Chapter 3: Cohomology

Felix Schremmer

Technical University of Munich Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients

1 Different notions of cohomology Cohomology from topology Cohomology from the topos

2 Cohomology with integral coefficients

3 Cohomology with real coefficients Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Notation

Throughout this talk, denote:

S ∈ Comp a quasi-compact Hausdorff (compact) topological space.

Cond(Set), Cond(Ab) the categories of condensed sets/abelian groups.

Aim: Discuss notions of H•(S, A) for A an abelian group. Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Singular cohomology

Start with the space S.

Consider the simplicial set

Sn = HomTop(n-Simplex, S).

Turn into a chain complex

C• : · · · → Z[S2] → Z[S1] → Z[S0] → 0,

where di is the alternating sum of the i + 1 face maps. • Then Hsing(S, A) = cohomology of HomAb(C•, A). Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Čech cohomology

Turn A into a constant sheaf on S: discrete Γ(U, A) = HomTop(U, A ) for all U.

For a finite open cover U = {Ui }i∈I on S, form a cosimplicial space a S0 := U, Sn = S0 × · · · × S0 . S S | {z } n+1 times

The alternating sum of the face (projection) maps Sn → Sn−1 give

0 → Γ(S0, A) → Γ(S1, A) → Γ(S2, A) → · · ·

H• of this complex is H• (U, A). H• (S, A) = lim H• (U, A). Čech Čech −→U Čech Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Sheaf cohomology

The functor

(abelian sheaves over S) −→Γ Ab

has right-derived functors.

• • Hsheaf (S, A) = R Γ(S, A).

Compute e.g. using injective resolution A → I•, then • • Hsheaf (S, A) = cohomology of Γ(S, I ). Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Comparison

Lemma. • ∼ • HČech(S, A) = Hsheaf (S, A). If S is a profinite set and A discrete,

0 ∼ 0 ∼ HČech(S, A) = Hsheaf (S, A) = HomTop(S, A). 0 ∼ Hsing(S, A) = HomSet(S, A).

Sheaf (Čech) cohomology is better suited for condensed mathemat- ics. Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients

1 Different notions of cohomology Cohomology from topology Cohomology from the topos

2 Cohomology with integral coefficients

3 Cohomology with real coefficients Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Condensed cohomology

Recall: Cond(Ab) ' category of abelian sheaves over Comp. Definition. • We define Hcond(S, ·) : Cond(Ab) → Ab to be the right-derived functors of Γ(S, ·). ∼ Since Γ(S, A) = HomCond(Ab)(Z[S], A), conclude

• ∼ • Hcond(S, A) = ExtCond(Ab)(Z[S], A).

May use a projective resolution of Z[S] or injective resolution of A. Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Resolving S

If S is extremally disconnected (projective), then ≥1 Γ(S, ·) : Cond(Ab) → Ab is exact =⇒ Hcond(S, ·) = 0. In general, we want a “resolution” in Comp

S• = (Sn)n≥0 + simplicial structure

with each Sn extremally disconnected. These should give rise to a projective resolution

· · · → Z[S2] → Z[S1] → Z[S0] → Z[S] → 0

in Cond(Ab). ∼ As HomCond(Ab)(Z[Sn], ·) = Γ(Sn, ·) is exact, Z[Sn] is projective in Cond(Ab). Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Let’s try Čech

Pick S0 → S surjective such that S0 is extremally disconnected (e.g. Stone-Čech compactification of Sdiscrete).

For n ≥ 1, let Sn = S0 ×S · · · ×S S0. | {z } n+1 times Usual arguments show the Čech complex

· · · → Z[S2] → Z[S1] → Z[S0] → Z[S] → 0

is exact! Problem: Sn not necessarily extremally disconnected for n ≥ 1. Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Hypercover

Need something better: Pick S0 → S surjective with S0 extremally disconnected. Pick S1  S0 ×S S0 with S1 extremally disconnected. Pick   d (u) = d (v),  1 1  S2  (u, v, w) ∈ S1 × S1 × S1 | d2(u) = d1(w), ,    d2(v) = d2(w) 

π1,2 where d1,2 is S1 → S0 ×S S0 −−→ S0.

Generally pick Sn+1  Coskeleton at level n + 1 of the truncated simplical set S0,..., Sn. Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients

• Computing Hcond(S, A)

Pick a hypercover S• → S such that each Sn is extremally discon- nected (at least, Z[Sn] is HomCond(Ab)(·, A)-acyclic).

This yields a projective (acyclic) resolution

· · · → Z[S2] → Z[S1] → Z[S0] → Z[S] → 0.

• • Then Hcond(S, A) = Ext (Z[S], A) is the cohomology of

0 → HomCond(Ab)(Z[S0], A) → Hom(Z[S1], A) → Hom(Z[S2], A) ···

=0 → Γ(S0, A) → Γ(S1, A) → Γ(S2, A) ··· ,

where each codifferential is the alternating sum of face maps. Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients

1 Different notions of cohomology

2 Cohomology with integral coefficients Case of profinite sets General case

3 Cohomology with real coefficients Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients The result

Theorem (Dyckhoff, 1976). There is an isomorphism

• ∼ • Hcond(S, Z) = Hsheaf (S, Z)

which is natural in S.

Observe that in particular the constant sheaf Z ∈ Cond(Ab) has infinite injective dimension (unlike Z ∈ Ab).

The proof should work for any discrete abelian group.

If S is a finite set, ( n n HomTop(S, Z), n = 0 Hcond(S, Z) = Hsheaf (S, Z) = . 0, n ≥ 1 Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients

1 Different notions of cohomology

2 Cohomology with integral coefficients Case of profinite sets General case

3 Cohomology with real coefficients Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients

• Computing Hsheaf(S, Z)

Let S = lim Sj be profinite. ←−j Then

H• (S, ) =∼ H• (S, ) =∼ lim H• (Sj , ). sheaf Z Čech Z −→j Čech Z

Eilenberg-Steenrod: Foundations of algebraic topology, Chapter X, Theorem 3.1

Now H≥1 (Sj , ) = 0. Thus H≥1 (S, ) = 0 and Čech Z sheaf Z

0 Hsheaf (S, Z) = Γ(S, Z) = HomTop(S, Z). Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients

• Computing Hcond(S, Z)

0 Certainly Hcond(S, Z) = HomTop(S, Z), so we have to show ≥1 Hcond(S, Z) = 0.

j Pick an e.d. hypercover S• → S, and for each S choose finite hypercover Sj → Sj such that S = lim Sj . • n ←−j n Then Sj is extremally disconnected, so that

j j j 0 → Γ(S , Z) → Γ(S0, Z) → Γ(S1, Z) → · · ·

is exact. Taking filtered colimits shows exactness of

0 → Γ(S, Z) → Γ(S0, Z) → Γ(S1, Z) → · · · .

≥1 Thus Hcond(S, Z) = 0. Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients

1 Different notions of cohomology

2 Cohomology with integral coefficients Case of profinite sets General case

3 Cohomology with real coefficients Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients The morphism

Consider morphism of topoi

α :(sheaves over Comp /S) → (sheaves on S).

For an abelian sheaf F over Comp /S, α∗(F) is the following abelian sheaf on S

U 7→ lim F(V ,→ S). ←− U⊇V closed in S

α∗ is left exact and Γsheaf (S, ·) ◦ α∗ = Γcond(S, ·). ∼ We have to show Rα∗Z = Z in D(abelian sheaves on S), as then

• • • Hcond(S, Z) =H (RΓcond(S, Z)) = H (RΓsheaf (S, ·) ◦ Rα∗(Z)) ∗ • • =H (RΓsheaf (S, Z)) = Hsheaf (S, Z). Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients

Towards Rα∗Z

Rα∗Z is a complex of abelian sheaves on S. 0 ∼ H (Rα∗Z) = α∗Z as abelian sheaves on S.

0 ∼ The global sections Γsheaf (S, H (Rα∗Z)) = Γcond(S, Z) induce a 0 morphism of sheaves Z → H (Rα∗Z)).

This yields a morphism of complexes of abelian sheaves

Z (concentrated in degree 0) → Rα∗Z.

We prove this is an isomorphism on stalks. Fix s ∈ S. Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients

Computing (Rα∗Z)s

(Rα ) = lim RΓ(U, Rα ) = lim RΓ (V , ). ∗Z s −→ ∗Z −→ cond Z s∈U open s∈V closed nbh

Pick a hypercover S• → S by profinite (extremally disconnected) sets.

For each closed V , (Sn ×S V )n≥0 → V is a hypercover by profinite sets. Hence RΓcond(V , Z) is isomorphic to

0 → Γ(S0 ×S V , Z) → Γ(S1 ×S V , Z) → Γ(S2 ×S V , Z) → · · ·

Taking the filtered colimit over V 3 s yields 0 → lim Γ(S × V , ) → lim Γ(S × V , ) → · · · −→ 0 S Z −→ 1 S Z V 3s V 3s ∼ = 0 → Γ(S0 ×S {s}, Z) → Γ(S1 ×S {s}, Z) → · · · ∼ ∼ = RΓcond({s}, Z) = Z. Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Proof summary

Proving the claim (or rather, acyclicity) for profinite sets allows us to use profinite hypercovers, which can be restricted to closed subsets V ⊆ S. ∼ Suffices to check Rα∗Z = Z.

Found a morphism Z → Rα∗Z and then checked isomorphism prop- erty on stalks (well, “checked”). Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients

1 Different notions of cohomology

2 Cohomology with integral coefficients

3 Cohomology with real coefficients Finite case Profinite case General case Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients The result

By R, we denote the condensed abelian group sending S ∈ Comp to C(S, R) = HomTop(S, R) with real topology. Theorem. We have

0 ≥1 Hcond(S, R) = C(S, R), Hcond(S, R) = 0.

More precisely, if S• → S is a profinite hypercover, the complex

0 → C(S, R) → C(S0, R) → C(S1, R) → · · ·

satisfies a quantified version of exactness: For f ∈ C(Si , R) with d(f ) = 0 and ε > 0, can write f = d(g) with

kgk := max |g(s)| ≤ (i + 2 + ε)kf k . ∞ s∈S ∞ Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Remarks

• • Of course, this is not Hsheaf (S, R) = H (S, Z) ⊗ R because we respect the topology on R.

≥1 For the sheaf F = C(·, R), we have Hsheaf (S, F) = 0 as F is soft. • • Thus Hsheaf (S, F) = Hcond(S, F).

ϕ For any morphism of compact spaces S −→ S0, the induced map ∗ 0 ϕ C(S , R) −→ C(S, R) has norm at most 1:

∗ 0 kϕ (f )k∞ = kf ◦ ϕk∞ ≤ kf k∞ for all f ∈ C(S , R). Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients

1 Different notions of cohomology

2 Cohomology with integral coefficients

3 Cohomology with real coefficients Finite case Profinite case General case Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Finite case

Let S and all Si be finite.

0 Another finite hypercover of S is given by Sj := S, all structure maps being the identity.

These two hypercovers are homotopy-equivalent as simplicial sets, inducing a homotopy equivalence of the complexes

0 → C(S, R) C(S0, R) C(S1, R) ···

id 0 id 0 → C(S, R) C(S, R) C(S, R) ··· Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Finite case, continued

0 → C(S, R) C(S0, R) C(S1, R) ···

id 0 id 0 → C(S, R) C(S, R) C(S, R) ···

In the lower complex, every f with df = 0 has preimage g with

kgk∞ = kf k∞.

The chain homotopy hi : C(Si , R) → C(Si−1, R) is the alternating sum of i + 1 pullback maps. Thus hi has norm ≤ i + 1.

Combining these results, each f ∈ C(Si , R) with df = 0 has a preimage g ∈ C(Si−1, R) with kgk∞ ≤ (i + 2)kf k∞. Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients

1 Different notions of cohomology

2 Cohomology with integral coefficients

3 Cohomology with real coefficients Finite case Profinite case General case Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Profinite case

Let S and each Si be profinite.

Write S = lim Sj and S = lim Sj with each Sj , Sj finite and ←−j i ←−j i i j j S• → S a hypercover.

By the previous case,

j j j 0 → C(S , R) → C(S0, R) → C(S1, R) → · · ·

is exact, and each cocycle f in degree i ≥ 0 can be written as f = dg

with kgk∞ ≤ (i + 2)kf k∞. Passing to the filtered colimit, the same holds for

0 → lim C(Sj , ) → lim C(Sj , ) → lim C(Sj , ) → · · · −→j R −→j 0 R −→j 1 R Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Completion

We have a morphism of complexes

0 lim C(Sj , ) lim C(Sj , ) lim C(Sj , ) ··· −→i R −→j 0 R −→j 1 R

0 C(S, R) C(S0, R) C(S1, R) ···

such that each lim C(Sj , ) → C(S , ) is an isometric and dense −→i i R i R embedding, i.e. completion map of normed vector spaces.

Let now f ∈ C(Si , R) satisfy df = 0. Pick a first approximation f (1) ∈ lim C(Sj , ) with kf (1) − f k “sufficiently small”. −→i i R ∞ Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Approximation

f ∈ C(Si , R) satisfies df = 0. Have an approximation f (1) ∈ lim C(Sj , ) with kf (1) − f k “small”. −→i i R ∞

(1) (1) (1) Then kdf k∞ = kd(f − f )k∞ ≤ (i + 2)kf − f k∞ is “small”, so we find (similar to previous proof) g (1) ∈ lim C(Sj , ) with −→j i−1 R (1) (1) (1) (1) kg k∞ ≤ (i + 2)kf k∞ and kdg − f k∞ “small”.

(1) In particular, we can make kf − dg k∞ arbitrarily small. Pick an approximation f (2) for f − dg (1) and repeat. (n) P (n) Now kg k∞ → 0 rapidly, so that g := n g exists in C(Si−1, R). Then dg = f and

X (n) X (n) kgk∞ ≤ kg k∞ ≤ (i + 2) kf k∞ ≤ (i + 2 + ε)kf k∞ n n if we choose our approximations sufficiently good. Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients

1 Different notions of cohomology

2 Cohomology with integral coefficients

3 Cohomology with real coefficients Finite case Profinite case General case Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Local approximations

Let S be a compact space and S• a profinite hypercover. Let f ∈ C(Si , R) satisfy df = 0.

Pick s ∈ S, we first find an approximation “near” s: The hypercover S• × {s} → {s} is handled by the above proof, so S we find gs ∈ C(Si−1 ×S {s}, R) with dg = f | , kg k ≤ (i + 2 + ε)kf | k . s Si−1×S {s} s ∞ Si ×S {s} ∞

Extend gs to a continuous function g˜s : Si−1 → R. Then (dg˜s − f )(Si ×S {s}) = 0, so there exists an open neighbour-

hood Us 3 s with k(dg˜s − f )|Si ×S Us k∞ “small”. By compactness, finitely many such neighbourhoods cover S:

Sn We can cover S = j=1 Uj with functions gj ∈ C(Si−1, R) such that k(dg − f )| k is “small” and kg k ≤ (i + 2 + ε)kf k . j Si ×S Uj ∞ j ∞ ∞ Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Glueing approximations

Sn We can cover S = j=1 Uj with functions gj ∈ C(Si−1, R) such that k(dg − f )| k is “small” and kg k ≤ (i + 2 + ε)kf k . j Si ×S Uj ∞ j ∞ ∞ Pick a partition of unity ρ for this cover, i.e. compactly supported P ρj ∈ C(Uj , [0, 1]) with 1 = j ρj . Pullback to a partition of unity on Si and Si−1.

(1) X g := gj ρj . j (1) =⇒ kg k∞ ≤ (i + 2 + ε)kf k∞, (1) X kf − dg k = k ρj (f − dgj )k∞ ≤ max k(dgj − f )|S × U k . ∞ j i S j ∞ j

Repeat for f (2) = f − dg (1). Similar arguments as before show that P (m) g := m g satisfies dg = f and kgk∞ ≤ (i + 2 + constant · ε)kf k∞. Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Summary and References

Singular cohomology is bad for our purposes.

Čech=sheaf cohomology is good for Z and C(·, R).

There is a quantitative exactness result for R.

The precise definitions of “small” are in Scholze’s lecture notes.

A gentle (but lengthy) introduction to simplicial sets, hypercovers and much more is https://math.stanford.edu/~conrad/papers/hypercover.pdf. Thank you for listening!