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 Different notions of cohomology 1 Cohomology from topology Cohomology from the topos Cohomology with integral coefficients 2 Cohomology with real coefficients 3

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 S n = Hom Top ( n -Simplex , S ) . Turn into a chain complex C • : · · · → Z [ S 2 ] → Z [ S 1 ] → Z [ S 0 ] → 0 , where d i is the alternating sum of the i + 1 face maps. Then H • sing ( S , A ) = cohomology of Hom Ab ( 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 ) = Hom Top ( U , A discrete ) for all U . For a finite open cover U = { U i } i ∈ I on S , form a cosimplicial space � S 0 := S n = S 0 × U , S · · · × S S 0 . � �� � n +1 times The alternating sum of the face (projection) maps S n → S n − 1 give 0 → Γ( S 0 , A ) → Γ( S 1 , A ) → Γ( S 2 , 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. Čech ( S , A ) ∼ H • = H • sheaf ( S , A ) . If S is a profinite set and A discrete, H 0 Čech ( S , A ) ∼ = H 0 sheaf ( S , A ) ∼ = Hom Top ( S , A ) . H 0 sing ( S , A ) ∼ = Hom Set ( S , A ) . Sheaf (Čech) cohomology is better suited for condensed mathemat- ics.

Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Different notions of cohomology 1 Cohomology from topology Cohomology from the topos Cohomology with integral coefficients 2 Cohomology with real coefficients 3

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 ) ∼ = Hom Cond( Ab ) ( Z [ S ] , A ), conclude cond ( S , A ) ∼ H • = 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 ⇒ H ≥ 1 Γ( S , · ) : Cond( Ab ) → Ab is exact = cond ( S , · ) = 0. In general, we want a “resolution” in Comp S • = ( S n ) n ≥ 0 + simplicial structure with each S n extremally disconnected. These should give rise to a projective resolution · · · → Z [ S 2 ] → Z [ S 1 ] → Z [ S 0 ] → Z [ S ] → 0 in Cond( Ab ). As Hom Cond( Ab ) ( Z [ S n ] , · ) ∼ = Γ( S n , · ) is exact, Z [ S n ] is projective in Cond( Ab ).

Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Let’s try Čech Pick S 0 → S surjective such that S 0 is extremally disconnected (e.g. Stone-Čech compactification of S discrete ). For n ≥ 1, let S n = S 0 × S · · · × S S 0 . � �� � n +1 times Usual arguments show the Čech complex · · · → Z [ S 2 ] → Z [ S 1 ] → Z [ S 0 ] → Z [ S ] → 0 is exact! S n not necessarily extremally disconnected for n ≥ 1. Problem:

Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Hypercover Need something better: Pick S 0 → S surjective with S 0 extremally disconnected. Pick S 1 ։ S 0 × S S 0 with S 1 extremally disconnected. Pick   d 1 ( u ) = d 1 ( v ) ,     ( u , v , w ) ∈ S 1 × S 1 × S 1 | d 2 ( u ) = d 1 ( w ) , , S 2 ։   d 2 ( v ) = d 2 ( w )   π 1 , 2 where d 1 , 2 is S 1 → S 0 × S S 0 → S 0 . − − Generally pick S n +1 ։ Coskeleton at level n + 1 of the truncated simplical set S 0 , . . . , S n .

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 S n is extremally discon- nected (at least, Z [ S n ] is Hom Cond( Ab ) ( · , A )-acyclic). This yields a projective (acyclic) resolution · · · → Z [ S 2 ] → Z [ S 1 ] → Z [ S 0 ] → Z [ S ] → 0 . cond ( S , A ) = Ext • ( Z [ S ] , A ) is the cohomology of Then H • 0 → Hom Cond( Ab ) ( Z [ S 0 ] , A ) → Hom( Z [ S 1 ] , A ) → Hom( Z [ S 2 ] , A ) · · · =0 → Γ( S 0 , A ) → Γ( S 1 , A ) → Γ( S 2 , A ) · · · , where each codifferential is the alternating sum of face maps.

Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Different notions of cohomology 1 Cohomology with integral coefficients 2 Case of profinite sets General case Cohomology with real coefficients 3

Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients The result Theorem (Dyckhoff, 1976). There is an isomorphism cond ( S , Z ) ∼ H • = 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, � Hom Top ( S , Z ) , n = 0 H n cond ( S , Z ) = H n sheaf ( S , Z ) = n ≥ 1 . 0 ,

Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Different notions of cohomology 1 Cohomology with integral coefficients 2 Case of profinite sets General case Cohomology with real coefficients 3

Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Computing H • sheaf ( S , Z ) − j S j be profinite. Let S = lim ← Then sheaf ( S , Z ) ∼ Čech ( S , Z ) ∼ H • = H • → j H • Čech ( S j , Z ) . = lim − Eilenberg-Steenrod: Foundations of algebraic topology , Chapter X, Theorem 3.1 Now H ≥ 1 Čech ( S j , Z ) = 0. Thus H ≥ 1 sheaf ( S , Z ) = 0 and H 0 sheaf ( S , Z ) = Γ( S , Z ) = Hom Top ( S , Z ) .

Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Computing H • cond ( S , Z ) Certainly H 0 cond ( S , Z ) = Hom Top ( S , Z ), so we have to show H ≥ 1 cond ( S , Z ) = 0. Pick an e.d. hypercover S • → S , and for each S j choose finite • → S j such that S n = lim hypercover S j − j S j n . ← Then S j is extremally disconnected, so that 0 → Γ( S j , Z ) → Γ( S j 0 , Z ) → Γ( S j 1 , Z ) → · · · is exact. Taking filtered colimits shows exactness of 0 → Γ( S , Z ) → Γ( S 0 , Z ) → Γ( S 1 , Z ) → · · · . Thus H ≥ 1 cond ( S , Z ) = 0.

Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Different notions of cohomology 1 Cohomology with integral coefficients 2 Case of profinite sets General case Cohomology with real coefficients 3

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 lim F ( V ֒ → S ) . U �→ ← − U ⊇ V closed in S α ∗ is left exact and Γ sheaf ( S , · ) ◦ α ∗ = Γ cond ( S , · ). We have to show R α ∗ Z ∼ = Z in D (abelian sheaves on S ), as then H • cond ( S , Z ) = H • ( R Γ cond ( S , Z )) = H • ( R Γ sheaf ( S , · ) ◦ R α ∗ ( Z )) ∗ = H • ( R Γ sheaf ( S , Z )) = H • sheaf ( S , Z ) .

Different notions of cohomology Cohomology with integral coefficients Cohomology with real coefficients Towards R α ∗ Z R α ∗ Z is a complex of abelian sheaves on S . H 0 ( R α ∗ Z ) ∼ = α ∗ Z as abelian sheaves on S . The global sections Γ sheaf ( S , H 0 ( R α ∗ Z )) ∼ = Γ cond ( S , Z ) induce a morphism of sheaves Z → H 0 ( R α ∗ Z )). This yields a morphism of complexes of abelian sheaves Z (concentrated in degree 0) → R α ∗ Z . We prove this is an isomorphism on stalks. Fix s ∈ S .