openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /ConeQuotientAugmentation.lean
| 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 | |