| import OAI.Algebra.Finitistic.Localization.EvaluationModule |
| import Mathlib.Algebra.Algebra.Tower |
|
|
| namespace OAI |
| noncomputable section |
| namespace LittleFinitistic.QuadraticEncoding |
| universe u v w |
| variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t) |
| variable {R : Type v} [Ring R] [Algebra k R] |
|
|
| / |
| def Relations.Satisfies (x : Fin d → R) : Prop := ∀ j, |
| (∑ h, ∑ l, P.quadratic j h l • (x h * x l)) + |
| (∑ h, P.linear j h • x h) + P.constant j • (1 : R) = 0 |
|
|
| namespace Representation |
| variable {P} {Y : Type w} [AddCommGroup Y] [Module k Y] |
|
|
| / |
| def ofAction (x : Fin d → R) (hx : P.Satisfies x) (ρ : R →ₐ[k] Module.End k Y) : |
| Representation P Y where |
| generator h := ρ (x h) |
| relation j := by |
| have h := congrArg ρ (hx j) |
| simpa only [map_add, map_sum, map_smul, map_mul, map_one, map_zero] using h |
|
|
| / |
| def ofModule (x : Fin d → R) (hx : P.Satisfies x) |
| [Module R Y] [IsScalarTower k R Y] : Representation P Y := |
| ofAction x hx (Algebra.lsmul k k Y) |
|
|
| @[simp] theorem ofModule_generator (x : Fin d → R) (hx : P.Satisfies x) |
| [Module R Y] [IsScalarTower k R Y] (h : Fin d) (y : Y) : |
| (ofModule (Y := Y) x hx).generator h y = x h • y := rfl |
| end Representation |
| end LittleFinitistic.QuadraticEncoding |
|
|
| end |
|
|
| end OAI |
|
|
