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construct serre spectral sequence for any map

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Floris van Doorn 2017-09-20 22:00:58 -04:00
parent f8157068e4
commit b31658c2f3
1 changed files with 13 additions and 2 deletions
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15
cohomology/serre.hlean
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@ -209,7 +209,9 @@ end unreduced_atiyah_hirzebruch
section serre
universe variable u
variables {X : Type} (x₀ : X) (F : X → Type) {X₁ X₂ : pType.{u}} (f : X₁ →* X₂)
variables {X : Type} (x₀ : X) (F : X → Type)
{X₁ X₂ : pType.{u}} (f : X₁ →* X₂)
{Z₁ Z₂ : Type.{u}} (g : Z₁ → Z₂)
(Y : spectrum) (s₀ : ℤ) (H : is_strunc s₀ Y)
include H
@ -235,6 +237,15 @@ section serre
end
qed
definition serre_convergence_of_map :
(λn s, uopH^-(n-s)[(x : Z₂), uH^-s[fiber g x, Y]]) ⟹ᵍ (λn, uH^-n[Z₁, Y]) :=
proof
converges_to_g_isomorphism
(serre_convergence (fiber g) Y s₀ H)
begin intro n s, reflexivity end
begin intro n, apply unreduced_cohomology_isomorphism, exact !sigma_fiber_equiv⁻¹ᵉ end
qed
definition serre_convergence_of_is_conn (H2 : is_conn 1 X) :
(λn s, uoH^-(n-s)[X, uH^-s[F x₀, Y]]) ⟹ᵍ (λn, uH^-n[Σ(x : X), F x, Y]) :=
proof
@ -248,7 +259,7 @@ section serre
(λn s, uoH^-(n-s)[X₂, uH^-s[pfiber f, Y]]) ⟹ᵍ (λn, uH^-n[X₁, Y]) :=
proof
converges_to_g_isomorphism
(serre_convergence_of_is_conn pt (λx, fiber f x) Y s₀ H H2)
(serre_convergence_of_is_conn pt (fiber f) Y s₀ H H2)
begin intro n s, reflexivity end
begin intro n, apply unreduced_cohomology_isomorphism, exact !sigma_fiber_equiv⁻¹ᵉ end
qed