File size: 2,534 Bytes
d7f9a33 | 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 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import OAI.Algebra.AuslanderReiten.Resolutions.BinaryKoszul
import OAI.Algebra.AuslanderReiten.Ext.ExtCentralGeneration
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian
open scoped ModuleCat.Algebra
namespace ArExplicit.ResolutionProducts
universe u
variable {k R : Type u} [Field k] [Ring R] [Algebra k R]
variable {X : ModuleCat.{u} R}
variable (a b : Ext X X 3)
variable (e : ∀ n : ℕ, Ext X X (3*n) ≃ₗ[k] BinaryKoszul.Piece k n)
variable
(ha : ∀ (n : ℕ) (x : Ext X X (3*n)),
e (n+1) (a.comp x (by omega)) = BinaryKoszul.mulX k n (e n x))
(hb : ∀ (n : ℕ) (x : Ext X X (3*n)),
e (n+1) (b.comp x (by omega)) = BinaryKoszul.mulY k n (e n x))
include ha hb
theorem cubicCommute_of_coefficients : a.comp b rfl = b.comp a rfl := by
have h0a := ha 0 (Ext.mk₀ (𝟙 X))
have h0b := hb 0 (Ext.mk₀ (𝟙 X))
simp only [Ext.comp_mk₀_id] at h0a h0b
apply (e 2).injective
rw [ha, hb, h0a, h0b]
exact LinearMap.congr_fun (BinaryKoszul.mul_commute k 0) _
theorem cubicGeneration_of_coefficients (n : ℕ) (x : Ext X X (3*(n+1))) :
∃ u v : Ext X X (3*n),
x = a.comp u (by omega) + b.comp v (by omega) := by
obtain ⟨⟨u,v⟩,huv⟩ := BinaryKoszul.row_surjective k n (e (n+1) x)
refine ⟨(e n).symm u, (e n).symm v, ?_⟩
apply (e (n+1)).injective
rw [map_add, ha, hb, LinearEquiv.apply_symm_apply, LinearEquiv.apply_symm_apply]
exact huv.symm
theorem cubicCentral_of_coefficients
(hzero : ∀ x : Ext X X 0,
a.comp x (by omega : 3+0=3) = x.comp a (by omega : 0+3=3) ∧
b.comp x (by omega : 3+0=3) = x.comp b (by omega : 0+3=3)) :
∀ (n : ℕ) (x : Ext X X (3*n)),
a.comp x (by omega : 3+3*n=3*(n+1)) = x.comp a (by omega) ∧
b.comp x (by omega : 3+3*n=3*(n+1)) = x.comp b (by omega) :=
cubicCentral_of_generation a b (cubicCommute_of_coefficients a b e ha hb)
hzero (cubicGeneration_of_coefficients a b e ha hb)
omit hb in
theorem cubicFirst_right_of_coefficients (n : ℕ) (x : Ext X X (3*n))
(hc : a.comp x (by omega : 3+3*n=3*(n+1)) = x.comp a (by omega)) :
e (n+1) (x.comp a (by omega)) = BinaryKoszul.mulX k n (e n x) :=
(congrArg (e (n+1)) hc.symm).trans (ha n x)
omit ha in
theorem cubicSecond_right_of_coefficients (n : ℕ) (x : Ext X X (3*n))
(hc : b.comp x (by omega : 3+3*n=3*(n+1)) = x.comp b (by omega)) :
e (n+1) (x.comp b (by omega)) = BinaryKoszul.mulY k n (e n x) :=
(congrArg (e (n+1)) hc.symm).trans (hb n x)
end ArExplicit.ResolutionProducts
end
end OAI
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