Download (Week 8: two classes) (5) A scheme is locally noetherian if there is

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2. SCHEMES
(Week 8: two classes)
(5) A scheme is locally noetherian if there is an affine cover by SpecAi where each Ai is
noetherian. A scheme is noetherian if there is a finite such cover.
Being a noetherian scheme is stronger than being a noetherian top space.
Example: Spec k[x1 , . . . ]/m2 is not a noetherian scheme, but it is of dimension 0.
Another example is R = Spec (k + yk[x, y]), it is not noetherian since there is an ascending
chain of ideals
(xy) ⊂ (xy, x2 y) ⊂ (xy, x2 y, x3 y) ⊂ · · ·
On the other hand, dim R = 2.
A fact: A scheme is locally noetherian iff every affine open subset has a noetherian coordinate
ring.
(6) A morphism f : X → Y is locally of finite type if there is an affine covering {SpecBi } of Y ,
such that f −1 (SpecBi ) is covered by open affine subsets {SpecAij }, and Aij is a finitely-generated
Bi -algebra. f is of finite type if f −1 (SpecBi ) can be covered by finite many {SpecAij }.
Example: Let k be an algebraically closed field. Regard a variety as a scheme, then it is of
finite type over k.
k.
Example: Speck[x](x) is not of finite type over k. SpecOP is not in general of finite type over
(7) A morphism f : X → Y is a finite morphism if there is an open covering {SpecBi } of Y , for
each i, f −1 SpecBi is isomorphic to SpecAi where Ai is a Bi -algebra which is a finitely generated
Bi -module.
In the definition of (locally) of finite type and finite, we can require the given properties for
every open covering of Y instead of some open covering.
A morphism is quasi-finite if for each point y ∈ Y , f −1 (y) is a finite set.
Example: A1 \ 0 → A1 is quasi-finite but not finite.
f : Speck[x1 , ...] → Speck is not locally of finite type. Actually, (quasi-finite)+(proper)=(finite).
(8) An open subscheme of a scheme X is a scheme U whose topological space is an open subset
of X and whose structure sheaf OU ∼
= OX |U . An open immersion is a morphism f : X → Y which
induces an isomorphism of X with an open subscheme of Y .
(9) A f : X → Y is a closed immersion if f (X) is a closed subset of the underlying space of Y ,
and f ] : OX → f∗ OY is surjective. A closed subscheme is an equivalent class of closed immersion
where f : Y → X and f 0 : Y 0 → X are equivalent if there is an isomorphism i : Y 0 → Y such that
f 0 = f ◦ i.
Example: Spec k → Spec k[x]/(x2 ) is a closed immersion, but Spec k[x]/(xn ) → Spec k is not.
On the other hand, Spec k[x]/(xn ) → Spec k[x] is a closed immersion, which is the infinitesimal
neighborhood of the origin in the line A1 defined by x = 0. In general, Spec A/I → Spec A is a
closed immersion, and every closed subscheme of Spec A arises in this way.
3. FIRST PROPERTIES OF SCHEMES
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Remark : Note that closed subscheme is more delicate than open subscheme in the sense that
it is not enough to specify the topological subset. It is convenient to specify a closed subscheme by
an ideal sheaf IY .
Homework: 3.1* (cf. EGA I, Prop 6.3.2 pp144), 3.11, 3.12, 3.13.
(10) Reduced induced closed subscheme structure, i.e. given a scheme X and a closed subset
Y , there is a ’smallest’ subscheme structure on Y (in the sense of closed immersion), which can be
constructed as follows.
√
First assume X = Spec A, for a subset Y defined by an ideal I, then I = ∩{p : p ⊇ I} defines
the reduced induced structure on Y .
For a general scheme X and a closed subset Y , take an affine cover Ui of X. Let Yi = Y ∩Ui and
give it the reduced induced structure. Those Yi glue together to give a reduced induced structure
on Y .
(11) Dimension. The dimension of a scheme is defined to be the dimension of the underlying
topological space.
Codimension of a irreducible closed subset Z of X: the supremum of integers n such that there
exists a chain
Z ⊂ Z1 ⊂ Z2 · · · ⊂ Zn
of distinct closed irreducible subsets of X.
For an arbitrary closed subset Y of X, define
codim(Y, X) := infZ⊆Y codim(Z, X)
where Z runs through all closed irreducible subsets of Y .
For affine integral scheme X of finite type over a field, any Y is a closed irreducible subset,
then
dimY + codim(Y, X) = dimX
Weird phenomenon (ex 3.21): Let R be a DVR containing its residue field(eg. k[[a]]). Let
X = Spec R[t] and Y be defined by at − 1. Then dimY = 0, codim(Y, X) = 1 and dimX = 2.
(12) Fiber product.
In category theory, given a category (eg the category of schemes) and two morphisms X →
S, Y → S in the category, the fiber product of X and Y over S is a space, denoted by X ×S Y
together with two projection maps X ×S Y → X and X ×S Y → Y for which the following diagram
commutes:
/Y
X ×S Y
²
²
/S
X
and the space X ×Y is universal in the sense that for any Z making the following diagram commute
Z
/Y
²
²
/S
X
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2. SCHEMES
there is a unique morphism Z → X ×S Y make the following diagram commute. (omit the diagram...)
Theorem 3.1. The fiber product exists for the category of schemes.
Proof. If existence is proven, uniqueness follows from the universal property of fiber products.
(1) For X = Spec A, Y = Spec B,Z = Spec C, then X ×Z Y = Spec (A ⊗C B). (Show
it first in the category of affine schemes, then the category of schemes. We need the fact that
Hom(R, O(S)) ∼
= Hom(S, Spec R))
(2) All we need to do is to patch the affine fiber products together. We cover S by affine charts
Wk = Spec Ck . Then cover the inverse of Wk in X (resp. Y ) by affine charts Uki (resp. Vkj ). Then
need to show Φkij := Uki ×Wk Vkj can be glued together to a scheme. For this, check that we can
naturally glue Φkij and Φk0 i0 j 0 along
(Uki ∩ Uk0 i0 ) ×S1 ∩S2 (Vkj ∩ Vk0 j 0 )
Then check the cocycle condition. c.f. EGA Ch1, pp.106-107.
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