ÉTALE COHOMOLOGY
NOTES TAKEN BY PAK-HIN LEE AND REVISED BY THE SPEAKERS
Abstract. These are notes from the (ongoing) Étale Cohomology Reading Seminar at
Columbia University in Fall 2015, which is organized by Remy van Dobben de Bruyn.
Contents
1. Lecture 1 (September 8, 2015): Remy van Dobben de Bruyn
Introduction: the Weil conjectures
2. Lecture 2 (September 15, 2015): Remy van Dobben de Bruyn
Étale morphisms; fpqc descent (affine case); Henselisations
2.1. Unramified morphisms
2.2. Étale morphisms
2.3. fpqc descent
2.4. Descent data
3. Lecture 3 (September 22, 2015): Remy van Dobben de Bruyn
Grothendieck topologies and sites
3.1. Grothendieck pretopologies
3.2. Čech cohomology
3.3. Sheafification
4. Lecture 4 (September 29, 2015): Remy van Dobben de Bruyn
Sheaves on the étale site
4.1. Change of site
4.2. Cohomology
4.3. Sheaves on the étale site
4.4. The étale site of Spec(K)
5. Lecture 5 (October 6, 2015): Sam Mundy
Cohomology on curves
6. Lecture 6 (October 13, 2015): Shizhang Li
Proper base change
6.0. Observation for finite morphisms
6.1. Reducing the scheme
6.2. Reducing the sheaf
6.3. Computing some cohomology
7. Lecture 7 (October 20, 2015): Ashwin Deopurkar
Local acyclicity of smooth morphisms; smooth base change
7.1. Prelude in algebraic topology
7.2. Statement of theorem and structure of the proof
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Last updated: October 22, 2015. Please send corrections and comments to [email protected].
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7.3. “Proof”
7.4. Application
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1. Lecture 1 (September 8, 2015): Remy van Dobben de Bruyn
Introduction: the Weil conjectures
PH: I missed the lecture.
2. Lecture 2 (September 15, 2015): Remy van Dobben de Bruyn
Étale morphisms; fpqc descent (affine case); Henselisations
Today’s plan is to talk about a few issues of commutative algebra and algebraic geometry
that we will be using throughout the semester. We will cover three topics today:
(1) étale morphisms,
(2) fpqc descent,
(3) Henselian rings.
The third topic will be treated minimally, and some good references are:
• Raynaud, Anneaux Locaux Henséliens (ALH).
• Stacks Project.
I will start with the notion of unramified morphisms.
2.1. Unramified morphisms.
Definition 2.1. Consider diagrams
/
A
f
/
B
where I 2
• f
• f
• f
=C
C/I
= 0. We say that
is formally unramified if there exists at most one lift,
is formally smooth if there exists at least one lift,
is formally étale if there exists exactly one lift.
This definition is not very practical as we cannot check this condition for all rings C.
Lemma 2.2. f : A → B is formally unramified if and only if ΩB/A = 0.
Proof. Let Cuniv = (B ⊗A B)/J 2 , where J = ker(B ⊗A B → B). Let Iuniv = J/J 2 . Then
Cuniv /Iuniv = B.
Let σ1 , σ2 : B → Cuniv be given by b 7→ b ⊗ 1 and b 7→ 1 ⊗ b respectively. They both lift the
identity B → B.
σ1 −σ2
Recall that Iuniv = J/J 2 ∼
= ΩB/A , and the universal derivation is B −→ Iuniv given by
b 7→ b ⊗ 1 − 1 ⊗ b.
Now if f is formally unramified, then σ1 = σ2 , so d = 0. Thus ΩB/A = 0.
Conversely, assume ΩB/A = 0. Then if τ1 , τ2 : B → C are two lifts, then we get a map
ϕ : B ⊗A B → C via b1 ⊗ b2 7→ τ1 (b1 )τ2 (b2 ), which factors through (B ⊗A B)/J 2 , since
ϕ(J) ⊆ I. But (B ⊗A B)/J 2 = Cuniv , and Iuniv = ΩB/A = 0. Thus, σ1 = σ2 , and
τ1 = σ1 ϕ = σ2 ϕ = τ2 .
Thus, there is at most one lift.
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Corollary 2.3. “Formally unramified” is local on the source and target.
Definition 2.4. A morhpism f : X → Y of schemes is unramified if f is locally of finite
type (l.f.t.) and formally unramified.
Lemma 2.5. Let f : X → Y be l.f.t.. Then f is unramified if and only if the diagonal
∆ : X → X ×Y X is an open immersion.
open
S
Proof. In general, if W = V ⊆X V ×Y V , then W ⊆ X ×Y X, and ∆ : X → W is a closed
affine
immersion. If I is the ideal sheaf, then I/I 2 = ΩX/Y .
Note that I is finitely generated: if B/A is generated by xi , then I is generated by
xi ⊗ 1 − 1 ⊗ xi . By Nakayama, V (I) is open if and only if I = I 2 (exercise).
Thus, ∆ is an open immersion if and only if ΩX/Y = 0.
`
Exercise. Let f : X → Spec K. Then f is unramified if and only if X = Spec Li , with
Li /K finite separable.
Corollary 2.6. Let f : X → Y be l.f.t.. Then f is unramified if and only if for all x ∈ X,
y = f (x), we have
mx = my OX,x
and k(x)/k(y) is finite separable.
Proof. Use properties of Ω, and Nakayama.
Summary: TFAE
(1) f is l.f.t. and formally unramified.
(2) f is l.f.t. and ΩX/Y = 0.
(3) f is l.f.t. and ∆ : X ,→ X ×Y X is an open immersion.
(4) f is l.f.t. and for all x ∈ X, y = f (x), we have mx = my OX,x and k(x)/k(y) is finite
separable.
Next we consider étale morphisms.
2.2. Étale morphisms.
Definition 2.7. f : X → Y is étale if it is l.f.t., flat and unramified.
`
Example 2.8. X → Spec K is étale if and only if X = Spec Li , for Li /K finite separable.
Example 2.9. A standard étale morphism is a ring map A → B given by
B = (A[x]/(f ))(f 0 )
for some f ∈ A[x] monic. We compute that
Ω(A[x]/(f ))/A = A dx/df = A dx/f 0 dx,
so once we invert f 0 , this is 0.
Example 2.10. Open immersions are étale.
Theorem 2.11. Locally, every étale morphism is given by a standard étale map.
A proof can be found in the Stacks Project or ALH.
Lemma 2.12. Let f : X → Y and g : Y → Z be étale. Then gf is étale.
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Proof. Clearly gf is l.f.p. and flat, because those properties are preserved under composition.
Finally, the “first” exact sequence
f ∗ ΩY /Z → ΩX/Z → ΩX/Y → 0
implies that ΩX/Z = 0.
Lemma 2.13. “Étale” is stable under base change.
Proof. This is immediate from the properties of Ω.
Here is a very important proposition.
Proposition 2.14. Let
f
X
/
g

h
Y
S
such that g is unramified. Then if h is étale, so is f .
Proof. The diagram
Γf
X
f
/
X ×S Y
∆Y
/ Y ×S Y
Y
is a pullback. Thus Γf is an immersion since ∆Y is. But f = π2 ◦ Γf , and π2 : X ×S Y → Y
is the base change of h along g.
/Y
X ×S Y
h
X
Thus f is étale, because both Γf are π2 are.
/
g
S
We only used the formal properties of being étale.
2.3. fpqc descent. I hope to give an accessible introduction to fpqc descent. Today we will
consider the affine case only, and once we understand the Grothendieck topology we will
look at the geometric picture. The reference is EGA IV, part 2, section 2.5–2.7.
(1) “Permanence properties under fpqc base change”,
(2) “Effectiveness of descent data”.
Throughout, A → B will be a faithfully flat ring homomorphism. The difference between
these is that in (1) we start with some object over A and try to descend properties of B,
whereas in (2), we have an object over B and some descent data, and prove that it descends
to A.
Faithfully flat means that any of the following equivalent conditions holds:
(1) B ⊗A − : A-Mod → B-Mod is faithful and exact.
(2) B is flat, and B ⊗A M = 0 ⇒ M = 0.
(3) B is flat, and 0 → M1 → M2 → M3 → 0 is exact if and only if 0 → B ⊗A M1 →
B ⊗A M2 → B ⊗A M3 → 0 is exact.
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(4) B is flat and Spec B → Spec A is surjective.
Lemma 2.15 (“Permanence properties of modules”). Let f : A → B be faithfully flat, M
an A-module. If B ⊗A M is
(1) finitely generated,
(2) finitely presented,
(3) flat,
(4) locally free of constant (finite) rank n,
(5) etc.,
then so is M . (These are separate statements.)
Proof.
P
(1) Let yi ∈ B ⊗A M be generators; say yi = j bij ⊗mij . Let M 0 ⊆ M be the submodule
generated by the mij . Then
• B ⊗A M 0 ,→ B ⊗A M (because B is flat),
• surjective as well.
Thus by (3) of f.f., M 0 ,→ M is surjective, i.e., M = M 0 .
(2) Apply (1) twice: K → An → M → 0. Apply it to K.
(3) Associativity of ⊗ and (3) of f.f.
(4) Locally free of locally constant finite rank ⇔ flat and f.p. By (2) and (3), this
descends. But if M has constant rank n on a component of Spec A, then B ⊗A M
also has constant rank n on the induced component.
Exercise (Permanence properties of algebras). If C is an A-algebra, and B ⊗A C is
(1) f.t.,
(2) f.p.,
(3) finite,
then so is C.
Lemma 2.16 (Permanence properties of morphisms). Let S = Spec A and T = Spec B. Let
f : X → Y be a morphism of A-schemes, and fT : XT → YT its base change to T . Then if
fT is
(1) l.f.t.,
(2) l.f.p.,
(3) flat,
(4) formally unramified,
(5) étale,
(6) etc.,
then so is f .
Proof. First we will do some reductions. Being f.f. is stable under base change, and each of
the properties is local on the source and target. Thus, by replacing S (resp. T ) with Y (resp.
YT ), or an affine in it, we may wlog assume S = Y affine and X affine; say X = Spec C.
(1) and (2) follow from the exercise.
(3) follows from the previous lemma.
(4): Since ΩB⊗A C/B = B ⊗A ΩC/A , we get ΩB⊗A C/B = 0 if and only if ΩC/A = 0 by (2) of
f.f.
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(5) follows from (2), (3) and (4).
These are just a few examples of permanence properties. EGA has a lot more.
Next we will study effectiveness of descent data.
2.4. Descent data.
Definition 2.17. Let f : A → B. A descent datum for f is a pair (N, ϕ), where N is a
B-module, and
∼
ϕ : N ⊗A B → B ⊗A N
is a B ⊗A B-linear isomorphism, such that the diagram of B ⊗A B ⊗A B-isomorphisms
ϕ02
N ⊗A B ⊗A B
/
)
ϕ01 =ϕ⊗B
B⊗
B ⊗A N
5 A
ϕ12 =B⊗ϕ
B ⊗A N ⊗A B
commutes.
A morphism f : (N, ϕ) → (N 0 , ϕ0 ) is a B-linear f : N → N 0 such that
N ⊗A B
f ⊗B
N ⊗A B
∼
ϕ
∼
ϕ0
/
/
B ⊗A N
B⊗f
B ⊗A N 0
commutes.
Example 2.18. For M an A-module, the canonical descent datum is (M ⊗A B, can), where
∼
can : (M ⊗A B) ⊗A B → B ⊗A (M ⊗A B)
m ⊗ b ⊗ c 7→ b ⊗ m ⊗ c.
This gives a functor
F : A-Mod → f -Desc.
Theorem 2.19 (fpqc descent). If f is f.f., then F is an equivalence.
Remark. Since B is f.f. over A, the functor F is faithful and exact.
Proof. This theorem is fairly involved. A proof can be found in the Stacks Project (Tag
023N), or BLR’s Néron models.
Here are some geometric remarks.
Remark. It says that we can glue modules along “fpqc opens”: given F on U → X and an
fpqc
∼
isomorphism F|U ×X U → F|U ×X U , with a cocycle condition on U ×X U ×X U , they glue to a
sheaf on X. (Next week we will see that a faithfully flat ring homomorphism is an example
of a covering in the fpqc topology.)
Unfortunately we ran out of time for Henselian rings. Please read a bit about them, e.g.
in ALH, the Stacks project, or Milne’s book Étale cohomology.
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3. Lecture 3 (September 22, 2015): Remy van Dobben de Bruyn
Grothendieck topologies and sites
Today we will discuss the formal categorical aspects of Grothendieck topologies. Next
week we will be more specific on étale sites and include examples.
3.1. Grothendieck pretopologies.
Example 3.1. Let X be a topological space. Let Top(X) be the category of open subsets
in X:
• objects: U ⊂ X open,
• morphisms: # Mor(U, V ) ≤ 1, with equality if and only if U ⊆ V .
A presheaf is a functor
Top(X)op → Set (or Ab, · · · ).
F is a sheaf if the diagram
F(U )
/
Y
/
F(Ui )
/
Y
F(Ui ∩ Uj )
is an equalizer, for any covering {Ui ⊆ U }i∈I .
Grothendieck pretopology generalizes this to any category.
Definition 3.2. Let C be a category. A Grothendieck pretopology on C is a collection Cov(C)
of coverings {Ui → U }i∈I such that:
(0) If U0 → U occurs in a covering, and V → U is arbitrary, then U0 ×U V exists.
(1) If {Ui → U } ∈ Cov(C), and V → U , then {Ui ×U V → V } ∈ Cov(C).
(2) If {Ui → U }i∈I and {Uik → Ui }k∈Ki (for all i) ∈ Cov(C), then {Uik → Ui →
U }i∈I,k∈Ki ∈ Cov(C).
∼
(3) If V → U , then {V → U } ∈ Cov(C).
Definition 3.3. A site is a category C with a pretopology on it.
Example 3.4. Top(X) is a pretopology; note that
∼
U1 ×U U2 → U1 ∩ U2 .
Example 3.5 ((small) (resp. large) étale site). Let X be a scheme. Let C = Ét/X (resp.
Sch/X ), and let
étale
Cov(C) = {{Ui → U } jointly surjective}.
Remark. By a lemma of last week, any Ui → U with
/
Ui
étale
X

U
étale
is étale.
Remark. {Ui → U } are jointly surjective if and only if
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`
Ui → U is surjective.
Example 3.6 ((large) fppf site). Let X be a scheme. Let C = SchX , and let
Cov(C) = {{Ui → U | flat of finite presentation} jointly surjective}.
Remark. U1 → U2 need not be flat in the diagram
/
U1
flat
X
~
U2
flat
Definition 3.7. A presheaf (with values in D) is a functor C op → D. (Often D = Set, Ab.)
Definition 3.8.
• A presheaf F is a sheaf if
F(U )
/
Y
/
F(Ui )
/
i
Y
F(Ui ×U Uj )
i,j
is an equalizer for all {Ui →
Q U } ∈ Cov(C).
• If we only know F(U ) ,→ i F(Ui ), then F is separated.
Definition 3.9. PshD (C) (resp. ShD (C)) is the category of presheaves (resp. sheaves).
Lemma 3.10. Suppose D is (co)complete. Then so is PshD (C), and (co)limits are “pointwise”:
(lim Fi )(U ) = lim(Fi (U )),
(colim Fi )(U ) = colim(Fi (U )).
Example 3.11. If f : F → G in PshD (C), then f is monic (resp. epic, an isomorphism) if
and only if each
f (U ) : F(U ) → G(U )
is monic (resp. epic, an isomorphism).
Proof. f : B → C is monic if and only if the diagram
B
1
B
1
f
/
/
B
f
C
is a pullback.
From now on, D = Ab (modify it yourself for Set).
3.2. Čech cohomology. Having the notion of coverings is enough for Čech cohomology.
Definition 3.12. Let F be a presheaf. Let U = {Ui → U }i∈I be a covering. The Čech
complex of F with respect to U is
Y
Č p (U, F) =
F(Ui0 ···ip ),
(i0 ,··· ,ip )∈I p+1
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where Ui0 ···ip = Ui0 ×U · · · ×U Uip . The maps Č p (U, F) → Č p+1 (U, F) are
(si )i∈I p+1 7→
p+1
X
(−1)j (resj (si0 ,··· ,iˆj ,··· ,ip+1 ))(i0 ,··· ,ip+1 )∈I p+2 .
j=0
(The usual argument shows that dp ◦ dp−1 = 0.) The Čech cohomology of F with respect to
U is
Ȟ p (U, F) = H p (Č • (U, F)).
This definition is only for one cover, and now we want to consider all covers at the same
time.
Definition 3.13. Let U = {Ui → U }i∈I , V = {Vj → U }j∈J be covers. Then V is a refinement
of U if there exist a map α : J → I and maps
ηj
Vj
U
/
Uα(j)
}
The data (α, (ηj )) is a refining morphism. We write V @ U.
This defines a map of complexes
Č • (U, F) → Č • (V, F)
induced by the restrictions along
Vj0 ···jp → Uα(j0 )···α(jp ) .
Lemma 3.14. Let (α, (ηj )), (β, (θj )) be refining morphisms V → U, Then they induce the
same maps on cohomology.
Proof. Construct a homotopy Č p+1 (U, F) → Č p (V, F) by
(si )i∈I p+2 7→
p
X
(−1)k resηj0 ···jk ×θjk ···jp (sα(j0 )···α(jk )β(jk )···β(jp ) )(j0 ,··· ,jp )∈I p+1
k=0
where the morphisms ηj0 ···jk : Vj0 ···jk → Uα(j0 )···α(jk ) and θjk ···jp : Vjk ···jp → Uβ(jk )···β(jp ) give
ηj0 ···jk × θjk ···jp : Vj0 ···jp → Uα(j0 )···α(jk )β(jk )···β(jp ) .
Corollary 3.15. If U @ V and V @ U, then
∼
Ȟ p (U, F) → Ȟ p (V, F).
1
Proof. The composition U @ V @ U can be computed using U @ U.
Remark. Thus, it suffices to look at the (possibly large) set
JU = Cov(U )/ ≡,
where U ≡ V if and only if U @ V and V @ U. Then JU is partially ordered by @. In fact,
JU is directed: for U, V, there exists a common refinement
U × V = {Ui × Vj → U }.
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Definition 3.16. The Čech cohomology of F on U is the colimit
Ȟ p (U, F) = colim Ȟ p (U, F)
U ∈JU
Warning. We only compute this if JU is a set.
3.3. Sheafification.
Remark. For f : V → U , the map JU → JV
U = {Ui → U } 7→ U × V = {Ui × V → V }
gives a map
Ȟ p (U, F) → Ȟ p (V, F).
p
Thus, the Ȟ p (−, F) give a presheaf Ȟp (F) (sometimes denoted Ȟ (F)).
Remark. The trivial cover {U → U } gives a map
F(U ) → Ȟ 0 (U, F).
This gives
ρ : F → Ȟ0 (F) =: F + .
This map is functorial in F.
Remark.
• If F is separated, then ρ : F → F + is injective.
∼
• If F is a sheaf, then ρ : F → F + is an isomorphism.
We shall later see these are if-and-only-if statements, which are not entirely clear because
colimits are taken.
Lemma 3.17. Let U ∈ ob C and s ∈ F + (U ). Then there exists U = {Ui → U }i∈I such that
ρ(si ) = s|Ui
for si ∈ F(Ui ).
Proof. An element s ∈ F + (U ) comes from some Ȟ 0 (U, F), i.e., (si ) ∈
si |Ui ×Uj = sj |Ui ×Uj . Then
F + (Ui ) 3 ρ(si ) = (si |Ui ×Uj )j = (sj |Ui ×Uj )j = s|Ui .
Q
i
F(Ui ) such that
Now comes the big theorem.
Theorem 3.18. Let F be a presheaf.
(1) F + is separated.
(2) If F is separated, then F + is a sheaf.
Proof.
(1) Let s ∈ F + (U ) such that s|Ui = 0 for some U. By the lemma, we know that there
exist V and sj ∈ F(Vj ) such that s|Vj = ρ(sj ). Then
ρ(sj |Vj ×Ui ) = ρ(sj )|Vj ×Ui = s|Vj ×Ui = 0 = ρ(0).
Therefore, (sj )j∈J becomes 0 in Ȟ 0 (U × V, F). Thus, s = 0.
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(2) By (1), F + is separated. Let si ∈ F + (Ui ) such that
si |Ui ×Uj = sj |Ui ×Uj .
Choose {Uik → Ui } such that si = ρ(sik ). Then
ρ(sik |Uik ×Ui0 k0 ) = ρ(sik )|Uik ×Ui0 k0 = si |Uik ×Ui0 k0 = si0 |Uik ×Ui0 k0 = ρ(si0 k0 |Uik ×Ui0 k0 ).
By injectivity of ρ, we get
sik |Uik ×Ui0 k0 = si0 k0 |Uik ×Ui0 k0 .
Thus,
s = (sik )i,k ∈ Ȟ 0 (W, F),
where W = {Uik → Ui → U }.
Corollary 3.19. Let F be a presheaf.
(1) F ++ is a sheaf.
(2) F is separated if and only if F ,→ F + .
∼
(3) F is a sheaf if and only if F → F + .
Proof.
(1) Clear.
(2) We saw ⇒. Conversely, if F ,→ F + , then F is a subpresheaf of separated presheaf,
thus separated.
∼
(3) We saw ⇒. Conversely, if F → F + , then F is separated by (2), so F + is a sheaf by
the theorem.
Lemma 3.20. Let F be a presheaf, G a sheaf. If g : F + → G is such that gρ = 0, then
g = 0.
Proof. Let s ∈ F + (U ). Then s|Ui = ρ(si ), so
g(s)|Ui = g(s|Ui ) = gρ(si ) = 0.
So g(s) = 0 because G is a sheaf.
Now we come to the big theorem.
Theorem 3.21. Let F be a presheaf, G a sheaf. Then
HomSh (F ++ , G) → HomPSh (F, G)
g 7→ gρρ
is an equivalence.
Definition 3.22. F ++ is the sheafification of F.
Corollary 3.23. Thus, we have an adjunction
(−)++
PSh(C) o
1
12
/
Sh(C).
Proof. By functoriality of ρ, we have
/
F
f
G
∼
/
/
F+
G+
∼
F ++
/
G ++
We need to show f factors uniquely through F ++ . The diagram gives existence. But if
f1
F ++
f2
/
/
G such that f1 ρρ = f2 ρρ, then applying the lemma twice gives f1 = f2 .
Corollary 3.24. Let D : I → Sh(C) be a diagram. Then the colimit (in Sh(C)) is given by
the sheafification of the pointwise (presheaf ) colimit.
Proof. Sheafification preserves colimits (it is a left adjoint), and is the identity on the subcategory of sheaves:
(colim Fi )++ = colim(Fi++ ) = colim(Fi ).
PSh
Sh
Sh
Example 3.25 (Cokernel). Let F → G. Then the sheaf cokernel is the sheafification of
U 7→ G(U )/F(U ).
Remark. Limits are just limits in PSh: if all Fi are sheaves, then so is lim Fi . (“Limits
commute with limits.”)
Corollary 3.26. Sh(C) is complete and cocomplete.
Theorem 3.27. Sh(C) is an abelian category (recall: we need JU to be small).
Lemma 3.28. Sheafification is exact, and commutes with restriction to a subcategory C 0 ⊆ C.
Proof.
• The second statement follows from the construction.
• Right exact: it is a left adjoint.
• Left exact: by hand (exercise).
4. Lecture 4 (September 29, 2015): Remy van Dobben de Bruyn
Sheaves on the étale site
4.1. Change of site. The setup is as follows. Let u : C → D be a functor (of sites). We
will assume at some point:
(*) C has and u preserves fibred products and a terminal object (hence all finite limits).
Example 4.1. Suppose f : X → Y is a continuous map of topological spaces, then
u : Top(Y ) → Top(X)
U 7→ f −1 (U )
satisfies condition (*).
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Lemma 4.2. The functor
up : PSh(D) → PSh(C)
F 7→ (U 7→ F(u(U )))
is exact.
Proof. (Co)limits are pointwise.
Definition 4.3. Define the comma category (A ↓ u) where A ∈ ob D by:
f
• objects: (A → u(U )) where U ∈ ob C and f ∈ D(A, u(U )),
• morphisms:
A
f
g
}
/
u(U )
u(φ)
!
u(V )
Definition 4.4. The functor
up : PSh(C) → PSh(D)
F 7→ A 7→ colim
f (U ) .
op
(U,f )∈(A↓u)
Remark. For topological spaces f : X → Y ,
(up F)(A) = colim F(U ) = colim F(U )
U ⊂Y
A⊆f −1 (U )
U ⊆Y
f (A)⊆U
is the classical inverse image functor.
Theorem 4.5. For F ∈ PSh(C), G ∈ PSh(D), we have
HomPSh(D) (up F, G) = HomPSh(C) (F, up G).
Proof. Exercise.
We want to do the same thing for sheaves. Let us recall the following
Lemma 4.6. If (*) holds, then up is exact.
Proof.
• Right exactness follows from the theorem.
• If (*), then (A ↓ u) is cofiltered: given (U, f ), (V, g) ∈ (A ↓ u), we have
A
A
u(U )
u(V )
and so
f ×g
A −→ u(U ) × u(V ) = u(U × V ).
14
So the pair (U × V, f × g) is the product in (A ↓ u), giving projections
(U × V, f × g)
π1
π2
w
'
(U, f )
(V, g)
etc (there are two more things we need to check; omitted).
Then (A ↓ u)op is filtered, and filtered colimits are exact.
Definition 4.7 (For us). A functor u : C → D of sites is continuous if (*) holds, and for
each covering {Ui → U } in C, the collection {u(Ui ) → u(U )} is a covering in D.
The Stacks Project has a more general version.
Example 4.8. If f : X → Y a morphism of schemes, then
Yét = Ét/Y → Xét = Ét/X
U 7→ U ×Y X
is continuous.
Example 4.9.
• Similarly for the big fppf, étale sites.
• Similarly for the (small) Zariski site.
Warning. Note the contravariance. We will fix this later.
Example 4.10. We get functors
XZar
(small)
/
Xét
(small)
/
Xfppf
(big)
Both of these are continuous.
Lemma 4.11. Let u : C → D be continuous. If F is a sheaf on D, then up F is a sheaf on
C.
Proof. The sheaf condition for up F on the covering {Ui → U } is just the sheaf condition for
F on {u(Ui ) → u(U )}.
Definition 4.12. The functor Sh(D) → Sh(C) is denoted us .
Definition 4.13. The composition
H0
up
(−)#
Sh(C) ,→ PSh(C) → PSh(D) → Sh(D)
is denoted us .
Theorem 4.14.
HomSh(D) (us F, G) = HomSh(C) (F, us G).
Proof. Use up a up and (−)# a H0 .
Lemma 4.15. (If (*) then) us is exact.
15
Proof.
• Clearly right exact.
• Each of
up
(−)#
Sh(C) ,→ PSh(C) → PSh(D) → Sh(D)
is left exact.
Now we will fix the contravariance.
Definition 4.16. A morphism of sites f : D → C is a continuous functor u : C → D (such
that us is exact).
Example 4.17.
• For f : X → Y a morphism of schemes, we get
f : Xét → Yét (fppf, ...).
• For a scheme X, we get morphisms
Xfppf → Xét → XZar .
Definition 4.18. If f : D → C is a morphism of sites, we write
f ∗ = us ,
f∗ = us .
Theorem 4.19. Let f : D → C be a morphism of sites. Then f ∗ a f∗ and f ∗ is exact.
Warning. For schemes, what we call f ∗ is usually called f −1 .
4.2. Cohomology.
Theorem 4.20. If C is small, then Sh(C) is a Grothendieck abelian category. In particular,
it has enough injectives.
Definition 4.21.
• The derived functors of Γ(U, −) are denoted H i (U, −). For the étale site, for now we
i
will write Hét
(U, −).
• The derived functors of PSh(C) → PSh(C) are denoted Hi . One checks that
Hi (F)(U ) = H i (U, F).
• If f : D → C is a morphism, the derived functors of f∗ are denoted Ri f∗ : Sh(D) →
Sh(C).
Theorem 4.22. The functors
Ȟi : PSh(C) → PSh(C)
form a universal δ-functor. That is, the Ȟi are the derived functors of Ȟ0 .
This is useful because we can then use spectral sequences.
Proof (Idea).
• For 0 → F1 → F2 → F3 → 0, we get
0 → Č • (F1 ) → Č • (F2 ) → Č • (F3 ) → 0,
proving that they are a δ-functor.
16
• For I injective, Ȟi (I) = 0 for i > 0.
Proposition 4.23. Let G : B → A have an exact left adjoint F . Then F preserves injectives.
Proof. Let I ∈ ob B be injective, and 0 → A → B in A. Then 0 → F A → F B by exactness
of F , so by the injectivity of I,
B(F B, I) A(B, GI) 
/

/
B(F A, I)
A(A, GI)
Thus GI is injective.
Example 4.24. The following functors have an exact left adjoint:
• H0 : Sh(C) ,→ PSh(C).
• For f : D → C a morphism of sites, we get
f∗ : Sh(D) → Sh(C).
Corollary 4.25 (Leray spectral sequence). Let f : D → C. Then
E2pq = H p (U, Rq f∗ (−)) ⇒ H p+q (u(U ), −),
E2pq = Hp (Rq f∗ −) ⇒ Hp+q (−).
Proof. From the Grothendieck spectral sequence for
Sh(D) → Sh(C) → Ab
and
Sh(D) → Sh(C) → PSh(C)
respectively.
Corollary 4.26. We have spectral sequences
E2pq = Ȟ p (U, Hq −) ⇒ H p+q (U, −),
E2pq = Ȟp (Hq −) ⇒ Hp+q (−).
Proof. From
Ȟ 0 (U,−)
Sh(C) → PSh(C) −→ Ab
and
Ȟ0 (U,−)
Sh(C) → PSh(C) −→ PSh(C).
Lemma 4.27. Ȟ0 (Hq (F)) = 0 for q > 0.
Proof. Recall Ȟ0 ⊆ (−)# = Ȟ0 (Ȟ0 ). So it suffices to prove
Hq (F)# = 0.
Let I • be an injective resolution. Then
Hq (F) = H q (H0 (I • )).
But (−)# is exact, so
H q (H0 (I • ))# = H q (H0 (I • )# ) = H q (I • ) = 0.
17
Corollary 4.28.
• H0 = Ȟ0 , and H 0 (U, −) = Ȟ 0 (U, −).
• H1 = Ȟ1 , and H 1 (U, −) = Ȟ 1 (U, −).
Technically we should write Ȟi (H0 (−)) instead of Ȟi , but we will simply identify a sheaf
with its underlying presheaf.
Proof.
• The statement about H0 is trivial.
• For H1 , use the spectral sequence and the lemma for q = 1.
Theorem 4.29. If X is quasiprojective over an affine, then
i
i
.
= Hét
Ȟét
4.3. Sheaves on the étale site.
Lemma 4.30. Let F be a presheaf on Xét (or Xfppf , etc.). Then F is a sheaf if and only if
(1) For each U , F|UZar is a Zariski sheaf.
(2) For each {V → U } with both U and V affine, the sequence
d0
0 → F(U ) → F(V ) → F(V ×U V )
is exact.
Proof. Exercise in Grothendieck topologies.
Proposition 4.31. Let f : A → B be faithfully flat. Then the sequence
0 → A → B → B ⊗A B → B ⊗A B ⊗A B → · · ·
(*)
coming form the (co?)simplicial ring
/
A
Bo
/
/
/
o
/
B ⊗A B o
/
B ⊗3 · · ·
is exact. Moreover, if M is an A-module, then also M ⊗A (*) is exact:
0 → M → M ⊗A B → M ⊗A B ⊗A B → · · ·
Proof.
• If f admits a retraction g : B → A, this contracts (*).
• Fppf-locally, f admits a retraction B ⊗A B → B given by b1 ⊗ b2 7→ b1 b2 . Use
permanence.
Proposition 4.32. Let Spec B → Spec A be faithfully flat. Let Z be any scheme. Then
Hom(Spec A, Z)
/
Hom(Spec B, Z)
/
/
Hom(Spec B ⊗A B, Z)
is an equalizer diagram.
Proof. We will consider Z = Spec C affine (general case: exercise). This follows from exactness of
0 → A → B → B ⊗A B.
Now we come to the big theorem.
18
Theorem 4.33. Let Z be a scheme. Then the presheaf
(Sch/X )op
fppf → Set
Y 7→ Hom(Y, Z)
is a sheaf for the fppf (or coarser, hence also étale) site. (“Representable presheaves are
sheaves.”)
Proof. By the lemma, we need it for Zariski covers and for one object affine covers. For
Zariski, it is “glueing of morphisms”; for {V → U } affine, it is the proposition.
Theorem 4.34. Let F be quasicoherent on X. For f : Y → X, set
F(Y ) = f ∗ F(Y ) (= f −1 ⊗ OY ).
Then the presheaf
(Sch/X )op
fppf → Ab
Y 7→ F(Y )
is a sheaf on the fppf site.
Proof. Similar: use the second statement of the proposition.
Theorem 4.35 (Hilbert 90). The natural map
1
1
1
(Pic(X) =) HZar
(X, Gm ) → Hét
(X, Gm ) (→ Hfppf
(X, Gm ))
is an isomorphism.
1
Proof. We know that H 1 = Ȟ 1 . An element of Ȟét
(X, Gm ) is an open covering {Ui → X}
with elements
sij ∈ Hom(Ui ×U Uj , Gm ) = O(Uij )×
such that sij |Uijk · sjk |Uijk = sik |Uijk .
Now the sij define glueing data for trivial line bundles Li on Ui :
∼
ϕij : Li |Ui ×Uj → Lj |Ui ×Uj
1 7→ sij
satisfying a 2-cocycle condition. By effectiveness of descent data, they come from a quasicoherent sheaf F on X. By permanence of “locally free of rank 1”, F is a line bundle. 4.4. The étale site of Spec(K). This will allow us to relate the theorem to the classical
Hilbert 90.
Given a sheaf F on Spec(K)ét , define
AF = colim F(L).
L/K
fin. sep.
Exercise. Check that AF is a discrete ΓK = Gal(K sep /K)-module.
Definition 4.36. Given a discrete ΓK -module A, define
!
a
Y
F
Spec Li =
AΓLi .
i
i
19
Exercise. Check that this is a sheaf on (Spec K)ét .
Exercise. Prove that these define an equivalence
/
Sh((Spec K)ét ) o
ΓK -Moddisc
Corollary 4.37.
i
i
(K, AF ).
(Spec K, F) = HGalois
Hét
Proof. They are the derived functors of Γ(Spec K, −) = (−)ΓK .
1
Thus we recover the classical Hilbert 90: Hét
(X, Gm ) is the Galois cohomology group
1
×
H (K, K̄ ), and the Picard group of Spec(K) is trivial.
5. Lecture 5 (October 6, 2015): Sam Mundy
Cohomology on curves
Sam: I fixed the following four points: 1) The fact DX is a sheaf requires Lemma 5.6. 2)
The proof of (3, affine) requires a snake lemma argument. 3) The map F in the proof of
Tsen’s theorem is not linear, but instead algebraic. 4) I added a remark at the end about the
proof of the fact before Definition 5.9.
Let k = k be an algebraically closed field, and X be a (projective or affine) smooth
connected curve over k, and n ∈ Z not divisible by char(k). We will calculate the étale
cohomology for the constant sheaves Z and Z/nZ.
Recall that Gm is a sheaf on Xét . Define n : Gm → Gm and µn = ker n.
Theorem 5.1 (Kummer theory). The sequence
n
0 → µn → Gm → Gm → 0
is exact.
Proof. The only thing we need to check is the surjectivity of the√map n. For all U → X
étale and a ∈ Γ(OU , U )× , we need to find V → U étale such that n a ∈ Γ(OV , V )× . Take
which has
√
n
a.
V = Spec(OU [T ]/(T n − a)),
We will use this sequence to prove the
Theorem 5.2 (Main theorem).
• For X projective,

µn (k)



Jac X[n]
H i (X, µn ) =

Z/nZ



0
if
if
if
if
i = 0,
i = 1,
i = 2,
i ≥ 3.
• For X affine,


µn (k)
i
H (X, µn ) = finite

0
20
if i = 0,
if i = 1,
if i ≥ 2.
This is what we expect in the complex case, i.e., cohomologies of Riemann surfaces.
Remark. µn ' Z/nZ is a sheaf. Later we will need
lim H i (X, Z/`n Z) ⊗Z` Q`
which will be Q1,2g,1
`
From the theorem we get the long exact sequence
(1)
/
0
/
(2)
/
H 0 (X, Gm )
n
/
H 1 (X, Gm )
β
H 0 (X, µn )
H 1 (X, Gm )
n
(3)
/
/
H 0 (X, Gm )
H 2 (X, µn )
(4)
(5)
/
α
/
H 1 (X, µn )
H 2 (X, Gm ).
(6)
We will fill in the following table. Some of the terms are already known.
Projective
Affine
1 µn (k) ' Z/nZ
µn (k)
2
k×
OX (X)×
3
4
Pic X
Pic X
5
6
α=0
β
(3, projective) is ker(n : Pic X → Pic X). We have
Pic X = Pic0 X × Z.
But Pic0 X = Jac X, so this kernel is, by definition, Jac X[n] ' (Z/nZ)2g (non-canonically).
Let us now calculate 6 for both affine and projective curves.
Let RX be the sheaf of rational functions on X (so RX (U ) = K(U )) and DX be the sheaf
of Cartier divisors.
Lemma 5.3. RX and DX are indeed sheaves.
Proof. To check, we need to show:
(1) RX is a Zariski sheaf on any U étale over X. (Exercise.)
(2) For all V → U étale covers with V and U affine, we have an exact sequence
0 → RX (U ) → RX (V ) → RX (V ×U V ).
DX will follow from Lemma 5.6 which does not require we know that DX is a sheaf.
Lemma 5.4. 0 → Gm → R×
X → DX → 0 is exact.
Proof. To check, it is enough to check exactness of the associated Zariski sheaves. We know
this from the definition of Cartier divisors.
Lemma 5.5. Let j : Spec K(X) → X be the inclusion. Then RX = j∗ Gm .
21
Proof. We have
j∗ Gm = (U 7→ Gm (U ×X Spec K(X))
= (U 7→ OU ×X Spec K(X) (U ×X Spec K(X))× )
= (U 7→ K(U )× )
×
#
= (R×
X ) = RX .
Lemma 5.6. DX =
L
x∈X
closed points
x∗ (Z).
Proof. Weil = Cartier.
Lemma 5.7. Ri j∗ Gm = 0 for i ≥ 1.
Proof. We need the fact
Ri j∗ Gm = (U 7→ H i (U ×X Spec K(X), Gm )#
(left as an exercise). Since
U ×X Spec K(X) =
a
Spec(Km ) = Spec
Y
Km ,
where Km /K(X) are finite separable extensions, we have
M
sep ×
H i (U ×X Spec K(X), Gm ) =
H i (GKm , (Km
) ).
By Tsen’s theorem, these vanish for i ≥ 1.
Lemma 5.8. H i (X, x∗ Z) = 0 for i ≥ 1.
Proof. Exercise.
We have the exact sequence
0
/
Gm
/
/
R×
X
'
0
/
Gm
/ j∗ Gm,Spec K(X)
/
L
/
0
/
0
DX
'
x closed
x∗ Z
By the Leray spectral sequence and Tsen,
H i (X, j∗ Gm ) ' H i (Spec K(X), Gm ) = 0
for i ≥ 1, and
M
H i X,
x∗ (Z) = 0
for i ≥ 1.
Therefore we know that (6, projective or affine) is 0.
(5, projective) is
coker(n : Pic X → Pic X) = coker(n : Z → Z) = Z/nZ.
For (β, affine), let X ⊂ X be a smooth projective curve. For all divisors D on X, there
exists D on X such that D is D on X and D has degree 0. This implies there exists
E ∈ Div X such that nE ∼ D, so β = 0.
Since (6, affine) is 0 and β = 0, the long exact sequence implies (5, affine) is zero.
22
Finally, for (3, affine), consider
OX (X)× = {nonzero rational functions on X with poles in X\X}.
Therefore, k × → OX (X)× has cokernel a subgroup D of the divisors with support in X\X.
Note that D is finitely generated. Consider the diagram
0
0
/
/
/
n
k×
/
×
OX
k×
/
n
×
OX
/
D
0
/
n
D
/
0.
n
Since k × → k × is surjective, the snake lemma gives
coker(n : OX (X)× → OX (X)× ) = coker(n : D → D),
which is finite. Also ker(n : Pic X → Pic X) is finite because ker(n : Pic0 X → Pic0 X) is
finite, so this gives (3, affine).
Projective
Affine
1
µn (k) ' Z/nZ
µn (k)
×
2
k
OX (X)×
3 Jac X[n] ' (Z/nZ)2g
finite
4
Pic X
Pic X
5
Z/nZ
0
6
0
0
α=0
β=0
Now we will discuss Tsen’s theorem in terms of Galois cohomology. Recall the following
Fact. If H 1 (GK(X) , (K(X)sep )× ) = H 2 (GK(X) , (K(X)sep )× ) = 0, then H i (GK(X) , (K(X)sep )× ) =
0 for all i ≥ 1.
We know that H 1 is zero by Hilbert 90. To calculate H 2 we need the
Definition 5.9. A central simple algebra (CSA) over a field K is a K-algebra which is
finite-dimensional and simple with center K.
Fact.
(1) If A is a CSA, then there exists a division ring D over K with center K such that
A ' Mm (D)
for some m.
(2) If A is a CSA, then there exists L/K finite separable such that
A ⊗K L ' Mn (L)
for some n.
Definition 5.10. We say two CSA’s are equivalent if their D’s in Fact (1) can be taken to
be the same. The Brauer group of K is
Br(K) := {CSA’s over K}/equivalence.
Fact.
23
(3) Br(K) ' H 2 (GK , (K sep )× ).
Thus we want to show Br(K(X)) = 0.
Let D be a division ring over K with center K. If we take L as in Fact (2), then we have
N = det : Mn (L) ' D ⊗K L → L
Use fpqc-descent to “glue” the N ’s to N : D → K. This N is the restriction of det :
Mn (K sep ) → K sep , so N is multiplicative, given by polynomials in n2 variables of degree n.
Definition 5.11. K is C1 if all homogeneous polynomials in m variables of degree d < m
have a root.
So Tsen’s theorem can be stated as follows.
Theorem 5.12 (Tsen). K(X) is C1 (if k is algebraically closed).
Proof. Let F be a homogeneous polynomial of degree d in m variables, with m > d. Let H
be a very ample divisor whose support contains the poles of the coefficients of F . Let q 0.
View F as a map
F : Γ(X, O(qH))m → Γ(X, O(dqH)).
View this as an algebraic map between two affine spaces. If dim Γ(X, O(qH))m > dim Γ(X, O(dqH)),
the fiber of this map at 0 has dimension the difference of the dimensions of these two spaces,
as long as it is nonempty. But the fiber contains 0 because F is homogeneous, so it suffices to
verify this inequality. Indeed by Riemann–Roch the two dimensions are mq deg H + m(1 − g)
and dq deg H + 1 − g respectively, so this inequality is satisfied if q 0.
Remark (Remy). Suppose H i (K, Gm ) = H i+1 (K, Gm ) = 0. Then we can prove that H i+1 (K, µn ) =
0 for all positive integers n ∈ K × . With some fiddling we can show H j (K, torsion) = 0 for
all j ≥ i + 1. This is done in Serre and Gille–Szamuely. This shows that the p-cohomological
dimension satisfies
cdp K ≤ i
for p ∈ k × . If p = char(k), then cdp K ≤ 1. It is a theorem that the strict cohomological
dimension satisfies scdp K ≤ cdp K + 1, so we have
scdp K ≤ i + 1.
It is not true that for dim X = r, K(X) being Cr implies cdK ≤ r. In Tsen’s theorem we
used Brauer groups explicitly.
Remark (Remy+Sam). Actually, to show the fact that H 1 (GK(X) , (K(X)sep )× ) = 0 and
H 2 (GK(X) , (K(X)sep )× ) = 0 implies H i (GK(X) , (K(X)sep )× ) = 0 for all i ≥ 1, one needs to
verify this on the finite level, i.e., for Gal(L/K(X)) where L is finite Galois over K(X).
We know H 1 (Gal(L/K(X)), L× ) = 0 by Hilbert 90, and so the inflation-restriction exact
sequence gives that H 2 (Gal(L/K(X)), L× ) injects into the Brauer group, which we know is
trivial by Tsen. Tate’s theorem implies then that H i (Gal(L/K(X)), L× ) = 0 for all i ≥ 1.
Taking (co)limits gives the fact.
24
6. Lecture 6 (October 13, 2015): Shizhang Li
Proper base change
The proper base change theorem states that
Theorem 6.1. Let f : X → S be proper with S Noetherian, and F be a sheaf of torsion
abelian groups on X. Then for all q ≥ 0,
Rq f∗ F|s = H q (Xs , Fs )
where s is a geometric point.
First we will reduce the theorem to the simpler case when X is a curve over a henselian
ring, and F = Z/nZ, and then we will give the proof.
s.h.
Remark. Spec(OS,s
) is an infinitesimal neighborhood of s, so we may assume S is the spectrum of a strictly henselian local ring.
6.0. Observation for finite morphisms. Observe that Theorem 6.1 holds for finite morphisms. This is justified by the following lemma in commutative algebra.
Lemma 6.2. Let A be a strictly henselian local ring.
(1) If A → B is finite, then
k
Y
B=
Bi ,
i=1
where the Bi ’s are strictly henselian local rings.
(2) For all étale sheaves F, we have
H i (Spec(A), F) = 0
for all i ≥ 1.
6.1. Reducing the scheme. By Chow’s lemma, a proper morphism is not too far from
being projective.
Lemma 6.3. Suppose we have a diagram
X̃
π
X

g
f
S
(1) If Theorem 6.1 holds for π and f , then it holds for g.
(2) If π is finite and Theorem 6.1 holds for g and F̃, then it holds for f and π∗ F̃.
Proof.
(1) Consider the spectral sequence
Ri f∗ (Rj π∗ F̃)|s ⇒ Ri+j g∗ F̃|s
25
By assumption, Ri f∗ (Rj π∗ F̃)|s ' H i (Xs , Rj π∗ F̃), and we have anothe spectral sequence
H i (Xs , Rj π∗ F̃) ⇒ H i+j (Xs , F̃),
which degenerates on page 1 or 2.
(2) We use the same spectral sequence and the fact that for π finite,
(
π∗ F̃ if i = 0,
Ri π∗ F̃ =
0
otherwise.
Remark. The assumption of π in (2) can be relaxed to only requiring π to be surjective, but
the proof will be a little more complicated (by running the argument for injective sheaves);
see the Stacks Project Tag 0A4C. In particular, (2) is true if π is a blow-up.
We will do an induction on the dimension of supp F. By Chow’s lemma, we have a diagram
X̃
finite
X

π
g proj.
f
proper
S
∼
Choose x ∈ supp F. There exists x ∈ U such that π −1 (U ) → U . Consider the exact sequence
π
∗
0 → K → F → π∗ π F → C → 0
By Lemma 6.3, we only need to prove the theorem for g : X̃ → S. Thus we have reduced to
the case when f is projective.
Next we will do an induction on dim X.
X̃
/
X

/
proj.
PnS o
So
P1S
BlPnS
}
The morphism X̃ → S factors through P1S , so we have reduced to the case when X is a curve
over S.
6.2. Reducing the sheaf.
Proposition 6.4. Let F be a sheaf on X and G be a finite abelian group. TFAE:
(1) F is represented by a finite étale cover Y → X, where G acts on Y with quotient X.
(2) There exists a covering {Ui → X} such that F|Ui = G.
In these cases, we call F locally constant.
Proof. (1) ⇒ (2): Structure theorem.
(2) ⇒ (1): Consider the trivial G-torsor
Vi = Ui × G → Ui → X.
Descent allows us to glue the Vi ’s to get Y using the sheaf conditions.
26
The locally constant sheaves do not form an abelian category, so we want to extend them
somehow.
Definition 6.5. F on X is constructible if there exists a stratification
X = X 0 ⊇ X1 ⊇ · · · ⊇ X N
such that Xi ’s are closed in X and F|Xi −Xi−1 is locally constant.
Proposition 6.6. The category of constructible sheaves is an abelian category.
In two weeks Linus will talk about constructible sheaves.
Lemma 6.7. Any torsion sheaf is a filtered colimit of constructible sheaves.
Proof. Choose ξ ∈ F(U ). Then the image of j! Z/nZU → F is a constructible subsheaf of
F.
Definition 6.8. Let C be an abelian category. A functor T : A → Ab is called effaceable if
u
for all objects C ∈ C and α ∈ T (C), there exists C ,→ M such that T (u)(α) = 0.
Lemma 6.9. H q (X, −) is effaceable on the category of constructible sheaves.
Proof. Consider the injection
Y
F ,→
ix,∗ Fx .
x∈|X|
Lemma 6.10. Let ϕ : T • → T 0 • be a natural transformation of cohomological functors.
Suppose T i is effaceable for i > 0, and there exists a sub-collection E of Ob(C) such that for
all C ∈ C, there exists C ,→ M ∈ E. Then TFAE:
(1) ϕ• is bijective on C.
(2) ϕ0 (M ) is bijective, and ϕq (M ) is surjective for all q > 0 and M ∈ E.
(3) ϕ0 is bijective and T 0 i are effaceable for i > 0.
This is proved by diagram-chasing.
Proposition 6.11. Let X0 ,→ X be a subscheme. Suppose for all n ≥ 0 and all X 0 → X
finite, H q (X 0 , Z/nZ) → H q (X00 , Z/nZ) is bijective when q = 0 and surjective when q > 0.
Then for all torsion F on X, we have
H q (X, F) ' H q (X0 , F).
Proof. We will prove for F constructible. Let E be the collection of constructible sheaves of
the form
Y
pi,∗ Ci
i
where pi :
Xi0
→ X is finite and Ci is Z/nZ on X 0 .
We have thus reduced the theorem to the case when X is a curve over the spectrum of a
strictly henselian ring, and F is given by Z/nZ.
Now we need to prove
H q (X, F) → H q (X0 , F)
27
is a bijection when q = 0 and surjection when q ≥ 1, where F = Z/nZ and X0 is a geometric
fiber:
/X
X0
/
s
S
6.3. Computing some cohomology.
6.3.1. Computing H 0 .
Proposition 6.12. Let (A, m) be a henselian local ring, f : X → Spec(A) be proper and
X0 = X ×A A/m. Then π0 (X) ' π0 (X0 ).
Proof. If A is strictly henselian, then we have an exact sequence
π1 (Spec(A)) → π0 (X0 ) → π0 (X) → π0 (Spec(A)).
The first and last terms are trivial.
This shows that the map on H 0 is an isomorphism.
6.3.2. Computing H 1 . Recall the
Proposition 6.13. H 1 (X, Z/nZ) ' {Y → XGalois covering with G = Z/nZ}
To prove surjectivity on H 1 , we want to extend any Galois covering Y0 → X0 to Y → X.
∼
This is done in Johan’s class: for X → Spec(A) with A henselian local, we have FÉtX →
FÉtX0 .
6.3.3. Computing H 2 . We can decompose
Z/nZ '
Y
Z/p`i i Z
and consider
p`−1
0 → Z/p`−1 Z → Z/p` Z → Z/pZ → 0,
so it suffices to compute H q (X0 , Z/pZ) for all primes p. There are two cases, depending on
whether the characteristic p divides n or not.
Suppose p | n. Recall the Artin–Schreier sequence
F −1
0 → Fq → Ga → Ga → 0,
where F is the Frobenius map. This gives a long exact sequence
0 → H 1 (Fq ) → H 1 (Ga ) → H 1 (Ga ) → H 2 (Fq ) → H 2 (Ga ) → H 2 (Ga )
Here is an important
Fact. If F is quasicoherent on X, then
•
•
•
Hét
(X, F) = HZar
(X, F) = Hfppf
(X, F).
Proof. This follows from the exactness of
0 → M → M ⊗A B → M ⊗A B ⊗A B → · · ·
for A → B faithfully flat and M an A-module.
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Note that Ga = OX and H 2 (Ga ) vanishes for the Zariski topology. Let V = H 1 (X, OX ),
which is a finite-dimensional k-vector space k g . Consider F − 1 : V → V . Note that F is
additive and satisfies F (λv) = λp v (F is called semi-linear).
Let W = ker(F ∞ ) ⊆ V . Then F induces an isomorphism V /W → V /W . Let us show
that we can take a basis {vi }ni=1 of V /W with F (vi ) = vi . Projectivizing gives
Frob
/
F
Pn−1
#
P; n−1
σ∈PGLn−1
Pn−1
where Frob is the geometric Frobenius. Then we only have to find one fixed point of F , but
n−1
they are the same as ΓF ∩ ∆ in Pn−1
degrees, we see there are exactly
P× P . Computing n−1
n−1
p
fixed points (corresponding to
ai vi , where (ai ) ∈ PFp ).
Remark. This proposition can be found on P.143 of Mumford’s Abelian Varieties, which gives
a different proof.
On W , we have F is nilpotent, so F − 1 : W → W is an isomorphism. On V /W , we have
F − 1 : V /W → V /W is surjective (by computing directly with the basis {vi }). Now we have
F −1
0 → H 1 (Fp ) → V V → 0
where H 1 (Fp ) is (n − 1)-dimensional. This shows H 2 (Fp ) = 0, so the map on H 2 is trivially
surjective.
The other case is p - n. Use the Kummer sequence
0 → µn → Gm → Gm → 0.
We have
/
Pic(X)
Pic(X0 )
//
H 2 (X, Z/nZ)
2
H (X0 , Z/nZ)
To prove that H 2 (X, Z/nZ) → H 2 (X0 , Z/nZ) is surjective, it suffices to prove Pic(X) →
Pic(X0 ) is surjective.
Proposition 6.14. In this situation, Pic(X) Pic(X0 ).
Proof. We will prove that every line bundle L0 on X0 lifts to a line bundle on X. Set
Xn = X0 ×S Spec(A/mn+1 ).
The exact sequence
∗
∗
0 → I → OX
→ OX
→0
n
n−1
gives the long exact sequence of Zariski cohomology
∗
∗
H 1 (X0 , I) → H 1 (Xn , OX
) → H 1 (Xn−1 , OX
) → H 2 (X0 , I) → · · · .
n
n−1
29
But X0 is 1-dimensional, so H 2 (X0 , I) = 0 (equivalently, Jac(X0 /k) is smooth). Note
∗
H 1 (Xn , OX
) = Pic(Xn ), so every line bundle on Xn−1 can be extended to Xn .
n
Xo
···
L2
L1
L0
··· o
X2 o
X1 o
X0
b Writing A
b = colim Aλ
By Grothendieck’s existence theorem, we get Lb over X = X ×S S.
as a colimit of finite type A-algebras, we have X = lim Xλ , so Lb is defined at some Xλ .
Now if A is excellent, then we have approximation: there exists Aλ → A such that
Lb
/
X
/
b
Spf(A)
Lλ
L
Xλ o
X
Spec(Aλ ) o
Spec(A)
By base changing Lλ to X, we get L on X.
7. Lecture 7 (October 20, 2015): Ashwin Deopurkar
Local acyclicity of smooth morphisms; smooth base change
The smooth base change theorem is impossible to prove! But it is possible to try proving
it.
7.1. Prelude in algebraic topology. In this section, all spaces are “nice” (paracompactness). For any space S, π : S × I n → S is a homotopy equivalence (where I = [0, 1]).
Consequentially, if F is a sheaf of abelian groups on S, then the maps
∼
H i (S, F) → H i (Y, π ∗ F)
are isomorphisms. Equivalently (by the Leray spectral sequence),
(
F if i = 0,
R i π∗ π ∗ F =
0 if i > 1.
In this case we say π is acyclic.
Proposition 7.1 (Smooth base change). Suppose we have a diagram
Xo
π
f0
X ×S T
So
f
π0
T
where f is “smooth”. Then for any shaef F on X, the maps
∗
f ∗ Ri π∗ F → Ri π∗0 (f 0 F)
are isomorphisms.
30
A morphism f is smooth if locally it looks like
∼
T ⊇U
/ In
×V
%
f
S⊇V
Being smooth is preserved under base change.
Proof. This follows from the key fact below.
Fact. Suppose π : Z1 → Z2 and G is a sheaf on Z1 . Then for every point t ∈ Z2 ,
(Ri π∗ G)t ' H i (tubular neighborhood of π −1 (t), G).
Using this, we can “zoom in to a neighborhood of t” and get a fibre diagram
π −1 (U ) o
π −1 (U ) × I n
Uo
U × In
I am going to prove this base change property for the étale topology assuming π is locally
acyclic.
Remark. In the situation of the key fact, (Ri π∗ G)t 6= H i (π −1 (t), G).
7.2. Statement of theorem and structure of the proof. All sheaves will be torsion,
with torsion coprime with the characteristic. Suppose π : Y → X.
Definition 7.2. We say that π is acyclic if for all X 0 → X finite étale and torsion sheaf F
on X 0 (torsion coprime to char(X)), we have
∼
∗
H i (X 0 , F) → H i (Y ×X X 0 , π 0 F),
where
Y o
π
Y 0 = Y ×X X 0
Xo
finite étale
π0
X0
Definition 7.3. We say π is universally acyclic if for all X-schemes X 0 , π 0 : Y ×X X 0 → X 0
is acyclic.
Y 0 = Y ×X X 0
Y o
Xo
X0
Definition 7.4. We say π is (universally) locally acyclic if for all geometric points y of Y
the map
sh
sh
Spec OY,y
→ Spec OX,y
is (universally) acyclic.
31
Theorem 7.5 (Smooth base change). Suppose
Y o
π
g0
Xo
Y0
g
π0
X0
where π is quasi-compact and g is smooth, and F is a torsion sheaf on Y . Then
∼
∗
g ∗ (Ri π∗ F) → Ri π∗0 (g 0 F)
are isomorphisms for all i.
This follows from the
Proposition 7.6.
(1) Same as above except we assume g is universally locally acyclic, instead of smooth.
(2) Smooth morphisms are locally (and hence universally locally) acyclic.
I am going to prove (1) and sketch (2).
7.3. “Proof ”. Since the general case can be reduced to constant sheaves, we may assume
F = Z/`Z. The set-up is
Y o
Y0
Xo
g
X0 o
Spec OX 0 ,t o
Spec k(t)
Proof of (1). We need to show that for all geometric points t of X 0 ,
∼
∗
g ∗ (Ri π∗ F)t → Ri π∗0 (g 0 F)t .
The RHS is
sh
H i (Y 0 ×X 0 Spec OX 0 ,t , Z/`Z) = H i ((Y ×X X 0 ) ×X 0 Spec OX
0 ,t , Z/`Z)
sh
= H i (Y ×X Spec OX
0 ,t , Z/`Z).
The LHS is
sh
(Ri π∗ F)t = H i (Y ×X Spec OX,t
, Z/`Z).
Note
sh o
Y ×X Spec OX,t
sh
Y ×X Spec OX
0 ,t
sh o
Spec OX,t
sh
Spec OX
0 ,t
is a Cartesian diagram and we are done.
Here are some preparations for (2).
Lemma 7.7. If f : X → Y and g : Y → Z are acyclic, then so is g ◦ f .
32
Proof. Consider the diagram
Xo
X0
Y o
Y0
Z
o
Z0
finite étale
Suppose F is a torsion sheaf on Z 0 . Then
H i (Z 0 , F) → H i (Y 0 , π ∗ F) → H i (Z 0 , π ∗ π ∗ F)
is a composition of isomorphisms and therefore an isomorphism.
Lemma 7.8. Suppose
finite étale 0
Y o
Y
X
If Y 0 → X is locally acyclic, then so is Y → X.
Proof. Étale morphisms give isomorphisms on étale local rings.
Lemma 7.9 (Key lemma). Any smooth morphism Y → X can be locally factored as
g0
g1
Y → AnX → An−1
→ · · · → A1X → X
X
where g0 is étale.
Lemma 7.10 (Key algebra lemma). Let A be a strictly Henselian local ring, and A{T } be
the Henselization of A[T ](mA ,T ) . Then
Spec A{T } → Spec A
is acyclic.
Sketch of proof. The steps are:
(
Z/`Z if i = 0,
• Ri π∗ π ∗ Z/`Z =
0
if i > 0.
• Check acyclicity by looking at geometric fibres.
• A{T } is the colimit of finitely generated algebras over A.
For the computation of cohomology:
• i > 1: Any geometric fibre Z of π is a colimit of a curve over a separably closed field.
This implies H i = 0 for all i > 1.
• i = 0: The fact that “A{T } ⊗A k is connected” gives H 0 .
• i = 1: “Abhyankar’s lemma”, where we need to use that ` is coprime to the characteristic of the residue field of A.
33
7.4. Application. Suppose π : Y → X is proper and smooth, and F is constructible and
locally constant with torsion coprime to char(X). Then Ri π∗ F is a locally constant sheaf.
If x0 specializes to x1 , then
H i (Yx0 , F|Yx0 ) → H i (Yx1 , F|Yx1 )
are isomorphisms.
34