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.