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b797a5b | 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 | import OAI.NumberTheory.PiExponent.LocalAlgebra.RegularSequenceResolution
namespace OAI
namespace PiExponentSiegelAux.W30
open scoped Pointwise
variable {R Y Z Q : Type*} [CommRing R] [AddCommGroup Y] [AddCommGroup Z]
[AddCommGroup Q] [Module R Y] [Module R Z] [Module R Q]
def coneQuotientAugmentation (π : Z →ₗ[R] Q) (r : R) : Z →ₗ[R] QuotSMulTop r Q :=
(r • (⊤ : Submodule R Q)).mkQ.comp π
theorem coneQuotientAugmentation_exact (g : Y →ₗ[R] Z) (π : Z →ₗ[R] Q) (r : R)
(hπ : Function.Surjective π) (hg : LinearMap.range g = LinearMap.ker π) :
LinearMap.range (coneBottom g r) = LinearMap.ker (coneQuotientAugmentation π r) := by
ext z
rw [LinearMap.mem_range, LinearMap.mem_ker]
constructor
· rintro ⟨⟨y, w⟩, rfl⟩
have hz : π (g y) = 0 := LinearMap.congr_fun (comp_zero_of_range_eq_ker g π hg) y
change (Submodule.Quotient.mk (π (g y + r • w)) : QuotSMulTop r Q) = 0
apply (Submodule.Quotient.mk_eq_zero (r • (⊤ : Submodule R Q))).mpr
simp only [map_add, map_smul, hz, zero_add]
exact Submodule.smul_mem_pointwise_smul (π w) r ⊤ (Submodule.mem_top)
· intro hz
have hmem : π z ∈ r • (⊤ : Submodule R Q) :=
(Submodule.Quotient.mk_eq_zero (r • (⊤ : Submodule R Q))).mp hz
obtain ⟨q, _, hq⟩ := (Submodule.mem_smul_pointwise_iff_exists _ _ _).mp hmem
obtain ⟨w, hw⟩ := hπ q
have hcycle : π (z - r • w) = 0 := by
simp only [map_sub, map_smul, hw, hq, sub_self]
have hrange : z - r • w ∈ LinearMap.range g := by
rw [hg]
exact hcycle
obtain ⟨y, hy⟩ := LinearMap.mem_range.mp hrange
refine ⟨(y, w), ?_⟩
change g y + r • w = z
rw [hy]
abel
theorem coneQuotientAugmentation_surjective (π : Z →ₗ[R] Q) (r : R)
(hπ : Function.Surjective π) : Function.Surjective (coneQuotientAugmentation π r) :=
(Submodule.mkQ_surjective _).comp hπ
end PiExponentSiegelAux.W30
end OAI
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