Chapter 3: Cohomology Felix Schremmer Technical University of - - PowerPoint PPT Presentation

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 H•

sing(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: Γ(U, A) = HomTop(U, Adiscrete) for all U. For a finite open cover U = {Ui}i∈I on S, form a cosimplicial space S0 :=

• U,

Sn = S0 ×

S · · · × S S0

• 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•

Čech(U, A). H• Čech(S, A) = lim

− →U H•

Čech(U, A).

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. H•

sheaf(S, A) = R•Γ(S, A).

Compute e.g. using injective resolution A → I•, then H•

sheaf(S, A) = cohomology of Γ(S, I•).

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

Comparison

Lemma. H•

Čech(S, A) ∼

= H•

sheaf(S, A).

If S is a profinite set and A discrete, H0

Čech(S, A) ∼

= H0

sheaf(S, A) ∼

= HomTop(S, A). H0

sing(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 H•

cond(S, ·) : Cond(Ab) → Ab to be the right-derived

functors of Γ(S, ·). Since Γ(S, A) ∼ = HomCond(Ab)(Z[S], A), conclude H•

cond(S, A) ∼

= Ext•

Cond(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 Γ(S, ·) : Cond(Ab) → Ab is exact = ⇒ H≥1

cond(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

• 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 S2 ։

    

(u, v, w) ∈ S1 × S1 × S1 | d1(u) = d1(v), d2(u) = d1(w), d2(v) = d2(w)

    

, where d1,2 is S1 → S0 ×S S0

π1,2

− − → 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 H•

cond(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 H•

cond(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 H•

cond(S, Z) ∼

= H•

sheaf(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, Hn

cond(S, Z) = Hn sheaf(S, Z) =

• HomTop(S, Z),

n = 0 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 H•

sheaf(S, Z)

Let S = lim ← −j Sj be profinite. Then H•

sheaf(S, Z) ∼

= H•

Čech(S, Z) ∼

= lim − →j H•

Čech(Sj, Z).

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

Now H≥1

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

H0

sheaf(S, Z) = Γ(S, Z) = HomTop(S, Z).

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

Computing H•

cond(S, Z)

Certainly H0

cond(S, Z) = HomTop(S, Z), so we have to show

H≥1

cond(S, Z) = 0.

Pick an e.d. hypercover S• → S, and for each Sj choose finite hypercover Sj

• → Sj such that Sn = lim

← −j Sj

n.

Then Sj is extremally disconnected, so that 0 → Γ(Sj, Z) → Γ(Sj

0, Z) → Γ(Sj 1, Z) → · · ·

is exact. Taking filtered colimits shows exactness of 0 → Γ(S, Z) → Γ(S0, Z) → Γ(S1, Z) → · · · . Thus H≥1

cond(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 → lim ← −

U⊇V closed in S

F(V ֒ → 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 H•

cond(S, Z) =H•(RΓcond(S, Z)) = H•(RΓsheaf(S, ·) ◦ Rα∗(Z)) ∗

=H•(RΓsheaf(S, Z)) = H•

sheaf(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. H0(Rα∗Z) ∼ = α∗Z as abelian sheaves on S. The global sections Γsheaf(S, H0(Rα∗Z)) ∼ = Γcond(S, Z) induce a morphism of sheaves Z → H0(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∗Z)s = lim − →

s∈U open

RΓ(U, Rα∗Z) = lim − →

s∈V closed nbh

RΓcond(V , Z). 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 ∋ s yields 0 → lim − →

V ∋s

Γ(S0 ×S V , Z) → lim − →

V ∋s

Γ(S1 ×S V , Z) → · · · ∼ = 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 H0

cond(S, R) = C(S, R),

H≥1

cond(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 g∞ := max

s∈S |g(s)| ≤ (i + 2 + ε)f ∞.

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

Remarks

Of course, this is not H•

sheaf(S, R) = H•(S, Z) ⊗ R because we

respect the topology on R. For the sheaf F = C(·, R), we have H≥1

sheaf(S, F) = 0 as F is soft.

Thus H•

sheaf(S, F) = H• cond(S, F).

For any morphism of compact spaces S

ϕ

− → S′, the induced map C(S′, R)

ϕ∗

− → C(S, R) has norm at most 1: ϕ∗(f )∞ = f ◦ ϕ∞ ≤ f ∞ 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. Another finite hypercover of S is given by S′

j := 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) · · · 0 → C(S, R) C(S, R) C(S, R) · · ·

id id

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

id id

In the lower complex, every f with df = 0 has preimage g with g∞ = f ∞. 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 g∞ ≤ (i + 2)f ∞.

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 ← −j Sj and Si = lim ← −j Sj

i with each Sj, Sj i finite and

Sj

• → Sj a hypercover.

By the previous case, 0 → C(Sj, R) → C(Sj

0, R) → C(Sj 1, R) → · · ·

is exact, and each cocycle f in degree i ≥ 0 can be written as f = dg with g∞ ≤ (i + 2)f ∞. Passing to the filtered colimit, the same holds for 0 → lim − →j C(Sj, R) → lim − →j C(Sj

0, R) → lim

− →j C(Sj

1, R) → · · ·

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

Completion

We have a morphism of complexes lim − →i C(Sj, R) lim − →j C(Sj

0, R)

lim − →j C(Sj

1, R)

· · · C(S, R) C(S0, R) C(S1, R) · · · such that each lim − →i C(Sj

i , R) → C(Si, R) is an isometric and dense

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 − →i C(Sj

i , R) with f (1) − f ∞ “sufficiently small”.

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 − →i C(Sj

i , R) with f (1) − f ∞ “small”.

Then df (1)∞ = d(f (1) − f )∞ ≤ (i + 2)f (1) − f ∞ is “small”, so we find (similar to previous proof) g(1) ∈ lim − →j C(Sj

i−1, R) with

g(1)∞ ≤ (i + 2)f (1)∞ and dg(1) − f (1)∞ “small”. In particular, we can make f − dg(1)∞ arbitrarily small. Pick an approximation f (2) for f − dg(1) and repeat. Now g(n)∞ → 0 rapidly, so that g :=

n g(n) exists in C(Si−1, R).

Then dg = f and g∞ ≤

• n

g(n)∞ ≤ (i + 2)

• n

f (n)∞ ≤ (i + 2 + ε)f ∞ 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} → {s} is handled by the above proof, so

we find gs ∈ C(Si−1 ×S {s}, R) with dgs = f |Si−1×S{s}, gs∞ ≤ (i + 2 + ε)f |Si×S{s}∞. Extend gs to a continuous function ˜ gs : Si−1 → R. Then (d ˜ gs − f )(Si ×S {s}) = 0, so there exists an open neighbour- hood Us ∋ s with (d ˜ gs − f )|Si×SUs∞ “small”. By compactness, finitely many such neighbourhoods cover S: We can cover S = n

j=1 Uj with functions gj ∈ C(Si−1, R) such

that (dgj − f )|Si×SUj∞ is “small” and gj∞ ≤ (i + 2 + ε)f ∞.

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

Glueing approximations

We can cover S = n

j=1 Uj with functions gj ∈ C(Si−1, R) such

that (dgj − f )|Si×SUj∞ is “small” and gj∞ ≤ (i + 2 + ε)f ∞. Pick a partition of unity ρ for this cover, i.e. compactly supported ρj ∈ C(Uj, [0, 1]) with 1 =

j ρj. Pullback to a partition of unity

• n Si and Si−1.

g(1) :=

• j gjρj.

= ⇒ g(1)∞ ≤ (i + 2 + ε)f ∞, f − dg(1)∞ =

• j

ρj(f − dgj)∞ ≤ max

j

(dgj − f )|Si×SUj∞. Repeat for f (2) = f − dg(1). Similar arguments as before show that g :=

m g(m) satisfies dg = f and

g∞ ≤ (i + 2 + constant · ε)f ∞.

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!