any subset of a scheme Y, it can be the image of a morphism: let X
be the disjoint union of spectra of the residue fields of all the points of S, and let
f : X → Y be the natural map. This is quite pathological, but in any reasonable
situation, the image is essentially no worse than arose in the previous example of
(x, y) ↦→ (x, xy). This is made precise by Chevalley’s theorem.
8.4.2. Chevalley’s Theorem. — If π : X → Y is a finite type morphism of Noetherian
schemes, the image of any constructible set is constructible. In particular, the image of π
is constructible.
(For the minority who might care: see §10.3.6 for an extension to locally finitely
presented morphisms.) We discuss the proof after giving some important consequences
that may seem surprising, in that they are algebraic corollaries of a seemingly
quite geometric and topological theorem.
8.4.3. Proof of the Nullstellensatz 4.2.3. The first is a proof of the Nullstellensatz. We
wish to show that if K is a field extension of k that is finitely generated as a ring,
say by x1, . . . , xn, then it is a finite field extension. It suffices to show that each xi
is algebraic over k. But if xi is not algebraic over k, then we have an inclusion of
rings k[x] → K, corresponding to a dominant morphism Spec K → A1 k of finite type
k-schemes. Of course Spec K is a single point, so the image of π is one point. But