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we don't need to assume that the map is pointed

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Floris van Doorn 2017-09-20 22:13:57 -04:00
parent b31658c2f3
commit ee4c9f989a
1 changed files with 11 additions and 13 deletions
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cohomology/serre.hlean
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@ -209,14 +209,12 @@ end unreduced_atiyah_hirzebruch
section serre section serre
universe variable u universe variable u
variables {X : Type} (x₀ : X) (F : X → Type) variables {X B : Type.{u}} (b₀ : B) (F : B → Type) (f : X → B)
{X₁ X₂ : pType.{u}} (f : X₁ →* X₂)
{Z₁ Z₂ : Type.{u}} (g : Z₁ → Z₂)
(Y : spectrum) (s₀ : ℤ) (H : is_strunc s₀ Y) (Y : spectrum) (s₀ : ℤ) (H : is_strunc s₀ Y)
include H include H
definition serre_convergence : definition serre_convergence :
(λn s, uopH^-(n-s)[(x : X), uH^-s[F x, Y]]) ⟹ᵍ (λn, uH^-n[Σ(x : X), F x, Y]) := (λn s, uopH^-(n-s)[(b : B), uH^-s[F b, Y]]) ⟹ᵍ (λn, uH^-n[Σ(b : B), F b, Y]) :=
proof proof
converges_to_g_isomorphism converges_to_g_isomorphism
(unreduced_atiyah_hirzebruch_convergence (unreduced_atiyah_hirzebruch_convergence
@ -237,29 +235,29 @@ section serre
end end
qed qed
definition serre_convergence_of_map : definition serre_convergence_map :
(λn s, uopH^-(n-s)[(x : Z₂), uH^-s[fiber g x, Y]]) ⟹ᵍ (λn, uH^-n[Z₁, Y]) := (λn s, uopH^-(n-s)[(b : B), uH^-s[fiber f b, Y]]) ⟹ᵍ (λn, uH^-n[X, Y]) :=
proof proof
converges_to_g_isomorphism converges_to_g_isomorphism
(serre_convergence (fiber g) Y s₀ H) (serre_convergence (fiber f) Y s₀ H)
begin intro n s, reflexivity end begin intro n s, reflexivity end
begin intro n, apply unreduced_cohomology_isomorphism, exact !sigma_fiber_equiv⁻¹ᵉ end begin intro n, apply unreduced_cohomology_isomorphism, exact !sigma_fiber_equiv⁻¹ᵉ end
qed qed
definition serre_convergence_of_is_conn (H2 : is_conn 1 X) : definition serre_convergence_of_is_conn (H2 : is_conn 1 B) :
(λn s, uoH^-(n-s)[X, uH^-s[F x₀, Y]]) ⟹ᵍ (λn, uH^-n[Σ(x : X), F x, Y]) := (λn s, uoH^-(n-s)[B, uH^-s[F b₀, Y]]) ⟹ᵍ (λn, uH^-n[Σ(b : B), F b, Y]) :=
proof proof
converges_to_g_isomorphism converges_to_g_isomorphism
(serre_convergence F Y s₀ H) (serre_convergence F Y s₀ H)
begin intro n s, exact @uopH_isomorphism_uoH_of_is_conn (pointed.MK X x₀) _ _ H2 end begin intro n s, exact @uopH_isomorphism_uoH_of_is_conn (pointed.MK B b₀) _ _ H2 end
begin intro n, reflexivity end begin intro n, reflexivity end
qed qed
definition serre_convergence_of_pmap (H2 : is_conn 1 X₂) : definition serre_convergence_map_of_is_conn (H2 : is_conn 1 B) :
(λn s, uoH^-(n-s)[X₂, uH^-s[pfiber f, Y]]) ⟹ᵍ (λn, uH^-n[X₁, Y]) := (λn s, uoH^-(n-s)[B, uH^-s[fiber f b₀, Y]]) ⟹ᵍ (λn, uH^-n[X, Y]) :=
proof proof
converges_to_g_isomorphism converges_to_g_isomorphism
(serre_convergence_of_is_conn pt (fiber f) Y s₀ H H2) (serre_convergence_of_is_conn b₀ (fiber f) Y s₀ H H2)
begin intro n s, reflexivity end begin intro n s, reflexivity end
begin intro n, apply unreduced_cohomology_isomorphism, exact !sigma_fiber_equiv⁻¹ᵉ end begin intro n, apply unreduced_cohomology_isomorphism, exact !sigma_fiber_equiv⁻¹ᵉ end
qed qed