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Stabilized ranks and the radical quotient

Lax342547.FlatRank · concepts/Lax342547/FlatRank.lean · lax-342547

proven

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    Natural Language Statement

    Lemma

    The flat-level and quotient steps of Lemma 5.2. These statements apply to arbitrary finite-dimensional symmetric bilinear forms; no positivity is used.

    Concept map
    1 concept
    100%
    Proven claimThis concept
    Evidence

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 exists_commuting_idempotents proven

    2 exists_flat_level proven

    3 quotient_restriction_surjective proven

    4 quotientPairing_nondegenerate proven

    Lean source view on GitHub

    1import Mathlib.LinearAlgebra.BilinearForm.Properties
    2import Mathlib.LinearAlgebra.Quotient.Bilinear
    3import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
    4
    5/-!
    6---
    7title: Stabilized ranks and the radical quotient
    8type: lemma
    9---
    10The flat-level and quotient steps of Lemma 5.2. These statements apply to
    11arbitrary finite-dimensional symmetric bilinear forms; no positivity is used.
    12-/
    13
    14namespace Lax342547.FlatRank
    15
    16axiom exists_flat_level (r : ℕ → ℕ) (A R J : ℕ) (hmono : Monotone r)
    17 (hbound : ∀ j ≤ J, r j ≤ R) (hroom : A + 2 * R + 2 ≤ J) :
    18 ∃ j, A ≤ j ∧ j + 2 ≤ J ∧ r j = r (j + 1) ∧ r (j + 1) = r (j + 2)
    19
    20variable {K V : Type} [Field K] [AddCommGroup V] [Module K V]
    21
    22noncomputable def quotientPairing (B : LinearMap.BilinForm K V) (hB : B.IsSymm) :
    23 LinearMap.BilinForm K (V ⧸ B.ker) :=
    24 LinearMap.IsRefl.liftQ₂ B B.ker hB.isRefl le_rfl
    25
    26axiom quotientPairing_nondegenerate (B : LinearMap.BilinForm K V) (hB : B.IsSymm) :
    27 (quotientPairing B hB).Nondegenerate
    28
    29axiom quotient_restriction_surjective [FiniteDimensional K V]
    30 (B : LinearMap.BilinForm K V) (hB : B.IsSymm) (U : Submodule K V)
    31 (hflat : Module.finrank K (LinearMap.range (B.restrict U)) =
    32 Module.finrank K (LinearMap.range B)) :
    33 Function.Surjective (B.ker.mkQ.comp U.subtype)
    34
    35axiom exists_commuting_idempotents [FiniteDimensional K V] {ι : Type}
    36 (B : LinearMap.BilinForm K V) (hB : B.IsSymm) (U : Submodule K V)
    37 (hflat : Module.finrank K (LinearMap.range (B.restrict U)) =
    38 Module.finrank K (LinearMap.range B))
    39 (shift : ι → U →ₗ[K] V)
    40 (hadj : ∀ a (x y : U), B (shift a x) y = B x (shift a y))
    41 (hidem : ∀ a (x y : U), B (shift a x) (shift a y) = B (shift a x) y)
    42 (hcomm : ∀ a c (x y : U), B (shift a x) (shift c y) = B (shift c x) (shift a y)) :
    43 ∃ M : ι → Module.End K (V ⧸ B.ker),
    44 (∀ a (x : U), M a (B.ker.mkQ x) = B.ker.mkQ (shift a x)) ∧
    45 (∀ a x y, quotientPairing B hB (M a x) y = quotientPairing B hB x (M a y)) ∧
    46 (∀ a, M a * M a = M a) ∧ ∀ a c, M a * M c = M c * M a
    47
    48end Lax342547.FlatRank
    49
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