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a72760f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | import OAI.NumberTheory.PiExponent.Approximation.ClosedImmersionSerreTransfer
import OAI.NumberTheory.PiExponent.Cohomology.EulerExact
namespace OAI
namespace PiExponent.ClosedImmersionSerreTransfer
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits CategoryTheory.Abelian
open TopologicalSpace Opposite PiExponentSeshadri.Geometry
variable {X Y : Scheme.{0}} (f : X ⟶ Y)
local instance closedPushforwardLinearSourceHasExt : HasExt.{1} X.Modules := schemeHasExt
local instance closedPushforwardLinearTargetHasExt : HasExt.{1} Y.Modules := schemeHasExt
lemma baseScalars_comp_actual (p : Y ⟶ Spec (CommRingCat.of ℂ)) :
baseScalars (f ≫ p) = f.appTop.hom.comp (baseScalars p) := by
ext c
simp only [baseScalars, Scheme.Hom.comp_appTop, CommRingCat.hom_comp,
RingHom.comp_apply]
lemma scalar_naturality (p : Y ⟶ Spec (CommRingCat.of ℂ)) (U : Y.Opens) (r : ℂ) :
restrictScalar X (f ⁻¹ᵁ U) (baseScalars (f ≫ p) r) =
f.app U (restrictScalar Y U (baseScalars p r)) := by
rw [baseScalars_comp_actual]
change X.presheaf.map (homOfLE le_top).op (f.appTop (baseScalars p r)) = _
exact (CategoryTheory.congr_fun (f.naturality (homOfLE le_top).op) (baseScalars p r)).symm
theorem complexLinearPushforward (p : Y ⟶ Spec (CommRingCat.of ℂ)) :
letI := sheafComplexLinear (f ≫ p)
letI := sheafComplexLinear p
(Scheme.Modules.pushforward f).Linear ℂ := by
let := sheafComplexLinear (f ≫ p)
let := sheafComplexLinear p
constructor
intro M N g r
ext U m
let y : Γ(N, f ⁻¹ᵁ U) := g.val.app (op (f ⁻¹ᵁ U)) m
change restrictScalar X (f ⁻¹ᵁ U) (baseScalars (f ≫ p) r) • y =
f.app U (restrictScalar Y U (baseScalars p r)) • y
rw [scalar_naturality]
variable [IsClosedImmersion f]
lemma cohomologyMap_comp {M N : X.Modules} {n m k : ℕ}
(x : cohomology M n) (y : Ext.{1} M N m) (h : n + m = k) :
cohomologyMap f N k (x.comp y h) =
(cohomologyMap f M n x).comp (y.mapExactFunctor (Scheme.Modules.pushforward f)) h :=
PiExponentSeshadri.ExtSectionComparison.mapFrom_comp
(Scheme.Modules.pushforward f) (structureMap f) x y h
def typedCohomologyMap (M : X.Modules) (n : ℕ) :
cohomology M n →+ cohomology ((Scheme.Modules.pushforward f).obj M) n :=
cohomologyMap f M n
lemma typedCohomologyMap_comp {M N : X.Modules} {n m k : ℕ}
(x : cohomology M n) (y : Ext.{1} M N m) (h : n + m = k) :
typedCohomologyMap f N k (x.comp y h) =
(typedCohomologyMap f M n x).comp
(y.mapExactFunctor (Scheme.Modules.pushforward f)) h :=
cohomologyMap_comp f x y h
lemma typedCohomologyMap_bijective (M : X.Modules) (n : ℕ) :
Function.Bijective (typedCohomologyMap f M n) :=
cohomologyMap_bijective f M n
def cohomologyLinearMap (p : Y ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) (n : ℕ) :
letI := Module.compHom (cohomology M n) (baseScalars (f ≫ p))
letI := Module.compHom (cohomology ((Scheme.Modules.pushforward f).obj M) n) (baseScalars p)
cohomology M n →ₗ[ℂ] cohomology ((Scheme.Modules.pushforward f).obj M) n := by
letI := Module.compHom (cohomology M n) (baseScalars (f ≫ p))
letI := Module.compHom (cohomology ((Scheme.Modules.pushforward f).obj M) n) (baseScalars p)
exact { toFun := typedCohomologyMap f M n
map_add' := (typedCohomologyMap f M n).map_add
map_smul' := by
intro r x
change typedCohomologyMap f M n ((baseScalars (f ≫ p) r) • x) =
(baseScalars p r) • (typedCohomologyMap f M n x)
rw [Ext.smul_eq_comp_mk₀, typedCohomologyMap_comp, Ext.smul_eq_comp_mk₀,
Ext.mapExactFunctor_mk₀]
congr 2
ext U m
change Γ(M, f ⁻¹ᵁ U) at m
change restrictScalar X (f ⁻¹ᵁ U) (baseScalars (f ≫ p) r) • m =
f.app U (restrictScalar Y U (baseScalars p r)) • m
rw [scalar_naturality] }
lemma cohomologyLinearMap_eq (p : Y ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) (n : ℕ) :
⇑(cohomologyLinearMap f p M n) = typedCohomologyMap f M n := rfl
def cohomologyLinearEquiv (p : Y ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) (n : ℕ) :
letI := Module.compHom (cohomology M n) (baseScalars (f ≫ p))
letI := Module.compHom (cohomology ((Scheme.Modules.pushforward f).obj M) n) (baseScalars p)
cohomology M n ≃ₗ[ℂ] cohomology ((Scheme.Modules.pushforward f).obj M) n := by
exact LinearEquiv.ofBijective (cohomologyLinearMap f p M n)
(by rw [cohomologyLinearMap_eq]; exact typedCohomologyMap_bijective f M n)
theorem finiteDimensional_of_pushforward
(p : Y ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) (n : ℕ)
(hfinite : letI := Module.compHom (cohomology ((Scheme.Modules.pushforward f).obj M) n) (baseScalars p)
FiniteDimensional ℂ (cohomology ((Scheme.Modules.pushforward f).obj M) n)) :
letI := Module.compHom (cohomology M n) (baseScalars (f ≫ p))
FiniteDimensional ℂ (cohomology M n) := by
let := Module.compHom (cohomology M n) (baseScalars (f ≫ p))
let := Module.compHom (cohomology ((Scheme.Modules.pushforward f).obj M) n) (baseScalars p)
have := hfinite
exact FiniteDimensional.of_injective (cohomologyLinearEquiv f p M n).toLinearMap
(cohomologyLinearEquiv f p M n).injective
lemma cohomologyDimension_pushforward
(p : Y ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) (n : ℕ) :
cohomologyDimension p ((Scheme.Modules.pushforward f).obj M) n =
cohomologyDimension (f ≫ p) M n := by
let := Module.compHom (cohomology M n) (baseScalars (f ≫ p))
let := Module.compHom (cohomology ((Scheme.Modules.pushforward f).obj M) n) (baseScalars p)
exact (cohomologyLinearEquiv f p M n).finrank_eq.symm
theorem euler_pushforward (p : Y ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) (d : ℕ) :
eulerCharacteristic p d ((Scheme.Modules.pushforward f).obj M) =
eulerCharacteristic (f ≫ p) d M := by
apply Finset.sum_congr rfl
intro n hn
rw [cohomologyDimension_pushforward]
end
end PiExponent.ClosedImmersionSerreTransfer
end OAI
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