Download Math 216A Homework 8 “...the usual definition of a scheme is not

Math 216A Homework 8
“. . . the usual definition of a scheme is not nicely suited to our proof.”
Nagata
Read the handout on Chow’s Lemma (which goes substantially beyond Exercise 4.10 in
Hartshorne’s Chapter 2 insofar as it relaxes properness to “separated of finite type”; this is
a very important improvement for later purposes), written by Matt Emerton when he was
the CA for a year-long schemes course during his last year of graduate school.
Do the following exercises.
Ch 2: 4.2* (give counterexample if X is not reduced), 4.4* (omit the noetherian hypothesis!),
4.6, 4.8*, 4.9*, 4.11*. For 4.11(a), see Theorem 11.7 in Matsumura (and its corollary) for
a proof of the Krull-Akizuki Theorem. For 4.11(b), only assume f is locally of finite type
for separatedness and explain why if S is locally noetherian and f : X → S is finite type
and separated, with X ×S AnS → AnS a closed map for all n ≥ 1, then f is proper (with
better results for finitely presented maps, the same theorem holds for f finitely presented
and separated, with arbitrary S).
Extra −1: Let f : X → S be a morphism of schemes, S 0 → S an fpqc map, f 0 : X 0 → S 0
the base changed map. Prove that f is proper if and only if f 0 is proper.
Extra 0: Let X1 , X2 be separated S-schemes, Ui open in Xi , and j : U1 ' U2 an Sisomorphism, so the gluing X of X1 and X2 along j makes sense as an S-scheme. Explain
carefully why X is separated over S if and only if the graph morphism Γj : U1 → X1 ×S X2
is a closed immersion.
Extra 1: Show the following variant on Exercise 4.1: all finite maps between affine schemes
are projective.
Extra 2: Let f : X → Y be a map between finite type k-schemes, for a field k. Assume f is
proper, Y is irreducible of dimension d, and the fibers are all irreducible of dimension d0 ≥ 0
(in particular, all fibers are non-empty). Then show that X is irreducible of dimension d+d0 .
If the properness hypothesis on f is dropped, show that X need not be irreducible.
Extra 3: Let k be a field, ks a separable closure, G = Gal(ks /k). We define a G-set to
be a finite set X together with a continious (left) action of G on X (i.e., we are given a
group homomorphism G → Aut(X) with open kernel). A finite étale k-algebra A is a finite
k-algebra A which is geometrically reduced over k (or equivalently, A ⊗k ks is ks -isomorphic
to a product of finitely many copies of ks , or equivalently A is a product of finitely many
finite separable extension fields of k).
(i) Given a G-set X, there is a natural G-action on the product ring Πx∈X ks ex , given by
g(aex ) = g(a)eg(x) . Define A(X) to be the invariant subring. Show that the natural map
ks ⊗k A(X) → Πx∈X ks ex is an isomorphism, so A(X) is a finite étale k-algebra. When the
characteristic of k is relatively prime to N and X is the set of N th roots of unity in ks ,
compute A(X). Do the same if X is a finite set with trivial G-action.
(ii) Given a finite étale k-algebra A, define the set of ring homomorphisms
X(A) = Homk (A, ks ) = Homks (ks ⊗k A, ks ).
Explain how this is naturally a G-set. When A = k[T ]/(T N − 1) and N is relatively prime
to the characteristic of k, describe the G-set X(A).
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(iii) Explain how A and X give an equivalence of categories between G-sets and finite étale
k-algebras, compatible with formation of products of G-sets and tensor products over k of
k-algebras. In particular, show that the theory of finite Galois representations over k (i.e.,
G-sets in which X is a commutative group compatibly with the G-action) is the ‘same’ as
the theory of finite étale commutative group schemes over k. In particular, check that the
k group scheme µN corresponds to the group of N th roots of unity in ks . Note how the
category of finite étale k-algebras does not explicitly mention ks , but the functors do use ks .
Explain what happens if we choose a different separable closure of k.
Optional Extra 4: The rest of this assignment is devoted to a very long optional exercise
developing an important synthesis of schemes with analytic geometry, providing a good
notion of “complex manifold with singularities”. There will be many future exercises on this
theme, so please think about the following if you have time.
An analytic space (over C) is a ringed space of C-algebras (X, OX ) which is locally isomorphic to one of the form (Y, i−1 (OU /(f1 , . . . , fr ))), where U is open in Cn , fj ∈ OCn (U )
are holomorphic, Y is the simultaneous zero set of the fj ’s in U , and i : Y → U is the
closed embedding map (so OU /(fj ) is supported exactly on Y ). Note that analytic spaces
are automatically locally ringed spaces with all residue fields canonically isomorphic to C,
so all maps between them are automatically locally ringed space maps.
(i) Explain why the category of complex manifolds is a full subcategory of the category of
analytic spaces (i.e., we get the same maps). We let Cn denote the usual analytic space we
know and love.
(ii) If f : X → Cn is a morphism of analytic spaces, then the global sections zj on Cn pull
back to ξj ∈ OX (X). Granting that the stalks OX,x are noetherian (for proof, read pp. 38-44
of the beautiful book Coherent Analytic Sheaves by Grauert and Remmert), prove that this
sets up a natural bijection between the sets Hom(X, Cn ) and OX (X)n , natural in X. Watch
out with the sheaves. It may be handy to look at completions of the stalks (here’s where
noetherianness is needed, and this is where things collapse with smooth functions).
Use this to show that the natural maps Cn+m → Cn and Cn+m → Cm exhibit Cn+m
as the product of Cn and Cm in the category of analytic spaces (not just the category of
complex manifolds).
(iii) Prove the existence of products in the category of analytic spaces. Then prove the
existence of fiber products in the category of analytic spaces. Construct X ×Z Y from X × Y
and ∆Z : Z → Z × Z.
Also, if f : X → Z and g : Y → Z are maps of analytic spaces and P = X ×Z Y is a fiber
product, with natural maps P → X, P → Y , show that the evident map from |P | to the
topological fiber product space
|X| ×|Z| |Y | = {(x, y) ∈ |X| × |Y | | f (x) = g(y)}
is a homeomorphism, where the above space is given the induced topology from the product
topology on |X| × |Y | (hint: begin with the special case in (ii), which is already where
the topology of product spaces goes batty in the algebraic setting; build up from this using
open/closed immersions and mapping properties).
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By being careful, do you see a way to make a completely unambiguous definition of the
fiber product, which on topological spaces is what we expect (you are to realize a ‘choice-free’
model for the fiber product, so it is not just something defined up to canonical isomorphism)?
Taking Y to be a point on Z with structure sheaf C, you can now begin to deal intelligently
with the fibers of a map f : X → Z, even when these fibers look topologically nasty; observe
that for z ∈ Z and x ∈ f −1 (z), the natural map f −1 (z) → X induces OX,x /mz OX,x ' Of −1 (z),x
(do you see why?).
(iv) You are now in a position to prove an analytic analogue of HW1, Exercise 1. For an
analytic space X and a point x ∈ X, we define a germ at x to be a pair (U, x) with U ⊆ X
an open analytic subspace containing x. For another analytic space Y and y ∈ Y , we define
a morphism of germs (U, x) → (V, y) to be a morphism U → V taking x to y. We say two
morphisms (U, x) → (V, y), (U 0 , x) → (V 0 , y) are equivalent if they coincide on some open
neighborhood of x in U ∩ U 0 . Show that any such equivalence class induces a well-defined Calgebra map OY,y → OX,x , and moreover that every C-algebra map OY,y → OX,x (necessarily
local!) arises in this way from a unique equivalence class of germ morphisms. In particular,
if f : OX,x ' OY,y as C-algebras, conclude there exists an isomorphism (U, x) ' (V, y) of
germs (inducing f ).
In other words, the local rings in analytic geometry encode ‘local’ knowledge of the space
around x in X. Here is one nice application (read for pleasure): if the local noetherian ring
OX,x is a regular ring, then X is a manifold near x (the converse is obvious . . . once we know
the miracle that OCn ,0 is noetherian). To prove this, say dim OX,x = n. Picking a C-basis of
mx /m2x and lifting to OX,x , (iv) gives rise to a map f : U → Cn on a neighborhood U of X,
taking x to 0. We expect this to be an isomorphism near x and 0. Well, f induces a local
map f ∗ : OCn ,0 → OX,x which induces a surjection (even isomorphism) on Zariski cotangent
spaces at the maximal ideals. On completions this is abstractly a map from C[[z1 , . . . , zn ]] to
itself which is surjective (why?) and therefore an isomorphism (Matsumura, Exercise 3.6).
If we could deduce that f ∗ had to be an isomorphism, then by (iv) again (!) we’d know f is
an isomorphism near x.
How to show f ∗ is an isomorphism? Note that the m0 -adic completion of OX,x is the same
as the mx -adic one, since mx = m0 OX,x . If we knew that f ∗ were a finite ring map, which is
to say that OX,x is a finite OCn ,0 -module, then the (isomorphism) map on completed rings is
b Cn ,0 ⊗O n to f ∗ . Therefore, f ∗ would
the same as applying the faithfully flat base change O
C ,0
have to have been an isomorphism to begin with, as desired. So how to show f ∗ is a finite
map? Consider the analytic fiber space f −1 (0) as constructed in (iii). On this space, the
local ring at x is isomorphic to OX,x /m0 OX,x = C, so Of −1 (0),x ' C. Since a one point
space with structure sheaf C is an analytic space, applying (iv) on f −1 (0) implies that x is
an isolated point (i.e., open point) in f −1 (0) = f −1 (f (x)) (why?). Keep in mind that the
analytic space f −1 (0) is topologically what we expect!
The fundamental Proposition on p. 63 in the Grauert-Remmert book asserts that if f :
X → Y is a map of analytic spaces with x isolated in f −1 (f (x)), then OY,f (x) → OX,x is a
finite ring map. This important property of analytic rings has the geometric intepretation
of saying that there should be some neighborhoods U of x and V of f (x) with f (U ) ⊆ V
and f : U → V a ‘finite’ map, in an appropriate sense (this can be made precise); from
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this point of view, analytic local rings are more analogous to complete local noetherian rings
rather than general local noetherian rings (consider the integer ring A of a number field K,
and a non-zero prime p ∈ Spec(A) over (p) ∈ Spec(Z); then Spec(A) → Spec(Z) is a finite
map and the local ring map Z(p) → Ap is generally not finite, but the map on completions
bp is finite).
Zp → A