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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 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
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