alternative lecture notes - Rational points and algebraic cycles

ARITHMETIC APPLICATIONS OF ÉTALE HOMOTOPY THEORY
YONATAN HARPAZ AND TOMER SCHLANK
These are unedited notes taken in real time by Bjorn Poonen, posted with permission of
the speakers. There is no guarantee that this reflects what was actually said or written by
the speakers. Use at your own risk! (Some corrections were made by Kęstutis Česnavičius.
René Pannekoek took the notes during the week of August 6.)
1. July 2 (Harpaz)
Set of equations: a variety X over a number field k. Set of solutions: the set X(k) of
k-rational points
Also one can consider a scheme X over Z or a number ring, and ask about the set X (Z)
of integral points.
Obstruction theory: a general method capable in principle of (sometimes) proving that no
solution exists.
Consider ax 2 +by 2 = 1, where a, b ∈ k × . If a solution exists, then the local Hilbert symbols
(a, b) v are all trivial. In fact, the converse holds too, so one says that we have a complete
obstruction.
The equation can be rewritten in the form X : x 2 − cy 2 = d. The equation x 2 − cy 2 = 1
defines an algebraic group G, in which multiplication is defined by thinking of (x, y) as
x + y √ c. There is an action G × X → X.
Definition 1.1. We will say that a nonempty k-variety with a G-action is a G-torsor over k
(or principal homogeneous space) if the morphism G × X → X × X sending (g, x) to (gx, x)
is an isomorphism of k-varieties.
Example 1.2. The translation action of G on itself defines a G-torsor.
G-torsors over k are classified by the nonabelian cohomology set H 1 et(k, G), a pointed set.
Let X be a scheme. Let ¯x 0 be a geometric point, i.e., a map Spec K → X for some
algebraically closed field K. Then one can define the étale fundamental group π1 et (X, ¯x 0 ),
which is a profinite group.
Example 1.3. If X is a smooth variety over C, then π et
1 (X, ¯x 0 ) is naturally isomorphic to
the profinite completion of the usual fundamental group π 1 (X(C), ¯x 0 ).
Example 1.4. If X = Spec k, then π et
1 (X, ¯x 0 ) ≃ Gal(k/k).
Let X be a geometrically connected k-variety. Let X := X × Spec k Spec k. Let ¯x 0 be a
geometric point of X. Then
X → X → Spec k
Date: July–August 2012.
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gives rise to an exact sequence
1 → π et
1 (X, ¯x 0 ) → π et
1 (X, ¯x 0 ) → Gal(k/k) → 1.
A k-point ι of X would give rise to a homomorphism Gal(k/k) → π1 et (X, ῑ), and there is an
isomorphism π1 et (X, ῑ) ≃ π1 et (X, ¯x 0 ) lying over Gal(k/k) that is well-defined up to conjugation
by an element of π1 et (X, ¯x 0 ).
Conjecture 1.5 (Grothendieck section conjecture). If X is a smooth projective curve over
a number field of genus at least 2, then the section obstruction is complete.
If true, the map from k-points to equivalence classes of sections is a bijection.
Also, if true, the question of deciding whether X has a k-point is decidable (see proof of
Ambrus Pal).
Let X be a geometrically connected variety over a number field k. If X(k v ) = ∅, then
X(k) = ∅. One can compute a finite set S of places such that X(k v ) is nonempty for all
v /∈ S.
Lind: The variety 2y 2 = x 4 − 17 has Q p -points for all p, but no Q-point.
Let A k = ∏ ′
v k v := {(x v ) ∈ ∏ v k v : x v ∈ O v almost always}. Then k ↩→ A k , so if
X(A k ) = ∅, then X(k) = ∅.
Brauer–Manin obstruction:
X(k) ⊆ X Br (A k ) ⊆ X(A k ).
Given u ∈ H 2 (X, G m ) and (x v ) ∈ X(A k ), we obtain x ∗ vu ∈ H 2 (K v , G m ) inv
↩→ Q/Z, and
((x v ), u) := ∑ inv(x ∗ v, u) ∈ Q/Z; the Hasse–Brauer–Noether theorem implies that if (x v )
comes from a k-point, then ((x v ), u) = 0. Define the Brauer set
This contains X(k).
X(A k ) Br := {(x v ) : ((x v ), u) = 0 for all u ∈ H 2 (X, G m )}.
Descent obstruction: Let G be an affine algebraic group over k. Let X be a base variety
over k. A G-torsor over X is a surjective morphism f : Y → X with a G-action over X
(i.e., G × Y → Y respects the projection to X) such that the map G × Y → Y × X Y is an
isomorphism. Isomorphism types are classified by Het(X, 1 G).
Given the class u ∈ Het(X, 1 G) of some torsor f : Y → X and given (x v ) ∈ X(A k ),
we obtain x ∗ vu ∈ H 1 (k v , G). We then ask: does the collection of elements (x ∗ vu) come
from an element of H 1 (k, G)? Let X(A k ) f be the set of (x v ) ∈ X(A k ) satisfying this
condition. Let X(A k ) desc be the intersection of X(A k ) f over all torsors under all affine
algebraic groups. Sometimes one restricts the set of groups considered, to define other sets:
X(A k ) conn , X(A k ) fin . Harari: X(A k ) conn = X(A k ) Br under certain reasonable conditions.
2. July 4 (Schlank)
Given a smooth geometrically connected variety X, a choice of base point gives
1 → π et
1 (X) → π et
1 (X) → Γ k → 1
where Γ k := Gal(k/k). Rational points give sections Γ k → π1 et (X) up to conjugation by
elements of π1 et (X).
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A functorial obstruction is given by specifying a subset X(A) obs between X(k) and X(A)
for each X such that for any morphism f : X → Y , we have f(X(A) obs ) ⊆ Y (A) obs .
Examples: X(A) Br , X(A) desc , X(A) fin , X(A) fin-Ab , X(A) conn , . . . .
Suppose that we have f : Y → X a G-torsor over X, where G is a finite étale group.
Given x 0 ∈ X(k), let c(x 0 ) be the class of f −1 (x 0 ) in H 1 (k, G). Then we can twist to obtain
f c(x 0
: Y c(x0) → X. We obtain
X(k) =
∐
f σ (Y σ (k)) ⊆ ⋃ f σ (Y σ (A)) ⊆ X(A).
σ∈H 1 (k,G)
Define X f (A) := ⋃ f σ (Y σ (A)). Given an obstruction obs, define
X f,obs (A) := ⋃ f σ (Y σ (A) obs ) ⊆ X(A) obs .
For example, taking obs = Br gives the étale-Brauer obstruction set, which in our notation
is X(A) fin,Br .
X(A) fin
X(A) fin-Ab
X(A)
X(k)
X(A) fin,Br
X(A) Br
Let p: E → B be a fiber bundle with nonempty connected fiber F (the spaces are CWcomplexes,
simplicial sets, or manifolds): i.e., B is covered by open sets U such that p −1 (U) ≃
F × B over B. We can lift any point of B to a point of E. After lifting points, given a
path in B connecting two points, we can lift it to a path between their lifts (first lift without
worrying about the end of the path, and then modify a small piece of the end so that it ends
where it should).
We may assume that B is triangulated so that each simplex is contained in some U. Let
be a solid triangle; then a section of F × → is a map → F . We are given the
section above the boundary of the triangle; to fill in the lift, the obstruction lives in π 1 (F ).
We get a 2-cochain in C 2 (B, π 1 (F )). But this is not exactly the obstruction to the existence
of a section, because we could also change the choices made along the way.
C 1 (B, π 1 (F )) → C 2 (B, π 1 (F )) d → C 3 (B, π 1 (F ))
The kernel modulo image at the center is the definition of cohomology H 2 (B, π 1 (F )). In fact,
the 2-cochain we constructed above lies in ker d. Also, changing the choice of lifts of paths
corresponds to modifying it by an element of C 1 (B, π 1 (F )), and its differential measures the
change in our 2-cochain. Thus we obtain a canonical obstruction lying in H 2 (B, π 1 (F )).
After lifting the 2-skeleton, to lift the 3-skeleton, we obtain an obstruction in H 3 (B, π 2 (F )).
Thus we get a sequence of obstructions, in H 2 (B, π 1 (F )), H 3 (B, π 2 (F )), H 4 (B, π 3 (F )),
. . . , each one defined if the previous ones vanish (and one makes a choice).
We would like to do the same for X → Spec Q, and even for arbitrary morphisms of
schemes X → S.
Let Sec(E → B) be the space of all sections, which is a topological space. We would
like to compute the homotopy groups π ∗ Sec(E → B). These will be computed by a second
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quadrant spectral sequence
We care only about the terms with t ≥ s.
E 2 st := Hs (B, π t (F )) =⇒ π t−s Sec(E → B).
Another way to think about sections of X → Spec k (for perfect k): they are elements of
X(k) Γ k .
Let X be a space with an action of a group G. We would like to understand the homotopy
groups of X G . This is impossible, because X G is not invariant up to homotopy, as the
following example shows:
Compare X = R with translation action of Z with a point with trivial action of Z. Here
X G is empty in the first case, and a point in the second case.
Solution: define homotopy fixed points.
A fixed point is the same as a G-equivariant map from a point (with trivial G-action) to
X. We should be allowed to replace the point with a contractible space with an action of
G. When are two such maps to be considered the same? To explain this, use the universal
contractible space EG with G-action: for any contractible space C with G-action, there is
a G-equivariant map EG → C, unique up to homotopy. The space EG is unique up to
homotopy. Another equivalent definition: EG is a contractible space with free G-action.
(Free means that for every x ∈ X there exists a neighborhood U of x such that gU ∩ U for
all g ≠ 1 in G.)
Example 2.1. EZ = R with translation action. E(Z × Z) = R 2 with translation action.
Example 2.2. S ∞ := ⋃ S n ⊆ R ∞ is E(Z/2Z): the nontrivial element acts as the antipodal
map.
Define X hG := Map G (EG, X), with the compact-open topology. Choosing a different EG
gives a homotopic space X hG . This is functorial in X.
Properties:
(1) There is a natural map X G → X hG .
(2) Let C be a contractible space with a (not necessarily free) G-action. Every G-map
C → X defines an element of π 0 (X hG ).
(3) If f : X → Y is a weak equivalence (i.e., isomorphism on all homotopy groups), then
f hG : X hG → Y hG is a weak equivalence. There is a second quadrant spectral sequence
E 2 st := Hs (G, π t (X)) =⇒ π t−s (X hG ).
We have an obstruction to the existence of a homotopy fixed point:
H 2 (G, π 1 (X)), H 3 (G, π 2 (X)), . . . .
This is a special case of what we did before: define the Borel construction
(X × EG)/G → EG/G =: BG = K(G, 1).
Here EG is the universal cover of BG. We have π 1 (BG) = G and π n (BG) = 0 for n > 1.
Also, H n (G, A) = H n (BG, A).
Now (X × EG)/G → BG is a fibration with fiber X. Sections are the same as homotopy
fixed points.
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We have
1 = π 2 (EG/G) → π 1 (X) → π 1 ((X × EG)/G) → G → 1
if X is connected. If there is a section, the surjection in the short exact sequence above
has a section. If π 1 (X) is abelian, the obstruction is given by an element in H 2 (G, π 1 (X)).
This specializes to the Grothendieck section obstruction. This motivates looking for an
obstruction in
H 3 (Γ k , π et
2 (X)),
whatever that means.
Plan: Build a functor F from k-varieties to topological spaces with Γ k -action such that
F (Spec k) is contractible (with some Γ k -action, not necessarily free). Then every section
of X → Spec k gives a Γ k -equivariant map F (Spec k) → F (X), which gives an element of
π 0 (F (X) hΓ k . So if π0 (F (X) hΓ k = ∅, then there is no section. In fact, one gets a sequence of
obstructions π 0 (F (X) hΓ k ), H 2 (Γ k , π 1 (F (X)), H 3 (Γ k , π 2 (F (X)).
3. July 9 (Schlank)
What is a limit of diagrams in a category?
Let C be a category. Let A, B be two objects of C. Recall that a product A × B is an
object with maps to A and B such that given any X with maps to A and B, arises by
composition with a unique map X → A × B. More generally, a fiber product of A → D and
B → D is an object X that is universal for maps to A and B commuting with the given
maps. The limit of · · · A 2 → A 1 → A 0 is an object X with compatible maps to all the A i
that is universal. Given a group G acting on X, the object of fixed points X G is an object
with a map to X that is unchanged when composed with any g ∈ G, and that is universal
for this property.
Let D be a small category. A diagram is a functor F : D → C. Given F , the limit lim F is ←−
an object of C equipped with a map α d : lim F → F (d) for all d ∈ D, compatible with each
←−
morphism in D (i.e., if d 1 → d 2 in D, then α d1 and α d2 form a commutative triangle with
F (d 1 → d 2 ), such that for any other X with maps β d : X → F (d) satisfying the compatibility
conditions, there is a unique X → lim ←−
F that when composed with the α d yield β d .
Given E → A and E → B, the colimit A ∪
E
B is an object X with maps A → X and
B → X compatible with the maps from E, such that for any other such X ′ with maps, the
maps arise from the maps to X by composition with a unique map X → X ′ . One gets a
general notion of colimit, by using any functor from a small category.
If D is a small category and C is any category, the functor category C D is a category
whose objects are functors F : D → C and whose morphisms are natural transformations.
A diagram is an object of C D , i.e., a functor from D to C. The limit of a diagram, if it exists,
is an object of C: we want a functor lim: C D → C.
←−
We have const: C → C D sending C 0 to the functor sending each d ∈ D to C 0 and each
morphism of D to id C0 . We want there to be a bijection
functorially for X ∈ C.
Hom C D(const(X), diag) ≃ Hom C (X, lim ←−
diag)
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Loosely speaking, given functors L: A → B and R: B → A, they are called adjoint
functors if there is a bijection Hom B (L(a), b) = Hom A (a, R(b)) varying functorially in a ∈ A
and b ∈ B.
Definition 3.1. A pair of functors L: A → B and R: B → A together with two natural
transformations u: id A → R ◦ L (called the unit) and c: L ◦ R → id B (called the co-unit) is
called an adjunction if for every a ∈ A the composition
is id L(a) , and for every b ∈ B the composition
is id R(b) .
and
Given an adjunction as above, we get
L(a) L◦u
−→ LRL(a) c◦L
−→ L(a)
R(b) u◦R
−→ RLR(b) R◦c
−→ R(b)
Hom(L(a), b) → Hom(RL(a), R(b) u → Hom(a, R(b))
Hom(L(a), b) c ← Hom(L(a), LR(b)) ← Hom(a, R(b)).
Example 3.2. Let U be the functor Sets ← Groups forgetting the group structure. (Do
not confuse U with the unit.) Let Fr: Sets → Groups be the functor sending a set to the
free group on the set. Then
and in fact we get an adjunction.
Hom Groups (Fr(S), G) = Hom Sets (S, U(G)),
Example 3.3. Let R be commutative ring. Let A be an R-module. Then ⊗A: R-modules →
R-modules and Hom(A, −) in the other direction are adjoint functors: in particular,
Hom(B ⊗ A, C) = Hom(B, Hom(A, C)).
Example 3.4. Let R → S be a ring homomorphism. Then ⊗ R S : R-modules → S-modules
is left adjoint to the functor U in the opposite direction that maps an S-module to the
R-module obtained by composing the action with R → S.
Let D be a small category. Let const: C → C D be the functor taking an object to the
constant diagram. A limit functor is a functor lim: C D → C that is right adjoint to const.
←−
Notation: Write L ⊣ R if (L, R) are a pair of adjoint functors. Thus const ⊣ lim. Similarly
←−
lim
−→ ⊣ const.
A limit functor does not always exist. In fact, individual limits need not exist. For
example: let D be the 3-object category with two morphisms to the same object. If C is the
same category, the identity functor D → C does not have a limit.
Call C complete if all limits of diagrams D → C exist, for every small category D. A
functor R: C → D is called continuous if it respects limits. There is always a morphism
R ( lim F ) → lim R◦F , from the universal property of the latter. To say that R is continuous
←− ←−
means that this morphism is always an isomorphism.
Theorem 3.5. If R has a left adjoint, then R is continuous.
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Proof. We have
Conclude by Yoneda.
Hom(A, lim R ◦ F ) = lim Hom(A, RF (d))
←− ←−
d∈D
= lim Hom(L(A), F (d))
←−
d∈D
= Hom(L(A), lim ←−
F (d))
= Hom(A, R lim ←−
F (d)).
□
Similarly, a functor is cocontinuous if it commutes with colimits. If L has a right adjoint,
then L is cocontinuous.
Does the converse to Theorem 3.5 hold? Not quite, but essentially yes.
Theorem 3.6. Let C, D be complete, and let R: C → D be continuous. If some smallness
requirement holds on C and D, then R has a left adjoint.
(Categories arising in practice satisfy the small requirement. The problem can be solved
also by using Grothendieck’s universes.)
Proof. Suppose that we have R: C → D. Given d ∈ D, define the comma category d/R
whose objects are pairs (c ∈ C, d → R(c)) and whose morphisms from c ′ 1 → R(c 1 ) to
c ′ f
2 → R(c 2 ) are morphisms c 1 → c2 such that the associated R(f): R(c 1 ) → R(c 2 ) makes a
triangle with the two given morphisms. Define L(d) as a limit over (c, d → R(c)) ∈ d/R of c;
this makes sense if the category d/R is small. We now construct the unit and co-unit. The
unit id D → R◦L is defined by d → R(lim c) = lim R(c). The co-unit L◦R → id C
←−d→R(c) ←−d→R(c)
is defined by lim ˜c → c.
□
←−R(c)→R(˜c)
Assume that D is a full subcategory of C. Let i: D → C be the inclusion functor. Assume
that C and D are complete and that i is continuous. Then we have a left adjoint L ⊣ i.
Then L is called a localization functor.
Example 3.7. Let C be the category of all groups. Let D be the full subcategory of abelian
groups. The inclusion functor is continuous (limits of diagrams of abelian groups in the
category of groups are abelian groups). Then L is the functor sending each group G to its
abelianization.
Example 3.8. If C is the category of presheaves and D is the full subcategory of sheaves,
then L is the sheafification functor.
Example 3.9. Let C be the category of topological spaces, and let D be the full subcategory
of compact Hausdorff topological spaces. Then L is the Stone-Čech compactification functor.
Example 3.10. Let C be the category of abelian groups. Let D be the full subcategory of
Q-vector spaces. Then L is ⊗Q.
f
Call c 0 → c1 in C a D-equivalence if L(f) is an isomorphism in D. Then inverting all the
D-equivalences in C yields the category D.
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4. July 11 (Harpaz)
Today: Ind-categories and pro-categories
Let Set be the category of sets. If A is a set, then A is the coproduct of singletons:
A = ∐ a∈A
{a}. It is also a colimit of finite sets.
Let Fin be the full subcategory of finite sets. Given A ∈ Set, let Fin /A be the category
of finite sets over A, i.e., the category of pairs (S, ρ) where S is a finite set and ρ: S → A;
morphisms (S, ρ) → (S ′ , ρ ′ ) are morphisms S → S ′ compatible with the morphisms ρ and
ρ ′ to A. This is an example of a comma category. Observe: A ≃ colim (S,ρ)∈Fin/A S, i.e., the
colimit of the diagram Fin /A → Sets sending (S, ρ) to S.
Definition 4.1. A category I is called filtered if
• I is not empty;
• for all two objects i, j ∈ I, there exists k ∈ I that admits maps i → k and j → k;
• for all f, g : i ⇒ j, there exist k ∈ I and a map h: j → k such that h ◦ f = h ◦ g.
Observation: If I has finite colimits, then I is filtered.
Set, Fin, Fin A are filtered.
Thus A is a filtered colimit.
A filtered diagram is a diagram I → C where I is a small filtered category.
Definition 4.2. Let C be a category that has filtered colimits (i.e., every filtered diagram
to C has a colimit in C). Say that c ∈ C is small (or compact) if for every filtered diagram
F : I → C, colim i∈I Hom(c, F (i)) → Hom(c, colim i∈I F (i)) is an isomorphism of sets.
Example 4.3. The small objects in Set are exactly the finite sets.
Example 4.4. The small objects in Groups are exactly the finitely presented groups.
Let F : I → Set and G: J → Set be filtered diagrams taking values in Fin. Then
Hom Set (colim i∈I F (i), colim j∈J G(j)) = lim
i∈I
Hom Set (F (i), colim j∈J G(j)) = lim
i∈I
colim j∈J Hom Fin (F (i), G(j))
This situation happens not only for Set; let’s generalize.
Definition 4.5. Let C be a category. Let Ind(C) be the category with
1) objects: filtered diagrams F : I → C
2) Given F : I → C to G: J → C,
Hom Ind(C) (F, G) = lim
i∈I
colim j∈J Hom C (F (i), G(j)).
Let {∗} be the one-point one-morphism category. The category C embeds as a full subcategory
of Ind(C): sending c to the filtered diagram {∗} → C sending ∗ to c.
Example 4.6. The functor colim: Ind(Fin) → Set is an equivalence of categories.
Example 4.7. Similarly, Ind(FinPresGroups) ≃ Groups. The same is true for abelian
groups, rings, small categories, modules.
Example 4.8. It turns out that the category Top of topological spaces is not Ind of any
category. If one takes Ind of the category of compact spaces, one does not quite get the
category of compactly generated topological spaces, but just some category that admits a
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(non-faithful) essentially surjective functor to the category of compactly generated topological
spaces. If one takes Ind of the category of Hausdorff compact spaces, one gets the
category of Hausdorff compactly generated topological spaces.
Basic properties of Ind(C):
1) (a) Ind(C) has filtered colimits. In fact, Ind(C) should be thought of as being obtained
by formally adding filtered colimits to C.
(b) If C has finite colimits, then Ind(C) has finite colimits, and the functor C → Ind(C)
preserves finite colimits. Later we will show that Ind C has all colimits.
2) Ind(C) is the universal category with filtered colimits and a filtered-colimit-preserving
functor from C to it. I.e., given another such functor C → D there is a filtered-colimitpreserving
functor from Ind(C) → D, unique in an appropriate sense.
3) The objects of C are small in Ind(C). (But Ind(C) may have more small objects.)
Example of the dual situation: the category ProFinGr of profinite groups. If Γ is a
profinite group, Let FinGr Γ/ be the category of (G, ρ) where G is a finite group and ρ: Γ → G
is a homomorphism of profinite groups (continuous). Then Γ → ∼ lim (G,ρ)∈FinGrΓ/ G.
Call I cofiltered if I op is filtered:
1) I is nonempty
2) For each i, j ∈ I, there exists k ∈ I with maps k → i, k → j.
3) for all f, g : j ⇒ i, there exist k ∈ I and a map h: k → j such that f ◦ h = g ◦ h.
Definition 4.9. Let C have cofiltered limits. An object c ∈ C is co-small if for every
cofiltered diagram F : I → C the natural map colim i∈I Hom(F (i), C) → Hom(lim i∈I F (i), C)
is an isomorphism of sets.
If F : I → ProFinGr and G: I → ProFinGr are cofiltered and take values in FinGr,
then
Hom ProFinGr (lim F (i), lim G(j)) = lim colim i∈I Hom FinGr (F (i), G(j)).
i∈I j∈J j∈J
Let us formalize this situation.
Definition 4.10. Let C be a category. Define Pro(C) as follows:
(1) The objects of Pro(C) are cofiltered diagrams F : I → C.
(2) Given two objects F : I → C and G: J → C, define
Alternatively, Pro(C) = Ind(C op ) op .
Hom Pro(C) (F, G) := lim
j∈J
colim i∈I Hom C (F (i), G(j)).
There is a canonical fully faithful embedding C → Pro(C).
1) Pro(C) has cofiltered limits, and is universal as such
2) If C has finite limits, then Pro(C) has finite limits and C → Pro(C) preserves them.
3) All the objects of C are co-small in Pro(C).
Suppose that (C i ) i∈I and (D j ) j∈J are cofiltered systems (also called inverse systems). How
do we compute their product? We can replace (C i ) i∈I by the isomorphic object (C i ) (i,j)∈I×J
of Pro(C) (the fact that they are isomorphic uses that J is cofiltered). Do the same to
(D j ) j∈J . Then (C i × D j ) (i,j)∈I×J is the product in Pro(C).
This can be generalized to show that that Pro(C) has finite limits.
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Theorem 4.11. If C has finite limits, then Pro(C) has all limits, i.e., it is complete.
Proof. Any limit can be expressed as an equalizer (limit of diagram A ⇒ B) of a pair of
morphisms between products: specifically, lim F (i) is the equalizer of
∏
F (i) ⇒
∏
F (i ′ )
i∈I
ρ=(i→i ′ )∈I
in which the first morphism is given by of ∏ i∈I F (i) pr i
−→ ′
F (i ′ ), and the second morphism is
given by of ∏ i∈I F (i) pr i
−→ F (i) −→ F (ρ)
F (i ′ ).
Thus it is enough to show that Pro(C) has arbitrary products (over any index set) and
equalizers. Since Pro(C) has finite limits, it has finite products and equalizers. Arbitrary
products are filtered limits of finite products:
∏
∏
A i =
A i .
i∈P
lim
finite S ⊆ P
Further claim: If C and D have finite limits and F : C → D preserves finite limits, then
Pro(F ): Pro(C) → Pro(D) preserves all limits.
Recall from last lecture that a functor preserving all limits (modulo smallness conditions)
has a left adjoint. So in the situation above, we obtain a left adjoint P L: Pro(D) → Pro(C)
of Pro(F ).
i∈S
5. July 16 (Schlank)
Today: Combinatorial and categorical approach to homotopy theory
Example 5.1. A circle, annulus, and solid torus are homotopy equivalent.
Homotopy theory is the study of the ways that contractible spaces can be glued together
to give global objects.
Definition 5.2. Let X be a space and let (U α ) φ∈A be a cover of X. Say that (U α ) is excellent
if for every finite nonempty subset I ⊆ A, the intersection U I := ⋂ i∈I U i is either contractible
or empty.
Example 5.3. The circle has an excellent cover given by three overlapping half-circles.
Proposition 5.4. Let X be a paracompact space. Let (U α ) be an excellent cover of X. Then
X is homotopically equivalent to the nerve of (U α ), where the nerve is the space constructed
by
(1) taking a point for every U α ,
(2) connecting U α and U β by a line segment if U α ∩ U β ≠ ∅,
(3) filling in a triangle if U α ∩ U β ∩ U γ ≠ ∅,
(4) etc.,
with the colimit topology of the Euclidean topologies on the simplices.
The circle, annulus, and solid torus have combinatorially equivalent excellent covers, so
they are homotopically equivalent.
10
□

Definition 5.5. Let X be a space and let (U α ) φ∈A be a cover of X. Say that (U α ) is good
if for every finite nonempty subset I ⊆ A, the intersection U I := ⋂ i∈I U i is a coproduct
(disjoint union) of contractible spaces.
Definition 5.6. We denote by ∆ 0 the category whose objects are nonempty finite ordered
sets [n] := {0 < 1 < 2 < . . . < n} and whose morphisms are strictly increasing maps.
Definition 5.7. We call the functor category Set ∆op 0
the category of semi-simplicial sets.
Given a semi-simplicial set X, we call X n := X([n]) the set of n-simplices of X.
Example 5.8. X 0 = {y, b}, X 1 = {L, R}, X n = ∅ for n ≥ 2, with morphisms [0]
1
[0] → [1] inducing two maps X 1 → X 0 .
Choose a well-ordering of the index set A of a cover. Define the nerve of a cover by
N((U α )) n :=
∐
π 0 (U I ).
Define
I⊆A, |I|=n
|∆ n | := {(x 0 , . . . , x n ) ∈ R n+1 : x i ≥ 0, ∑ x i = 1}.
Given a semi-simplicial set X := ∆ op
0 → Set.
⎛
|X| := coequalizer ⎝
∐
f : [k]→[m]
X([m]) × ∆ k ⇒ ∐ [n]
⎞
X([n]) × ∆ n ⎠
0
→ [1]
in the category CGHS (compactly generated Hausdorff topological spaces, the Ind-category
of the category of compact Hausdorff spaces). The two maps in the coequalizer diagram
above: given f, use id ×∆(f): X([n]) × ∆ k → X([n]) × ∆ n or X(f) × id: X([n]) × ∆ k →
X([k]) × ∆ k .
Example 5.9. According to the definitions, there is no map of semi-simplicial sets from a
solid triangle to a point, because the point has no 2-simplices.
Definition 5.10. Let ∆ be the category whose objects are [n] (the same as for ∆ 0 ), but
whose morphisms are (weakly) increasing maps.
Definition 5.11. The category of sisets is defined to be Set ∆op .
Define |X| as before.
The point in Set ∆op is the functor sending [n] to a point for all n.
We have that | |: Set ∆op → CGHS commutes with finite limits. In particular, |X × Y | ≃
|X| × |Y |. (The analogue for semi-simplicial sets is false, because the product of semisimplicial
sets X and Y has simplices only in dimensions that X and Y have.
For 0 ≤ i ≤ n+1, define the “skip i” map d i : [n] → [n+1]. These are called the face maps.
For 0 ≤ i ≤ n, define the “repeat i” map s i : [n + 1] → [n]. These are called the degeneracy
maps. These generate the category ∆. Say that an n-simplex x ∈ X n is nondegenerate if it
is not in the image of any X(s i ): X n−1 → X n .
11

Define the nerve N ∆ ((U α )) of a covering U := ∐ α∈A U α → X as the simplicial set with
⎛
⎞
N ∆ ((U α )) n := π 0
⎜
⎝ U × · · · × U⎟
X X
⎠ .
} {{ }
n+1
Given f, g : X → Y , we say that f ∼ g if there exists H : I × X → Y such that H| 0 = f
and H| 1 = g.
Define the simplicial set ∆ n : ∆ op → Set by ∆ n ([m]) := Hom ∆ ([m], [n]). For example, ∆ 0
is a point.
Definition 5.12. Let A and B be simplicial sets, and let f : A → B. We say that f ∼ st
g
if there exists H : A × ∆ 1 → B such that H| 0 = f and H| 0 = g. This is reflexive, but not
symmetric or transitive. Define the relation ∼ as the equivalence relation generated by ∼.
st
Definition 5.13. Let A and B be simplicial sets. Define the mapping space Map(A, B) ∈
Set ∆op
as the simplicial set given by Map(A, B) n := Hom Set ∆ op (∆ n × A, B).
For X, Y, Z ∈ CGHS, we have the exponential law
Map(X × Y, Z) ≈ Map(Y, Map(X, Z)).
Let A, B be simplicial sets. Let f, g : A → B. The following are equivalent:
(1) f ∼ g
(2) f and g (or rather, the realization of the corresponding two points in Map(A, B) 0 )
lie in the same connected component of | Map(A, B)|.
If A and B are simplicial sets, there is a map
φ: | Map(A, B)| → Map Top (|A|, |B|).
The map from a circle to a circle that wraps around twice does not come from a simplicial
map (unless one subdivides). Dan Kan solved this problem as follows.
Definition 5.14. Let Λ n k be ∆n minus the interior and k th face. A Kan simplicial set is a
simplicial set A such that for every diagram of the form
the diagonal map exists.
Λ n k
∆ n
A
∃
Theorem 5.15. If B is Kan, then φ above is a homotopy equivalence.
6. July 18 (Harpaz)
If C and D are categories with finite limits, and F : C → D is a functor that preserves
them, then Pro(F ): Pro(C) → Pro(D) preserves all limits.
The inclusion functor FinGr → Groups preserves only finite limits, but we get an adjoint
P L: Pro(Groups) → Pro(FinGr).
12

Let F : C → Set be a functor. Suppose that F preserves all limits. Then F has a left
adjoint L: Set → C. Let ∗ be a singleton. What is L(∗)? We have
This proves the nontrivial half of
Hom C (L(∗), c) ≃ Hom Set (∗, F (c)) ≃ F (c).
Theorem 6.1. Let C be a category with all limits. A functor F : C → Set is representable
if and only F it preserves all limits.
What if F preserves only finite limits? Then Pro(F ): Pro(C) → Pro(Set) preserves all
limits, and we get P L: Pro(Set) → Pro(C). For c ∈ C ↩→ Pro(C),
Hom Pro(C) (P L(∗), c) ≃ Hom Pro(Set) (∗, Pro(F )(c)) ≃ F (c).
A functor F : C → Set is pro-
Theorem 6.2. Let C be a category with finite limits.
representable if and only if F preserves finite limits.
Claim 1: The functor T : Pro(C) → (Set C ) op sending (C i ) to Hom Pro(C) (C i , −) is fully
faithful.
Claim 2: The essential image of T consists of all left exact functors. (Left exact means
“preserves finite limits”.)
Claim 3: For every C with finite limits, the category of left exact functors C → Set is
complete and cocomplete.
Proof.
(a) Set C is complete and cocomplete.
(b) If E is a complete cocomplete category and E ′ ⊆ E is a full subcategory closed under
limits, then E ′ is cocomplete.
□
Back to simplicial sets.
Take a convenient category of nice spaces, such as Top. Let S be the category of simplicial
sets. We have the realization functor S → Top.
Definition 6.3. For X, Y ∈ S , let [X, Y ] S
classes for ∼).
be the set of homotopy classes (equivalence
How does this compare with [|X|, |Y |] Top ? There is an obvious map [X, Y ] S → [|X|, |Y |] Top .
Theorem 6.4. If B is Kan, this map is a bijection of sets.
Theorem 6.5. There exists a functor Ex ∞ : S → S and a natural transformation id →
Ex ∞ such that
1) Ex ∞ X is Kan
2) The map X → Ex ∞ (X) induces a homotopy equivalence on realizations.
Example 6.6. If X is the circle having one nondegenerate 0-simplex and one nondegenerate
1-simplex, Ex ∞ (X) adds a 1-simplex that wraps around the circle n times, for each n, and
adds appropriate 2-simplices, etc.
13

Let X and Y be two nice spaces. A weak equivalence is a map f : X → Y that induces
an isomorphism on π 0 , and on π n with respect to all x ∈ X (and f(x) ∈ Y ). This is not an
equivalence relation. Two spaces are weakly equivalent if they can be connected by a zigzag
of weak equivalences.
Definition 6.7. Let Sing : Top → S be the functor sending a topological space Z to
Hom Top (|∆ n |, Z).
Sing is right adjoint to the realization functor | |.
Claim 1: The counit map | Sing(Z)| → Z is a weak equivalence.
Claim 2: If X, Y ∈ S are such that |X|, |Y | are weakly equivalent, then |X|, |Y | are
homotopy equivalent.
Definition 6.8. A relative category is a category C together with a subcategory W ⊆ C
such that W contains all objects (and all identities).
Examples 6.9.
(1) C = Top and W is the subcategory of all homotopy equivalences
(2) C = Top and W is the subcategory of all weak equivalences
(3) C = S and W contains f : X → Y if |f| is a homotopy equivalence (or equivalently
by Claim 2, weak equivalence)
(4) C is the category of chain complexes of abelian groups, and W is the subcategory of
chain equivalences (there are variants: bounded below, bounded above, unbounded
on both sides)
(5) C is the category of chain complexes of abelian groups, and W is the subcategory of
quasi-isomorphisms
(6) C is the category of G-spaces and W contains f : X → Y if the underlying map of
spaces is a weak homotopy equivalence.
Say that (C, W ) is weakly homotopical if
(i) W contains all isomorphisms, and
7. July 23 (Schlank)
(ii) for every pair of composable maps X → f Y
is the third.
Examples 7.1.
g → Z, if two of f, g, g ◦ f are in W , then so
(1) C = Top and W is the subcategory of all homotopy equivalences
(2) C = S and W contains f : X → Y if |f| is a homotopy equivalence (or equivalently
by Claim 2, weak equivalence)
(3) C is the category of chain complexes of abelian groups, and W is the subcategory of
quasi-isomorphisms
(4) C is the category of G-spaces and W contains f : X → Y if the underlying map of
spaces is a weak homotopy equivalence.
(5) C is (Set ∆op ) G (the category of simplicial sets equipped with a G-action) and W is
the subcategory of weak equivalences on the underlying category Set ∆op .
14

A relative functor F : (D, W 1 ) → (C, W 2 ) is a functor D → C such that F (W 1 ) ⊆
W 2 . Denote by RelCat the category of relative categories. There is a forgetful functor
F : RelCat → Cat (forget W ). There is also a functor U : Cat → RelCat sending C to
(C, Iso(C)). The functor U has a left adjoint Ho: RelCat → Cat:
Hom RelCat ((C, W ), U(D)) = Hom Cat (Ho(C), D)
(Here Ho(C) means Ho(C, W ).) Another notation: C[W −1 ] = Ho(C), because it is the
universal category in which the morphisms of W become isomorphisms.
We get equivalences of categories Ho(Set ∆op ) ↔ Ho(Top), but something is lost: these
categories do not have limits and colimits. Also, passing to the homotopy category forgets
too much: the morphisms in this category are just π 0 (Map(X, Y )), but we may be interested
in π 1 (Map(X, Y )), etc.
Let TopCat be the category of categories whose Hom sets are topological spaces and such
that composition is continuous.
Dwyer–Kan localization of a relative category (also called hammock localization): We define
L H : RelCat → TopCat, sending C to a category L H C with the same objects and whose
morphisms between a, b ∈ C are a simplicial set Map L H
C
(a, b). Take the set of zigzags
a ← c 1 → c 2 ← · · · c n → b
where all the left arrows are in W up to equivalence, where equivalence includes
• reversing isomorphisms (replacing an isomorphism by its inverse in the other direction)
• replacing pairs of consecutive maps by their composition (and vice versa)
• adding or removing identity morphisms
An edge is an equivalence class of diagrams
a
c 0
W
c 1
W
· · · c n
W b
c ′ 0 c ′ 1
· · · c ′ n
in which the directions of the arrows in the top row are arbitrary but match the directions
in the bottom row. A 2-simplex is an equivalence class of diagrams obtained by “composing”
two edges (to form a wider hammock), and so on.
We have π 0 (L H C ) ≃ Ho(C). (Here π 0(L H C ) is the category with the same objects, but in
which each Hom space has been replaced by its set of connected components.)
The space Map L H
C
(a, b) is called the derived mapping space. There is a map of sets
Hom C (a, b) → π 0 (Map L H
C
(a, b)).
Definition 7.2 (Quillen 1969). A model category is (C, W, FIB, COF) where
(1) C is a complete co-complete category,
(2) W , FIB, COF are subcategories with the same objects as C and containing the
subcategory of isomorphisms
(3) W satisfies the 2-out-of-3 property
15

(4) W , FIB, COF are closed under retracts
(5) COF ∩W ⊥ FIB, COF ⊥ FIB ∩W
(6) Every map f : A → B can be functorially decomposed in either of the following two
ways:
C
∼
A
f
B
or
where terminology is as below.
Given
A
C
g
∼
C
f
A
D B D
where the horizontal compositions are the identity on C and D, respectively, we call g a
retract of f. To say that W is closed under retracts means that for any such diagram, if
f ∈ W , then g ∈ W .
Say that f has the left lifting property with respect to g or that g has the right lifting
property with respect to f (and write f ⊥ g) if for all horizontal maps completing a diagram
f
f
A C
∃ g
B D,
a diagonal map exists.
We write A → ∼ B for weak equivalence, A ↠ B for FIB, A ↩→ B for COF, A ↠ ∼ B for
FIB ∩W , and A ↩→ ∼
B for COF ∩W .
(The definition above is self-dual.)
Definition 7.3. An object A ∈ C is called cofibrant if ∅ → A is in COF. An object A ∈ C
is called a fibrant if A → ∗ is in FIB.
Lemma 7.4. If C is a model category, then every object A ∈ C is weakly equivalent to an
object that is simultaneously cofibrant and fibrant.
Proof. Factor A → ∗ as
A
∼
A f
f
Dually define A c . Then A is weakly equivalent to (A f ) c , which is both fibrant and cofibrant.
□
16
B
C
g
∗

Example 7.5. C = Top, W is the subcategory of weak equivalences, FIB is the subcategory
of Serre fibrations (have right lifting property with respect to all A ↩→ A × I where A is a
CW-complex), and COF = ⊥ (FIB ∩W ).
Example 7.6. Fix a ring R. C is the category of ≥ 0 complexes of R-modules, W is the
subcategory of quasi-isomorphisms, FIB is the subcategory of maps that are onto for k > 0,
COF is the subcategory of monomorphisms with projective cokernel for k ≥ 0.
Example 7.7. Fix a ring R. C is the category of ≤ 0 complexes of R-modules, W is
the subcategory of quasi-isomorphisms, FIB is the subcategory of maps that are onto with
injective kernelfor k ≤ 0, COF is the subcategory of maps that are monomorphisms for
k < 0.
Example 7.8. C is the category of simplicial sets, W is the subcategory of weak equivalences,
FIB is the subcategory of Kan fibrations (maps of simplicial sets that have the right lifting
property with respect to all horns Λ n k ↩→ ∆n ; then being fibrant is the same as being a Kan
simplicial set), COF is the subcategory of monomorphisms levelwise.
Example 7.9. C is the category of simplicial sets with G-action, W is the subcategory
of weak equivalences, FIB is the subcategory of Kan fibrations, COF is the subcategory of
monomorphisms with free action on B − im(A).
Example 7.10. C is the category of simplicial sets with G-action, W is the subcategory of
weak equivalences, COF is the subcategory of monomorphisms, FIB = (COF ∩W ) ⊥ .
8. July 25 (Schlank)
If A, B ⊆ C are subcategories, write C = B ◦ A if every morphism of C can be factored
as a morphism in A followed by a morphism in B (not necessarily uniquely, or functorially).
Lemma 8.1.
(a) COF = ⊥ (FIB ∩W )
(b) FIB = (COF ∩W ) ⊥
(c) COF ∩W = ⊥ FIB
(d) FIB ∩W = COF ⊥ .
Proof. We prove the first statement (the others are similar). Suppose that f : A → B is in
⊥ (FIB ∩W ). It can be factored as A ↩→ C ∼ ↠ B. Make a commutative square:
A C
l
B B,
∼
Then
A
A
A
B l C ∼ B
Then f is in R(A ↩→ C), and A ↩→ C is in COF, so f is in COF.
f
17
f
□

It is common when describing a model category to specify W and only one of FIB and
COF — the lemma shows that the other is determined.
Let A ∈ C. Let f : A ∐ A → A. It factors as A ∐ A ↩→ A × I ∼ ↠ A. (This is the definition
of A × I, which is a single symbol; there is no I.) This A × I is called the universal cylinder
object of A.
More generally, given a factorization A ∐ A → A ∧ I ∼ → A, call the object A ∧ I a cylinder
object.
Given two maps f, g : A → B, we will say that f is strictly left homotopic to g by A ∧ I
and write f l ∼ g if there exists H completing the diagram below:
A ∐ f ∐ g
A B
H
A ∧ I.
The dual definition: Let X ∈ C. Let ∆: X → X × X. We have X ∼
↩→ X I ↠ X × X and
X I is called the universal path object. Given any decomposition X ∼ → X ∧I → X × X, call
X ∧I a path object.
Given f, g : Y → X, write f r ∼ g (right homotopic) if there exists H completing the diagram
below:
Y
X ∧I
H
X × X
Proposition 8.2. Let A ∈ C be cofibrant, let X ∈ C be fibrant, and let f, g : A → X. Then
(1) If f l ∼ g by some A ∧ I, then f l ∼ g by A × I.
(2) l ∼ is an equivalence relation on maps A → X
(3) f l ∼ g if and only if f r ∼ g.
Proof.
(1) Given A ∐ A → A ∧ I ∼ → A, factor the second map as A ∧ I ∼
↩→ A ∧ I ′ ↠ A; then the
last morphism must also be in W .
Then
A ∐ A
A ∧ I
∼
A ∧ I ′
H
f ∐ g
X
∗
A ∐ A A ∧ I
′
∼
∼
A × I A
18

The diagonal map must be in W .
(2) For reflexivity, use A ∧ I := A. For symmetry, compose with the involution A ∐ A →
A ∐ A interchanging the factors. Now we prove transitivity. Then we have the commutative
diagram.
A
A ∐ f ∐ g
g ∐ h
A
X
A ∐ A
H 1 H 2
A × I
A × I
Let P be the pushout of the two maps out of A to A × I. From the diagram, we obtain
P → X. We have compatible maps A × I → A, so we get P → A. We want to show
that P → A is a weak equivalence.
Claim: COF and W ∩ COF are preserved by cobase change, i.e., if A → B has the
property, then any pushout C → D arising from A → C also has the property. Dually,
FIB and W ∩ FIB are preserved by base change.
More general claim: Let M be any class of maps. Then ⊥ M is closed under retracts
and cobase change and composition, and M ⊥ is closed under retracts and base change
and composition.
Proof: Composition: Suppose that we have A → C and C → B in ⊥ M.
Cobase change:
∈ ⊥ M
Retracts: equally easy, left as exercise.
A X
g
C X
B
A C X
B
C ∐
B Y
A
We return to the proof of the transitivity.
∈M
A
A ∐ A
A × I
(3)
∅
A
A ∐ A → A × I H → X
19

A f → X c → X I → X × X
×{0}
A
c◦f
X I
(s,t)
A ×{1} A × I X × X
A
Lemma 8.3. Let A and X be both simultaneously fibrant and cofibrant. Then a map f : A →
B is in W if and only if it has a homotopy inverse, i.e., there exists g : B → A such that
f ◦ g ∼ id X and g ◦ f ∼ id A .
Proof. Skipped.
Using the lemma, we can prove the following important result:
Theorem 8.4. Let A and X be in C, and let A c be a cofibrant replacement for A. Let X f
be a fibrant replacement for X. Then
Hom C (A c , X f )
≃ Hom Ho(C,W ) (A, X).
∼
Example 8.5. Let A and B be simplicial sets. Every object is cofibrant. Fibrant objects
are Kan, so we take the Kan replacement.
f
X
9. July 30 (Schlank)
We now give two model categories of bounded complexes.
Example 9.1. In Ch ≥0
R-mod
(the category of chain complexes of R-modules in nonnegative
degrees), being cofibrant is being levelwise projective, and all objects are fibrant.
Example 9.2. In Ch ≤0
R-mod
(the category of chain complexes of R-modules in nonpositive
degrees), being fibrant is being levelwise injective, and all objects are cofibrant.
Let A be an R-module. Define K(A, n) ∈ Ch ≥0
R-mod
{
by
A if m = n,
(K(A, n)) m =
0 if m ≠ n.
Then
Hom Ch
≥0 (K(A, n) cof , K(B, m))/ ∼= Ext m−n
R (A, B).
R-mod
Let D be a small category, and let M be a model category. Objective: Define a model
category on M D .
Let G, F : D → M be two functors. A natural transformation h: G → F is a morphism in
M D . Say that h is a levelwise weak equivalence (or just a weak equivalence) if for all d ∈ D,
the morphism h d : G(d) → F (d) is a weak equivalence. Define levelwise fibration and levelwise
cofibration similarly.
20
□
□

Definition 9.3. A model category structure on M D is called projective if the weak equivalences
are levelwise weak equivalences and fibrations are levelwise fibrations.
When does a projective model category structure on M D exist? The exact answer is
technical, so let us just say that it exists when M is a model category of chain complexes or
of simplicial sets.
Definition 9.4. A model category structure on M D is called injective if the weak equivalences
are levelwise weak equivalences and cofibrations are levelwise cofibrations.
Let M be the model category of simplicial sets SSet. Let G be a group viewed as a
category (i.e., one object, and all morphisms are isomorphisms). Then SSet G is the category
of simplicial sets equipped with a G-action.
Proposition 9.5. In the projective model structure on SSet G , every cofibrant object has free
action levelwise.
Proof. Consider the simplicial set EG with G i+1 in level i. This is the coskeleton cosk 0 (G).
Then EG is Kan, contractible, and has free action of G.
□
∅ EG
∼
A ∗
Exercise 1: A is cofibrant if and only if A has free action.
We have the space of fixed points X G = Hom(∗, X). Let Ex ∞ X be the Kan replacement
of X. We have
[∗, X] Ho(SSet G ) = Map(EG, Ex ∞ X)/ ∼ .
An acyclic map is a map that is a weak equivalence. (Usually this terminology is used only
for fibrations and cofibrations.)
A model category M is enriched over SSet if there is a functor M op × M → SSet such
that
(1) Map(A, B) 0 ≃ Hom M (A, B) as functors on M op × M
(2) there exists a natural transformation Map(A, B) × Map(B, C) → Map(A, C) that
is associative and identity-respecting, compatible with Hom(A, B) × Hom(B, C) →
Hom(A, C).
Say that M is powered and over SSet if it has a functor SSet op ×M → M written A, X ↦→
X A such that Map M (X, Y A ) ≃ Map SSet (A, Map M (X, Y )) functorially in A, X, Y .
Say that M is copowered over SSet if it has a functor SSet ×M → M written A, X ↦→
A × X such that Map SSet (A, Map M (X, Y )) ≃ Map M (A × X, Y ) functorially in A, X, Y .
A model category M is said to be simplicial if
(1) M is enriched, powered, and co-powered over SSet.
(2) For every A ↩→ B in SSet and X ↠ Y in M, the map X B → X A × Y A Y B in M is a
fibration, and it is acyclic if either A ↩→ B or X → Y is.
We won’t try to justify this definition completely, but let us explore some consequences of
these conditions.
21

Example 9.6. Take Y = ∗, A = ∗, and B = [0, 1], and the map A → B sending a point to
0 ∈ [0, 1] = B; this is a cofibration. Then the condition implies that if X is fibrant, then the
map X line → X sending a path to its starting point is a fibration.
Example 9.7. Let A = ∅. Then A → B is a cofibration for every B. (Every simplicial set
is cofibrant.) The condition says that if X ↠ Y is a fibration, then X B → Y B is a fibration.
(Here X A and Y A are ∗.)
(Map M (X, Y )) n = Hom SSet (∆ n , Map M (X, Y )) ≃ Hom M (∆ n × X, Y )
Knowing any one of the enriched structure, power structure, and copower structure is
enough to determine all three uniquely. (But specifying one of these structures does not
necessarily mean that the other structures exist.) We will use this for the copower structure.
Let us give some examples of simplicial model categories.
Example 9.8. Let D be a small category, and consider SSet D with the projective/injecture
model structure. Then SSet D is a simplicial model category with copower structure SSet × SSet D →
SSet D sending A, F to A × F : d ↦→ A × F (d).
Theorem 9.9 (originally proved by Dwyer? Appears in Hovey, Model categories). Let M
be a simplicial model category and let A, X ∈ M. Let A cof be the cofibrant replacement of A.
Let X fib be the fibrant replacement of X. Then Map(A cof , X fib ) ∈ SSet is (naturally) weakly
equivalent to the Map L H
M
(A, X).
Example 9.10. Take M = SSet G in the theorem. Then Map L H
SSet G(∗, X) ∼ Map(EG, Ex∞ X).
The simplicial set on the right is called X hG .
We will never use L H M
from now on.
10. August 2 (Harpaz)
Let A be an abelian category. Let Ch(A) ≥0 be the category of complexes in degrees ≥ 0.
Define Ch(A) ≥0 similarly. If A has enough projectives, then we have a model category on
Ch(A) ≥0 in which weak equivalences are quasi-isomorphisms, all objects are fibrant, and the
cofibrant objects are the projective complexes (so cofibrant replacement is given by projective
resolution).
If A has enough injectives, then we have a model category on Ch(A) ≥0 in which weak
equivalences are quasi-isomorphisms, all objects are cofibrant, and the fibrant objects are
the injective complexes, (so fibrant replacement is given by injective resolution).
Suppose that C • , D • ∈ Ch(A) ≥0 . Then Hom(C • , D • ) ∈ Ch(Ab) is the unbounded complex
with
Hom(C • , D • ) n := ∏ Hom A (C k , D k+n )
k
and differential
∂ n ({f k : C k → D k+n }) := {∂ ◦ f k + (−1) k−1 f k−1 ◦ ∂ : C k → D k+n−1 }.
Definition 10.1. The co-Postnikov functor
P ≥0 : Ch(Ab) → Ch(Ab) ≥0
22

sending E = (· · · → E 1 → E 0 → E 1 → · · · ) ∈ Ch(Ab) to P ≥0 (E) with
{
P ≥0 E n , if n ≥ 1
(E) n :=
ker(∂ 0 ), if n = 0.
There is a natural map P ≥0 (E) → E that is an isomorphism on H n for n ≥ 0. Then
P ≥0 (Hom(C • , D • )) 0 = Hom Ch(A) ≥0(C • , D • ).
Theorem 10.2 (Dold–Kan correspondence). Let A be an abelian category. Then the category
Ch(A) ≥0 is equivalent to the category of A ∆op .
In particular, Ch(Ab) ≥0 ≃ Ab ∆op . Moreover, if C • corresponds to Γ(C • ) and if Γ(C • ) ∈
Set ∆op
is the underlying simplicial set, then H n (C • ) ≃ π n (Γ(C • ), e).
Theorem 10.3. Let C proj
• be a projective resolution of C • (cofibrant replacement). Then
Γ(P ≥0 (Hom(C proj
• , D • ))) ≃ Map der (C • , D • )
(The latter is notation for the derived mapping space.) For n ≥ 0, we get
H n (Hom(C proj
• , D • )) ≃ π n (Map der (C • , D • )).
and the former is Ext n (C • , D • ).
Let C be a Grothendieck site. Let Sh(C) be the category of sheaves of sets on C. Let
Sh(C) ∆op be the category of simplicial sheaves. Given F ∈ Sh(C) ∆op , define π n (F ) by taking
sections over each U to get a simplicial set, take its π n to get a presheaf, and sheafify.
Definition 10.4. Let f : F → G be a map of simplicial sheaves. Say that f is a local
equivalence (stalkwise) if
1) π 0 (f) := π 0 (F ) → π 0 (G) is an isomorphism of sheaves.
2) For all n ≥ 1, for all U ∈ C, for all x 0 ∈ F (U), the map π n (F | U , x 0 ) → π n (G| U , f(x 0 )) is
an isomorphism of sheaves of groups.
Theorem 10.5 (Joyal). There exists a simplicial model structure on Sh(C) ∆op such that
1) The weak equivalences are the local equivalences.
2) The cofibrations are the levelwise and objectwise injective maps.
3) The fibrations are determined by the right lifting property with respect to trivial cofibrations.
We want to describe sheaf cohomology in terms of the derived mapping space for this
model category.
Let F be a sheaf of abelian groups on C. Let H n (C, F ) be sheaf cohomology. Let F be the
complex of sheaves with F in degree 0, and 0 elsewhere. Let A be the category of sheaves
of abelian groups on C. Then F ∈ Ch(A) ≥0 . Suspension produces Σ n F , the complex of
sheaves with F in degree n, and 0 elsewhere. We have Γ as before, but defined on complexes
of sheaves of abelian groups instead of complexes of abelian groups. Then Γ(Σ n F ) ∈ A ∆op .
Let ·: A ∆op → Sh(C) ∆op be the functor forgetting the group structure. Let ∗ ∈ Sh(C) ∆op be
the terminal object, which is levelwise the one-point sheaf.
Claim:
π 0 (Map der (∗, Γ(Σ n F ))) ≃ H n (C, F )
23

Notation for adjunction: F : C
Lemma 10.6. Let F : C
following are equivalent:
D : U
D : U
(1) F preserves cofibrations and trivial cofibrations
(2) U preserves fibrations and trivial fibrations
be an adjunction between model categories. Then the
Proof. Recall that in a model category, a morphism is a fibration (resp. trivial fibration) if and
only if it has the right lifting property with respect to trivial cofibrations (resp. cofibrations).
Now assume that F preserves cofibrations and trivial cofibrations. Let f : X → Y be a
fibration.
A U(X)
trivial cofibration
B U(Y )
corresponds under adjunction to
So U(f) is a fibration.
trivial cofibration
F (A) X
F (B) Y
Definition 10.7. An adjunction satisfying the equivalent conditions of the lemma above is
called a Quillen adjunction.
F ′
We can compose adjunctions: C F
D E .
U U ′
Give SSet the Kan model structure. Give Top the Quillen model structure discussed
earlier, in which weak equivalences are weak equivalences and fibrations are Serre fibrations.
Let | · | be the realization functor. (Exercise: it preserves cofibrations.) Then
is a Quillen adjunction.
| · |: SSet
f
Top : Sing
Remark 10.8. F preserves weak equivalences between cofibrant objects.
11. August 6 (Schlank) — notes taken by René Pannekoek
Let D be a small category and M a model category. The colimit defines a functor colim :
M D → M that is left adjoint to the constant functor const : M → M D .
Theorem 11.1. If we endow M D with the projective model structure, then
□
becomes a Quillen adjunction.
colim: M D
M : const
24

Proof. It suffices to check that const preserves fibrations and trivial fibrations. This follows
simply from the definition of projective model structure. If f is a fibration, then const(f)
is level-wise just f. Since it is level-wise a fibration, it is a fibration in the projective model
structure. Similarly for trivial fibrations.
□
Dually, the functors const and lim define a Quillen adjunction between M and M D endowed
with the injective model structure:
const: M
M D : lim
Theorem 11.2. Let C, D be Grothendieck sites and f : D → C a morphism of sites. Let
Sh(D) ∆op and Sh(C) ∆op be the categories of simplicial sheaves of abelian groups on D and
C, both endowed with the Joyal model structure. Then
is a Quillen adjunction.
f ∗ : Sh(D) ∆op
Sh(C) ∆op : f ∗
Example 11.3. Let D be the trivial site (i.e. the Zariski site of the empty set). Then Sh(D)
is Set. Theorem 11.2 says that, for any site C, we get a Quillen adjunction
Γ ∗ : Set ∆op
Sh(C) ∆op : Γ ∗
where Γ ∗ = const is taking the constant sheaf and Γ ∗ is taking global sections.
Example 11.4. Let D be as in Example 11.3. Let C be a category endowed with the trivial
Grothendieck topology. This means that the covering families are the isomorphisms. The
sheaf condition now becomes vacuous. Then all presheaves are sheaves, so that Sh(C) =
Set C , and Sh(C) ∆op = (Set ∆op ) C . Now the Joyal model structure on Sh(C) ∆op coincides
with the injective model structure on (Set ∆op ) C . The Quillen adjunction takes the form
const: Set ∆op
(Set ∆op ) C : lim
where the functor on the right is taking the limit over C.
We briefly explain why the Joyal model structure and injective model structure on Sh(C) ∆op
coincide when C has the trivial topology. For this, we may check that the cofibrations and
weak equivalences coincide. In both model structures, the cofibrations are the levelwise and
objectwise injective maps. One can further show that the topology being trivial implies that
the level-wise weak equivalences coincide with the stalk-wise weak equivalences.
Example 11.5. Let M be a simplicial model category and let C ∈ M be a cofibrant object.
Then there is a Quillen adjunction between SSet and M:
• × C : SSet
M : Map(C, •)
This is simply a restatement of the fact that M is copowered over SSet.
Dually:
Example 11.6. Let M be a simplicial model category and let F ∈ M be a fibrant object.
Then there is a Quillen adjunction between SSet and the opposite category of M:
F • : SSet
M op : Map(•, F )
This is simply a restatement of the fact that M is powered over SSet.
25

Note that the model structure on M op is obtained from the one on M by switching the
roles of fibrations and cofibrations.
Definition 11.7. Let L: C D : R
derived functor LL : C → D of L by
be a Quillen adjunction. We define the total left
LL(c) = L(c cof ).
Definition 11.8. Let L: C D : R
derived functor RR : D → C of R by
be a Quillen adjunction. We define the total right
RR(d) = R(d fib ).
(It is important for the well-definedness of LL and RR that taking cofibrant and fibrant
replacements is a functorial construction, given as part of the model category structure.)
Fact: both LL and RR preserve weak equivalences. (Recall from Remark 10.8 that L
does not have to preserve weak equivalences, but does preserve weak equivalences between
cofibrant objects. Dually, R preserves weak equivalences between fibrant objects.) In general,
LL and RR do not form an adjunction. However, since LL and RR preserve weak
equivalences, we get functors Ho(LL) : Ho(C) → Ho(D) and Ho(RR) : Ho(D) → Ho(C).
Moreover, these are adjoint to each other:
Ho(LL): Ho(C)
Ho(D) : Ho(RR)
In fact, LL and RR satisfy an even stronger property than this: we have natural isomorphisms
of derived mapping spaces
Map der
D (LL(c), d) = Map der (c, RR(d))
This property is summarized by saying that LL and RR are homotopy adjoint to each other.
Example 11.9. Return to the case where f : D → C is a morphism of Grothendieck sites.
We have an adjunction
C
f ∗ : Ch ≤0 (ShAb(D))
Ch ≤0 (ShAb(C)) : f ∗
where both categories are endowed with the injective model structure (see Example 9.2 for
this). In this case the total derived functors Lf ∗ and Rf ∗ are the usual derived functors
in the sense of derived categories. In particular, since all objects of Ch ≤0 (ShAb(D)) are
cofibrant, the left derived functor Lf ∗ is just f ∗ .
Example 11.10. Consider the Quillen adjunction from Theorem 11.1, where M D has the
projective model structure:
colim: M D
M : const
L colim is sometimes called the homotopy colimit. The homotopy limit is defined dually.
26

Example 11.11. (Colimits do not preserve weak equivalences in general.) Let M = SSet
and D be the category defined by the following graph
d 1
d 2
We consider M D with the projective model structure. Let A = (A 1 , A 2 , A 3 ) ∈ M D be the
object defined by A 1 = {pt ′ , pt ′′ }, A 2 = A 3 = pt. The colimit of A is a single point. Let
object B = (B 1 , B 2 , B 3 ) ∈ M D be defined by B 1 = {pt ′ , pt ′′ }, B 2 = B 3 = line segment,
where the non-identity maps send pt ′ , pt ′′ to the end points of both line segments. The
colimit of B is a circle. There is an obvious element of Hom M D(B, A) that collapses the
line segments to points. This map is a weak equivalence, since it is a weak equivalence
levelwise. The induced map colim B → colim A collapses a circle onto a point, so is not a
weak equivalence. Therefore, colim does not preserve weak equivalences in general.
Exercise: Let A be an object in M D . If A 1 is a cofibrant object and A 1 → A 2 and
A 1 → A 3 are cofibrations in M, then A is cofibrant in M D with the projective model
structure. (Hint: use that the cofibrations are exactly the morphisms satisfying the LLP
with respect to trivial fibrations.)
From the exercise it follows that the colimit of the B of Example 11.11 is in fact the
homotopy colimit.
Example 11.12. Here is how to take the cofibrant replacement A cof of A as in Example
11.11. Set A cof
1 = A 1 , and define A cof
2 and A cof
3 respectively as the mapping cylinders of
A 1 → A 2 and A 1 → A 3 . There are canonical inclusions A cof
1 → A cof
2 and A cof
1 → A cof
3 .
Taking the colimit can be described geometrically as gluing the two mapping cylinders along
the images of A 1 .
Dually, if B is a diagram in M D , the homotopy limit turns out to be
holim(B) = B 2 × h B 1
B 3
= {(b 2 , b 3 , p) : b 2 ∈ B 2 , b 3 ∈ B 3 , p : [0, 1] → B 1 : p(0) = b 2 , p(1) = b 3 } .
Let G be a group, considered as a category with one object. Then we have the Quillen
adjunction
colim: SSet G
SSet: const
The colimit maps a simplicial G-set to its orbit space. Recall that an object of SSet G is
cofibrant if and only if the action is free. To compute the homotopy colimit of X, one
replaces X by X ×EG, which is a cofibrant replacement since EG is contractible. Hence the
homotopy colimit of X is weakly equivalent to (X × EG)/G. This is the Borel construction
which we encountered in Section 2. When X is a point with trivial G-action, we get that
the homotopy colimit of X is BG, the classifying space of G.
Dually, we have:
const: SSet SSet G : lim
27
d 3

The limit maps a simplicial G-set to its fixed points. The homotopy limit or homotopy fixed
points functor takes the fixed points of the fibrant replacement. Easy fact: if X is in SSet G ,
then Map(EG, X) is a fibrant replacement of X. Thus the homotopy fixed points of X up
to homotopy are
Map(EG, X) G = Map G (EG, X).
This is the X hG from Section 2.
The next examples finally bring us within the realm of arithmetic geometry.
Example 11.13. Let k be a perfect field and let Γ k = Gal(k/k). Consider the category
Sh(Spec(k) et ) ∆op of simplicial sheaves on the étale site of Spec(k). Passing to the stalk at
Spec(k) → Spec(k) considered with its natural Γ k -action gives an equivalence
α : Sh(Spec(k) et ) ∆op → Γ k Set ∆op ,
where Γ k Set ∆op
is the category of simplicial Γ k -sets such that every simplex has an open
stabilizer. We consider the Joyal model structure on Sh(Spec(k) et ) ∆op . Since Sh(Spec(k)) has
enough points, the weak equivalences are the stalkwise weak equivalences. Since Sh(Spec(k))
only has the point given by Spec(k) → Spec(k), the weak equivalences are precisely the
morphisms that give weak equivalences between underlying simplicial sets. Further, f ∈
Hom Γk Set
(X, Y ) corresponds to a cofibration iff f Γ L
: X Γ L
→ Y Γ L
are monomorphisms
∆op
for all finite and separable field extensions L/k. Hence, f is a cofibration iff the map on the
underlying simplicial sets is a levelwise monomorphism (i.e., a cofibration in the usual model
structure on SSet).
Summarizing: a morphism in the model category Sh(Spec(k) et ) ∆op is a weak equivalence
(cofibration) if the underlying map on simplicial sets is a weak equivalence (cofibration).
This description of the Joyal model structure on Sh(Spec(k) et ) ∆op resembles the injective
model structure on SSet Γ k
, which disregards the topology of the group.
Consider Γ k Set ∆op with the Joyal model structure via the identification with Sh(Spec(k) et ) ∆op .
From the discussion above it follows that we have a Quillen adjunction:
const: SSet
Γ k Set ∆op : lim
Indeed, the functor const preserves (trivial) cofibrations.
Example 11.14. Continuing the notation of the previous example, let M be a Γ k -module
and n a non-negative integer. Let K(M, n) be the corresponding Eilenberg–MacLane space.
(This is a simplicial Γ k -set that is obtained as follows: take the constant complex in Ch(Mod Γk )
with M in degree n, take the object in (Mod Γk ) ∆op
that corresponds to it under the Dold–
Kan correspondence, and forget group structure.) This has the following property:
π i (K(M, n) hΓ k
) = H n−i (Γ k , M).
12. August 8 (Harpaz) — notes taken by René Pannekoek
Definition 12.1. Let Γ be a profinite group. Let ΓSet be the category of sets equipped
with a Γ-action such that all stabilizers are open. Let ΓSet ∆op
be the category of simplicial
objects in ΓSet. Alternatively, this is the category of simplicial sets equipped with a Γ-action
such that the stabilizer of each simplex is open.
28

If G is a group, then we have the categories GSet of G-sets and SSet G = GSet ∆op
of
simplicial G-sets. We had adjunctions:
const: SSet
where the limit takes fixed points, and
GSet ∆op : lim
colim: GSet ∆op SSet : const
where the colimit takes G-orbits. We can turn these two adjunctions into Quillen adjunctions
by endowing GSet ∆op
= (Set ∆op ) G with the injective resp. projective model structure. In
the injective (projective) model structure, the weak equivalences and cofibrations (weak
equivalences and fibrations) are the weak equivalences and cofibrations (weak equivalences
and fibrations) on the underlying simplicial sets.
The adjunctions persist when the role of the discrete group G is taken over by a profinite
group Γ, with ΓSet ∆op interpreted as in Definition 12.1. In the first case, when Γ = Gal(k/k),
we saw in Example 11.13 that a close variant of the injective model structure could be found
and gave a Quillen adjunction. The argument given there actually works for every profinite
group Γ.) We are less lucky with the second adjunction.
Proposition 12.2. If Γ is infinite, there is no projective model structure on ΓSet ∆op .
Proof. The basic idea is that there are not enough cofibrant objects. Suppose a projective
model structure does exist. Claim: a cofibrant object X has a free Γ-action. Take a continuous
finite quotient Γ ↠ G. Let EG be a Kan and contractible simplicial G-set with free
G-action (such an object exists); so EG ∼ ↠ ∗ is a trivial fibration. Inflating the G-action on
EG gives a Γ-action. Now ∅ ↩→ X has the left lifting property in the diagram below:
∅ EG
∼
X ∗
Since the G-action on EG was free, all stabilizers on X are contained in the kernel of
Γ ↠ G. But G was arbitrary, so the Γ-action on X is free. But such an X does not exist in
ΓSet ∆op .
□
We need some sort of substitute for the projective model structure, and in particular the
cofibrant replacement. For this, we turn to the Pro-category. Given a category C with
subcategories W and F that “behave as weak equivalences and fibrations” (precise definition
below), the cofibrant replacement of an object X should be a morphism ˜X ∼ ↠ X that is
initial among morphisms
X ′ ∼
↠ X.
However, these morphisms do not describe a filtered system, so even in the Pro-category we
need to perform additional tricks.
Definition 12.3. Let C be a category with finite limits. Let W, F ⊂ C be subcategories
containing all isomorphisms. We will say that (C, W, F ) is a weak fibration category if:
(1) W has the 2-out-of-3 property;
(2) W, F are closed under pull-backs;
29

(3) F ◦ W = C.
Examples 12.4.
(1) Any model category.
(2) ΓSet ∆op where W consists of the morphisms inducing weak equivalences on underlying
simplicial sets, and F consists of the morphisms inducing fibrations on underlying
simplicial sets.
(3) More generally: let C be a Grothendieck site with enough points. Then we give
Sh(C) ∆op the structure of a weak fibration category as follows. Let W C be the morphisms
that give weak equivalences on the stalks, and F C the morphisms that give
Kan fibrations on the stalks.
The next object is, given a weak fibration category (C, W, F ), to put a model structure on
Pro(C). We require the morphisms in W and F to become weak equivalences and fibrations
in Pro(C) via the full embedding C ↩→ Pro(C). The other weak equivalences and fibrations
we want to add in a minimal way:
Definition 12.5. Let (C, W, F ) be a weak fibration category. Pro(C) is said to have a
(F, W )-generated model structure if there exists a model structure on Pro(C) such that COF =
⊥ (F ∩ W ).
If Pro(C) has such a model structure, it follows that further triv FIB = ( ⊥ (F ∩ W )) ⊥ ,
triv COF = ⊥ F and FIB = ( ⊥ F ) ⊥ .
Definition 12.6. Let (C, W ) be a relative category and let X, Y ∈ Pro(C). A morphism
f : X → Y is called a levelwise weak equivalence if there exist isomorphisms X ∼= → X ′ and
Y ∼= → Y ′ in Pro(C) and f ′ ∈ Hom(X ′ , Y ′ ) such that the diagram
X
∼= X ′ f ′
Y
f
∼= Y ′
commutes, X ′ , Y ′ have the same indexing set I, f ′ is defined levelwise for each object in I
(i.e. is induced by a natural transformation), and the morphisms f ′ (i) : X ′ (i) → Y ′ (i) are
in W for all i ∈ I. The set of all levelwise weak equivalences is denoted Lw ∼= (W ).
Theorem 12.7 (Schlank, Barnea). Let (C, W, F ) be a weak fibration category. If the class
Lw ∼= (W ) of levelwise weak equivalences is closed under composition and satisfies the 2-outof-3
property, then the (F, W )-generated model structure on Pro(C) exists. In this case, the
class of weak equivalences coincides with Lw ∼= (W ).
Example 12.8. The levelwise weak equivalences in Pro(ΓSet ∆op ) satisfy the hypotheses of
this theorem, if W consists of the levelwise weak equivalences in ΓSet ∆op .
Starting from the adjunction
we get an adjunction between Pro-categories
• Γ : ΓSet ∆op
SSet : const
Pro(• Γ ): Pro(ΓSet ∆op )
30
Pro(SSet) : Pro(const)

which we would like to be Quillen. We have to check that Pro(const) preserves fibrations
and trivial fibrations. This is taken care of by:
Proposition 12.9 (Schlank, Barnea). Let F : C → D be a functor between weak fibration
categories, and assume that Pro(C), Pro(D) have (F, W )-generated model structures. If F
preserves fibrations, trivial fibrations and finite limits, then Pro(F ) : Pro(C) → Pro(D)
preserves fibrations, trivial fibrations and all limits.
Let C be a Grothendieck site with enough points that is locally connected, i.e., there exists
a functor π 0 fitting into an adjunction
π 0 : Sh(C) ∆op
SSet : const
(Example: if C = Spec(k) et , then π 0 is the lower-shriek functor f ! .) Here, the category
on the left is considered with the weak fibration category structure mentioned in Example
12.4(3). Since const surely preserves fibrations and trivial fibrations, and also limits since it
is a right adjoint,
is a Quillen adjunction.
Pro(π 0 ): Pro(Sh(C) ∆op )
Pro(SSet) : Pro(const)
Remark 12.10. For the result to hold it is not necessary to require that C has enough points.
However, we only had a definition of weak fibration category structure on Sh(C) ∆op in that
case (Example 12.4(3)). Also, the local connectedness requirement is not essential.
Definition 12.11. L Pro(π 0 )(∗) ∈ Pro(SSet) is called the shape or realization of C.
13. August 13 (Schlank)
Let C be a Grothendieck site with enough points. Let M C = Pro(Sh(C) ∆op ).
Example 13.1. If C is trivial (one object), then M C = Pro(Set ∆op ) (pro-spaces).
Example 13.2. If C = (Spec k) et , then M C = Pro((Γ k -set) ∆op ) (pro-Γ k -spaces).
Theorem 13.3 (Barnea, S.). There exists a model structure (M C , W, COF, FIB) on M C
such that
(1) W = Lw ≃ W C , where W C are the stalkwise weak equivalences
(2) COF = ⊥ (F C ∩ W C ), where F C are the stalkwise fibrations.
Theorem 13.4 (Barnea, Schlank). Let f : C → D be a map of sites (we use the convention
that this means that f is a functor C → D). Then there is a Quillen adjunction
L: M D M C : Pro(f ∗ )
Every site has a unique map Γ: ∗ → C so we get
L: M C Pro(Set ∆op ) : Pro(Γ)
Define the shape of C as
|C| := LL(∗ MC ) ∈ Pro(Set ∆op ).
Given a (pro) space X and an abelian group A, we have
H n (X, A) ≃ [X, K(A, n)] Pro-Set ∆ op .
31

Thus
H n (|C|, A) = [|C|, K(A, n)]
= [LL(∗ MC , K(A, n)] Pro-Set ∆ op
= [∗ MC , RΓ ∗ (K(A, n))] Pro-Sh(C) ∆ op
= [∗ MC , K(Γ ∗ A, n)] Pro-Sh(C) ∆ op
= [∗ MC , K(Γ ∗ A, n)] Sh(C) ∆ op
= H n (C, A),
where Γ ∗ A ∈ Sh(C) ∆op .
Let X be a scheme. Then |X et | ∈ Pro(Set ∆op ). If A is a constant sheaf, then Het(X, n A) =
H n (|X et |, A). Also, if X is noetherian, π 1 (|X et |) ≃ π1 et (X) as pro-groups. (In particular,
π 1 (|X et |) is profinite when X is noetherian.)
Artin and Mazur (1969) define Et(X) ∈ Pro-Ho(Set ∆op ). There is an obvious functor
Pro(Ho): Pro-Set ∆op → Pro-Ho(Set ∆op ).
Theorem 13.5. There is a natural isomorphism Pro(Ho)(|X et |) ≃ Et(X).
Motivation: Let Y be a compactly generated Hausdorff space and let (U i ) i∈I be a good
cover of Y (recall that this means that for every finite nonempty J ⊆ I, the intersection
U J := ⋂ i∈J U i is a disjoint union of contractible spaces). Recall that the nerve of (U i ) is the
simplicial set N((U i )) with
N((U i )) n := π 0 (U × Y · · · × Y U) ∈ Set ∆op .
Then Y is weakly equivalent to N((U i )).
Now let X be a regular noetherian scheme. We cannot expect to find a good cover in the
sense above. For example, if C is a curve of positive genus, then there are no “contractible”
étale opens in C. (Here “contractible” should be some property that implies that C(C) in
the analytic topology is a contractible topological space.) The nerve of each étale covering
of X should be viewed as an approximation to the étale homotopy type.
Étale homotopy type (version 1, beta): take in Pro-Ho(Set ∆op ) the inverse system of N(U)
for all étale covers U → X. This presumes that the category of étale covers is cofiltered, but
this is not true (equalizers do not exist).
Definition 13.6. If U is an object of the étale site C, we obtain a functor h U : C op → Sets
defined by h U (O) = Hom(O, U) for O ∈ C, and it is a sheaf, called the represented sheaf.
Definition 13.7. A hypercover in a site C with enough points is an object U • ∈ C ∆op such
that the represented simplicial sheaf of U • is stalkwise fibrant and stalkwise constructible.
Lemma 13.8. Let U → X be an étale cover. Then the simplicial object Č(U) defined by
the fibered powers of U over Y is a hypercover.
Proof. Let x ∈ X be a point. Then the stalk of h U at x is just f −1 (x). Thus the stalk of
Č(U) at x is the simplicial set . . ., f −1 (x) × f −1 (x), f −1 (x) is Kan and contractible. □
Étale homotopy type (version 2): Take in Pro-Ho(Set ∆op ) the inverse system of π 0 (U • )
for hypercovers U • → X.
32

Definition 13.9. Let U • , V • ∈ C ∆op and let f, g : U • → V • , we say that f and g are strictly
homotopic if there exists H : U • × ∆ 1 → V • (a morphism in Sh(C) ∆op ) such that H(−, 0) =
f(−) and H(−, 1) = g(−). Say that f and g are homotopic if they can be connected by a
finite chain in which every two adjacent morphisms are strictly homotopic.
Proposition 13.10. The category of hypercovers and maps up to homotopy is cofiltered.
Definition 13.11. Étale homotopy type (version 3, final): Take in Pro-Ho(Set ∆op ) the inverse
system of π 0 (U • ) for hypercovers U • → X, in which the maps in the inverse system are maps
up to homotopy.
Let X be a smooth variety. We have a map of sites (Spec k) et → X et and
Now, take
L X/k : M X et
M (Spec k) et : Pro(Γ k -spaces)
LL X/k (∗ X ) ∈ Pro-(Γ-Set ∆op ).
We call this the relative shape of X over Spec k.
14. August 15 (Harpaz)
Let us recall the setup of the previous lecture. Let S be a base scheme (e.g., Spec k) Let
p: X → S be a scheme over S (e.g., a k-variety).
p ! : Pro(Sh(X et ) ∆op )
Define the relative étale shape of X over S to be
Pro(Sh(S ∗ et ) ∆op )
Et /S (X) := Lp 1 (∗ X ) ∈ Pro(Sh(S et ) ∆op ),
where ∗ X is the terminal sheaf considered as a pro-simplicial set.
Special cases:
(1) If S = Spec k where k = k, then Sh(S et ) = Set and Et /S (X) = Et /k (X) is a prosimplicial
set that coincides with the shape of X et , so the image in Pro(Ho(SSet)) is
the étale homotopy type of S.
Brief digression: profinite completion. We have the Kan homotopy category Ho(SSet).
Let Fin ⊆ Ho(SSet) be the full subcategory of all the simplicial sets such that all their
homotopy groups are finite.
It can be shown that the inclusion Pro(Fin) ⊆ Pro(Ho(SSet)) admits a left adjoint
̂•: Pro(Ho(SSet)) → Pro(Fin).
For X ∈ Ho(SSet) ⊆ Pro(Ho(SSet)), we call ̂X ∈ Pro(Fin) the profinite completion of
X. We have a natural map X → ̂X inducing an isomorphism of cohomology with finite
coefficients.
More special cases:
(1) For X a variety over K = C, the image of Et /C (X) in Pro(Ho(C)) is isomorphic to
the profinite completion of X(C) (Artin–Mazur comparison theorem).
33

(2) Let S = Spec k and Γ = Gal(k sep /k). Then Sh(S et ) = Γ-Sets. For X → S, we have
that Et /k (X) is an inverse system of simplicial Γ-sets. The underlying pro-simplicial
set of Et /k (X) is weakly equivalent to Et /k
(X k
).
In general, given X → S and f : S ′ → S, and X ′ := X × S S ′ , there is a map
Et /S ′(X ′ ) → f ∗ Et /S (X).
If f is proper, then this is a weak equivalence. Also, for f : Spec k → Spec k, it is a weak
equivalence; here f ∗ Et /k (X) is just taking the underlying pro-simplicial set.
Upshot: If K is a number field with i: K ↩→ C and X is a K-variety, then the image of
Et /K (X) in Pro(Ho(SSet)) is the profinite completion X(C) endowed with a Galois action.
Example 14.1. Let K = Q and X = G m = Spec Q[t, t −1 ]. What is
f ! : Pro(Sh((G m ) et ) ∆op ) → Pro(Γ-Set ∆op )?
There is a functor f ! : Sh(X) → Γ-Sets sending the sheaf h U for U → X to the Γ-set
π 0 (U Q ) =: π 0/Q (U).
Given n ∈ Z >0 , we have p n : G m → G m sending t to t n . Let Č(p n) k be the k + 1 fold fiber
p n→
power of G m Gm , which is equal to G m × µ k n. This is an object of Xet ∆op ↩→ Sh(X et ) ∆op ,
and it is acyclically fibrant. Applying Lf ! yields {π 0/Q (Č(p n))}, and π 0/Q (Č(p n)) k = µ k n.
Given a group G, let BG be the (homotopy type) of the simplicial set whose set at level k is
G k (with suitable face and degeneracy maps); with G k+1 instead one would get EG — the
realizations of these simplicial sets are what we defined long ago, up to homotopy. Thus in
our setting we get Bµ n , and Et /Q (G m ) is the inverse system {Bµ n } n∈Z>0 .
On the other hand, G m (C) = C ∗ ≃ S 1 ≃ BZ, and ̂BZ is the profinite completion of BZ,
which is the inverse system of B(Z/nZ), whose π 1 is Ẑ.
Given p: X → S, the set X(S) is the set of sections s: S → X of p. Let ∗ ∈ Sh(X et ) ∆op
be the terminal sheaf. We have ∗ cof = p ∗ p ! (∗ cof ). Then s ∗ (∗ cof ) → s ∗ p ∗ p ! (∗ cof ) = p ! (∗ cof )
since s ∗ p ∗ = (p ◦ s) ∗ = id. From the trivial fibration ∗cof → ∗, we get a trivial fibration
s ∗ (∗ cof ) → s ∗ (∗) = ∗ Set . Since s ∗ (∗ cof ) is equivalent to ∗ Set , this map determines a well-defined
element of π 0 (Map der (∗ Set , Et /S (X))). So we get a map
X(S) → π 0 (Map der (∗ Set , Et /S (X))).
Call the latter set the set of derived sections. If there are no derived sections, then X has
no S-points; in this case, we say that there is a homotopy obstruction to the existence of an
S-point.
This is all very abstract. How do we show that π 0 (Map der (∗ Set , Et /S (X))) is empty?
1) The one-sheaf-at-a-time approach: find one simplicial sheaf in the inverse system
Et /S (X) that does not admit a derived section.
Let S = Spec k. If X is a simplicial Γ-set, then Map der (∗, X) = X hΓ , the homotopy fixed
point space. There exists a sequence of obstructions a n ∈ H n+1
Gal (Γ, π n(X)), each defined if
the previous one vanishes. The sequence starts with HGal 2 (Γ, π 1(X)) (for simplicity, let us
assume that π 1 (X) is abelian).
Next time we will apply this theory to x 2 − ay 2 = b for a, b ∈ Q × .
34

15. August 20 (Schlank)
Let k be a field with char k ≠ 2. Let a, b ∈ k × . Let X be the variety x 2 − ay 2 =
b. Let U be the variety defined by t 2 = a, s 2 = b, z 2 − aw 2 = s. Define U → X by
(s, t, z, w) ↦→ (z 2 + aw 2 , 2zw). In fact, this is a torsor under the dihedral group of order 8,
acting by (s, t, z, w) ↦→ (−s, −t, tw, z/t) and (s, t, z, w) ↦→ (s, −t, z, w). Let X := ¯π 0 (Č(U)) ∈
Γ k -Set ∆op . The first obstruction lives in H 2 (Γ k , π 1 (X)).
Let us now discuss some constructions related to group cohomology, as a warmup. Let G
be a finite group. Let E be a G-simplicial set such that E ∼ ∗ and G acts freely on E. (All
groups act on the left.) Then
{
0, if n ≠ 1
π n (E/G) =
G, if n = 1.
Example 15.1. If G is a set with free G-action, then the simplicial set cosk 0 (L) consisting
of powers of L is G-free and ∼ ∗. Also, cosk 0 (L) is Kan.
Example 15.2. Take L = G. Define EG := cosk 0 (G) and BG = EG/G.
Let 1 → K → L → M → 1 be a short exact sequence of finite groups. Then
{
0, if n ≠ 1
π n (EL/K) =
K, if n = 1.
Also, M acts on EL/K. Is (EL/K) hM ≠ ∅? This holds if and only if the obstruction o 1 ∈
H 2 (M, π 1 (EL/K)) = H 2 (M, K) is 0 (if o 1 is 0, then all the higher obstructions automatically
vanish). Claim: o 1 = 0 if and only if there is a section M → L.
(EL/K) hM = Map der
M-SSet(∗, EL/K). Here EL/K is fibrant. Replace ∗ by EM, which is
cofibrant. We compute maps from EM to EL/K levelwise.
.
M × M × M −→ K\L × L × L
M × M −→ K\L × L
M −→ K\L
By connectedness, we may assume that the last map sends e to e and hence m to m ∈
M = K\L. The next map is M-equivariant and respects endpoints. By M-equivariance, it
suffices to say where (e, m) goes for each m; say it maps to (l 1 , l 2 ). Here l 1 ∈ f −1 (e) = K
and l 2 ∈ f −1 (m). Because we are taking the quotient by K, we may assume l 1 = e.
Define γ(m) := l 2 . This is a section of sets M → L, and we claim that it is a section
of groups. To extend to level 2, it is enough to specify the values on (e, m 0 , m 0 m 1 ) since
the map is M-equivariant. We may suppose it maps to (e, l 0 , l 2 ) where f(l 0 ) = m 0 and
f(l 2 ) = m 0 m 1 , because of compatibility with vertices. Let p be the map EM → EL/K
we are trying to describe. For compatibility with level 1, we need p(e, m 0 ) = (e, l 0 ), so
l 0 = γ(m 0 ); and p(e, m 0 m 1 ) = (e, l 2 ), so l 2 = γ(m 0 m 1 ); and p(m 0 , m 0 m 1 ) = (l 0 , l 2 ), so
m 0 p(e, m 1 ) = (l 0 , l 2 ), so m 0 (e, l 1 ) = (l 0 , l 2 ) so (γ(m 0 ), γ(m 0 )l 1 ) = (l 0 , l 2 ) in K\L × L and
γ(m 0 ) = l 0 , so l 2 = γ(m 0 )l 1 .
35

Back to algebraic geometry. Recall our D-torsor U → X, where D is the dihedral group
of order 8. Recall that U has 4 connected components on which Gal(K( √ a/ √ b)/K) acts
transitively and freely (assuming that this extension has degree 4). Then Č(U) is
U × D × D
U × D
The action of D on t and s defines a surjective homomorphism
U
D → Gal(K( √ a, √ b)/K);
let C be the kernel, a group of order 2. Taking π 0 yields
So X := ¯π 0 (Č(U)) = ED/C, so
π n (X) =
C\D × D × D
C\D × D
C\D.
{
0, if n ≠ 1
Z/2Z = C, if n = 1.
So the obstruction o 1 ∈ H 2 (Γ k , π 1 (X)) = H 2 (Γ k , Z/2Z) would be the pullback by Γ k
Gal(K( √ a, √ b)/K) of the class in H 2 ((Z/2Z) 2 , Z/2Z) related to
→
1 → Z/2Z → D → (Z/2Z) 2 → 1.
But now it is an easy calculation to see that the corresponding element in H 2 (Z/2Z ×
Z/2Z, Z/2Z) is the cup product of the two generators of H 1 (Z/2Z × Z/2Z, Z/2Z). Thus we
get that o 1 ∈ H 2 (Γ k , π 1 (X)) = H 2 (Γ k , Z/2Z) equals (a) ∪ (b).
Generalization: Given a 1 , . . . , a n ∈ K × , if there is a solution to a 0 x 2 0 + · · · + a n x 2 n = 1,
then the cup product ∪(a i ) ∈ H n+1 (K, Z/2Z) is 0.
Now let X be any k-variety. Let U • → X be a hypercover. For any set A, let Z 1 A be the
set of degree 1 formal integer combinations of elements of A. Extend this to Γ k -simplicial
sets. There is a map ¯π 0 (U • ) → Z 1¯π 0 (U).
Properties of Z 1 A for a Γ k -simplicial set A.
(1) If Z 1 A hΓ k = ∅, then A
hΓ k = ∅.
(2) π n (Z 1 A) = ˜H n (A, Z). (There is a variant: for n ≥ 1, π n ((Z/nZ) 1 A) = H 2 (A, Z/n).)
It would be nicer if Z 1 A were a group. Define ZA to be the simplicial set obtained by
taking formal integer combinations at each level. Then ZA = ⊕ i∈Z Zi A, and it is a simplicial
Γ k -module.
By the Dold–Kan correspondence, studying homotopy fixed points of ZA modulo equivalence
is the same as studying Galois hypercohomology of the corresponding complex N(ZA).
36

16. August 22 (Harpaz)
Let K be a field. Let X be a K-variety. Let Γ = Gal(K sep /K). We constructed Et /K (X) ∈
Pro(Γ- Set ∆op ). As we have seen, if π 0 (Map der (∗, Et /K (X))) = ∅, then X(k) = ∅.
Definition 16.1. We will say that a simplicial set X is bounded if there is an n such that
for all basepoints x and all N > n, we have π N (X, x) = 0. A pro-simplicial set is bounded if
every simplicial set in the diagram is bounded (the n may depend on the simplicial set).
Claim: There is a canonical functor X → X ♮ from Pro(Set ∆op ) to the full subcategory of
bounded pro-simplicial sets. For each X, there is a morphism X → X ♮ with the property
that for each bounded pro-simplicial set Y , there is a weak equivalence from the derived
mapping space Map der (X ♮ , Y ) to Map der (X, Y ).
Remark 16.2. Given a continuous Γ-action on X, there is a continuous Γ-action on X ♮ .
Applying the functor twice yields an X ♮♮ weakly equivalent to X ♮ . From now on, we assume
that this functor has been applied to all objects, so all pro-objects have been implicitly ♮-
ized. So we work with Et ♮ /K(X), but from now on we drop the ♮. (This possibly weakens the
obstructions.)
So now we have an obstruction theory to the nonemptiness of
X(hK) := π 0 (Map der (∗, Et /K (X)))
with obstructions that live in
Hcont n+1 (Γ, π n (Et /K (X)))
For n = 1, we get
Hcont(Γ, 2 π 1 (Et /K (X))) = Hcont(Γ, 2 π1 et (X))
Claim: The n = 1 obstruction is exactly the nonsplitting of the Grothendieck short exact
sequence.
Last week, Tomer gave the example ∑ n
i=0 x2 1 = −1 over R, for which there is a nonzero
obstruction in H n+1 (Z/2Z, π n (Et /R (X))).
There is a functor Y ↦→ P n (Y ) from Set ∆op to Set ∆op such that π k (P n (Y )) = 0 for
k > n and π k (P n (Y )) ≃ π k (Y ) for k ≤ n. Then (Y ) ♮ = (P n (Y )) n∈N ; this is called the
Postnikov tower. For each k, there is an isomorphism π k (Y ) ≃ π k ((P n (Y )) n∈N . Similarly,
π k ((Y α ) α ) ≃ π k ((Y α ) ♮ ) := π k ((P n (Y α )) α,n ).
In fact, (Y α ) ♮ is defined to be (P n (Y α )) α,n .
Let K be a number field. Let X be a k-variety. We defined
X(hK) := π 0 (Map der (∗, Et /K (X))).
There is a map X(k) → X(hK) (not necessarily injective or surjective). Because of cohomological
obstruction, this might not give an obstruction much stronger than the Grothendieck
section obstruction, however.
For each place v, consider f v : Spec K v → Spec K. Then f ∗ v Et /K (X) ∈ Pro(Γ v - Set ∆op ).
(It turns out that this is the same as Et /Kv (X Kv ); this is a general property of pullback by
field extensions.) We get
X(K v ) → π 0 (Map der (∗, f v Et /K (X)) =: X(hK v ).
37

There is a commutative diagram of sets
X(K)
X(hK)
X(K v ) X(hK v )
Say that an element of X(K v ) homotopy rational if its image in X(hK v ) is in the image of
X(hK). If there are no homotopy rational K v -points, then X(K) = ∅.
We can also use all v at once:
X(K)
∏
X(K v )
v
X(hK)
∏
X(hK v )
We say that the tuple (x v ) ∈ ∏ v X(k v) is homotopy rational if its image in ∏ v X(hK v)
comes from a single element of X(hK). We can make it even stronger by using adelic points:
X(K)
v
X(hK)
X(A)
h
X(hA)
Let X h (A) ⊆ X(A) be the set of homotopy rational adelic points. If X(A) ≠ ∅ and
X h (A) = ∅, then we say that there is a homotopy obstruction to the local-global principle.
Let Y be a simplicial set. Let ZY ∈ Ab ∆op
be given by (ZY ) n = ZY n , the free abelian
group with basis Y n . This is left adjoint to the forgetful functor Ab ∆op → Set ∆op . In
particular, there is a map from Y to the simplicial set ZY obtained from ZY by forgetting
the group structure.
Then π n (ZY ) can be identified with the homology group H n (Y ). Furthermore, π n (Y ) →
π n (ZY ) ≃ H n (Y ) coincides with the Hurewicz map.
Now suppose that we have a continuous Γ-action on Y . Then there is continuous Γ-action
on ZY and on ZY .
Apply the Dold–Kan correspondence to the abelian category Γ Mod to get an equivalence
of categories
Γ Mod N ∆op Ch ≥0 (Γ Mod)
Γ
(different Γ here!)
Claim: Let Z be a simplicial Γ-module. Then
π 0 (Z hΓ ) ≃ H 0 (Γ, N(Z))
and
π n (Z hΓ ) ≃ H −n (Γ, N(Z)).
Let us apply this to the relative étale shape. Prolong the functor Z to
Z: Pro(Γ Set ∆op ) → Pro(Γ Mod ∆op ).
38

Also prolong the forgetful functor in the opposite direction. There is a map
and
Et /K (X) → Z Et /K (X)
X(K) → π 0 (Map der (∗, Et /K (X))) → π 0 (Map der (∗, Z Et /K (X))) = H 0 cont(N(Z Et /K (X))).
17. August 27
Theorem 17.1. Let Y be a simplicial Γ-module. Then the Dold–Kan correspondence induces
a natural isomorphism of sets
π 0 (Y hΓ ) ≃ H 0 (Γ, N(Y )).
Theorem 17.2. Let (Y α ) α∈I ∈ Pro(Γ- Mod ∆op ). Then by Dold–Kan,
π 0 (Map der (∗, (Y α ))) ≃ H 0 cont(Γ, N(Y α )).
X(K)
h
H 0 cont(K, N(Z Et /K (X)))
X(A)
loc
∏h
v H0 cont(K v , N(Z Et /K (X)))
The image of the bottom horizontal map lands in a subgroup H 0 (A, N(Z Et /K (X))). Define
Then X(K) ⊆ X(A) Zh ⊆ X(A).
X(A) Zh := {(a v ) ∈ X(A) : h(a v ) ∈ im(loc)}.
Theorem 17.3 (Harpaz, Schlank). If X is smooth and connected, then X(A) Zh = X(A) Br .
Remark 17.4. The only reason that X is assumed to be connected is that if X = Spec K ∐ Spec K,
for instance, and K has at least one complex place, then X(A) Br ≠ X(K).
Let U → X be a hypercover.
X(K)
h
H 0 cont(K, N(Z Et /K (X)))
ψ U
lim U H 0 (K, N(Z¯π 0 (U)))
X(A)
loc
h ∏
v H0 cont(K v , N(Z Et /K (X)))
loc
ψ U ∏
lim U v H0 cont(K v , N(Z¯π 0 (U)))
Proposition 17.5. For every (a v ) v ∈ X(A), h(a v ) ∈ im(loc) if and only if for all U → X,
we have ψ U (h(a v )) ∈ im loc U .
Sketch of proof. Cohomology of local fields is finite and the hyper-Tate–Shafarevich group
(kernel of the right loc U ) is finite and profinite groups do not have lim
1 .
□
←−
One can replace the products by adelic versions.
Definition 17.6. H q (A, C) is the restricted direct product of H q (K v , C) with respect to the
images of the maps H i (Ẑ, CI ) → H i (K v , C), where I is the inertia.
39

Let C U := N(Z¯π 0 (U)). Then we have
X(K)
H 0 (K, C U )
X(A)
loc U
h U
H 0 (A, C U )
We showed that an element (a v ) ∈ X(A) is in X(A) Zh if and only if for every hypercover
U • → X we have h U ((a v )) ∈ im(loc U ).
Poitou–Tate for a finite Galois module M: Let ̂M := Hom(M, G m ) be the Cartier dual.
We have a pairing
M × ̂M → G m
and thus
So we have a pairing
H i (K v , M) × H 2−i (K v , ̂M) → H 2 (K v , G m ) inv
↩→ Q/Z.
H i (A, M) × H 2−i (K, ̂M) → ⊕ v
By Brauer–Hasse–Noether, the image of
Q/Z
loc: H i (K, M) × H i (K, A)
∑
→ Q/Z
is contained in the left kernel of this pairing. (Use Tate modified H 0 from now on.) Poitou–
Tate says that the image of loc equals the left kernel. For a torsion abelian group A, let
A ∗ := Hom(A, Q/Z) be the Pontryagin dual. Then we have the 9-term Poitou–Tate exact
sequence
0 → H 0 (K, M) → H 0 (A, M) → H 2 (K, ̂M) ∗
→ H 1 (K, M) → H 1 (A, M) → H 1 (K, ̂M) ∗
→ H 2 (K, M) → H 2 (A, M) → H 0 (K, ̂M) ∗ → 0.
Now, for a complex C of finitely generated Galois modules, let Ĉ := Hom(C, G m). We
can still define
H i (A, C) × H 2−i (K, Ĉ) → Q/Z.
It is still true by Brauer–Hasse–Noether that the image of of
loc: H i (K, C) × H i (K, Ĉ)
is contained in the left kernel of this pairing. (Use Tate modified H 0 from now on.) Poitou–
Tate again says that the image of loc equals the left kernel. Then we have a long exact
Poitou–Tate exact sequence
· · · → H i (K, C) → H i (A, C) → H 2−i (K, Ĉ)∗ → · · ·
(not just 9 terms). If C is a bounded complex, finitely generated as an abelian group in every
level, with finite homologies, then there is a long exact sequence
· · · → H i (K, C) → H i (A, C) → H 2−i (K, Ĉ)∗ → · · ·
40

Our complex C U almost satisfies these conditions: it does, except that it has Z in its
homology at level 0. In particular, we get
H 0 (A, C U ) × H 2 (K, C U ) → Q/Z
and
lim H
←− 0 (A, C U ) × lim H 2 (K, C U ) → Q/Z.
−→U
U
To conclude, we have a natural pairing
X(A) × lim H 2 (K, ˜C U ) → Q/Z
−→U
and X(A) Zh is the left kernel of this pairing.
Let G m be the usual étale sheaf on Spec K (corresponding to the Galois module K × . Let
t: X → Spec K be the structure map. Define G const
m
:= t ∗ G m . (On each étale U → X, it
gives the constant invertible functions.) There is a natural map G const
m → G m . There is a
pairing
X(A) × Het(X, 2 G const
m ) → Q/Z.
Let X(A) “Br” be the left kernel. Then X(K) ⊆ X(A) “Br” ⊆ X(A).
Lemma 17.7. X(A) Br = X(A) “Br” .
Proof. The pairings are compatible with respect to the map φ: Het(X, 2 G const
m ) → Het(X, 2 G m ),
which is surjective (work with µ n ). So X(A) Br ⊆ X(A) “Br” . But X(A) “Br” ⊆ X(A) Br since
φ is surjective.
□
To identify the homotopy obstruction with the Brauer obstruction, we need to show that
lim H
−→ 2 (K, ĈU) ≃ Het(X, 2 G const
m ).
U
18. August 29
41