| import Mathlib.Algebra.BigOperators.Group.Finset.Basic |
| import Mathlib.LinearAlgebra.Quotient.Basic |
| import Mathlib.RingTheory.Length |
|
|
| namespace OAI |
|
|
| namespace PiExponentJets.W22 |
|
|
| open scoped BigOperators |
|
|
| variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M] |
|
|
| theorem length_filtration_step (A B : Submodule R M) (hAB : A ≤ B) : |
| Module.length R B = Module.length R A + Module.length R (B ⧸ A.submoduleOf B) := by |
| have h := Module.length_eq_add_of_exact (A.submoduleOf B).subtype |
| (A.submoduleOf B).mkQ (Submodule.subtype_injective _) |
| (Submodule.mkQ_surjective _) (LinearMap.exact_subtype_mkQ _) |
| rw [(Submodule.submoduleOfEquivOfLe hAB).length_eq] at h |
| exact h |
|
|
| theorem length_eq_sum_filtration |
| (N : ℕ → Submodule R M) (hmono : ∀ i, N i ≤ N (i + 1)) |
| (hzero : N 0 = ⊥) (n : ℕ) : |
| Module.length R (N n) = |
| ∑ i ∈ Finset.range n, Module.length R (N (i + 1) ⧸ (N i).submoduleOf (N (i + 1))) := by |
| induction n with |
| | zero => |
| simp only [Finset.range_zero, Finset.sum_empty] |
| have hlen := congrArg (fun A : Submodule R M => Module.length R A) hzero |
| exact hlen.trans Module.length_eq_zero |
| | succ n ih => |
| rw [length_filtration_step (N n) (N (n + 1)) (hmono n), ih, |
| Finset.sum_range_succ] |
|
|
| theorem module_length_eq_sum_filtration |
| (N : ℕ → Submodule R M) (hmono : ∀ i, N i ≤ N (i + 1)) |
| (hzero : N 0 = ⊥) (n : ℕ) (htop : N n = ⊤) : |
| Module.length R M = |
| ∑ i ∈ Finset.range n, Module.length R (N (i + 1) ⧸ (N i).submoduleOf (N (i + 1))) := by |
| have hlen := congrArg (fun A : Submodule R M => Module.length R A) htop |
| have hlen' : Module.length R (N n) = Module.length R M := |
| hlen.trans Module.length_top |
| exact hlen'.symm.trans (length_eq_sum_filtration N hmono hzero n) |
|
|
| end PiExponentJets.W22 |
|
|
| end OAI |
|
|
