| import OAI.AlgebraicGeometry.StableCoordinate.ChartIntersection | |
| import OAI.AlgebraicGeometry.StableCoordinate.DeterminantCharts | |
| import OAI.AlgebraicGeometry.StableCoordinate.ObstructionDomains | |
| namespace OAI | |
| /-! | |
| # The actual coefficient-ring intersection for the determinant hypersurface | |
| This is the two-chart coefficient-membership step in source Section 5, now | |
| instantiated on the exact quotient `k[a,d,b,c,u]/(ac-bd-1)`. The chart maps, | |
| their injectivity, the domain property, and independence of `a,d,u` are all | |
| proved constructions. No chart or domain assumption remains in the theorems. | |
| -/ | |
| namespace StableCoordinate.DeterminantRing | |
| open MvPolynomial | |
| variable (k : Type*) [Field k] | |
| /-- If an element of the determinant ring is a coefficient in each of its two | |
| polynomial charts, it belongs to the actual polynomial subring `k[a,d,u]`. -/ | |
| theorem exists_base_of_constant_charts {q : C k} | |
| {v : BaseAway k 0} {w : BaseAway k 1} | |
| (hA : chartMapA k q = Polynomial.C v) | |
| (hD : chartMapD k q = Polynomial.C w) : | |
| ∃ r : Base k, baseMap r = q := by | |
| exact ChartIntersection.exists_base_of_two_ring_charts | |
| (baseMap (k := k)).toRingHom (MvPolynomial.X_ne_zero (0 : Fin 3)) | |
| (StableCoordinate.isRelPrime_X_of_ne (k := k) (i := (0 : Fin 3)) | |
| (j := (1 : Fin 3)) (by decide)) | |
| (chartMapA k).toRingHom (chartMapA_injective k) | |
| (chartMapD k).toRingHom (chartMapD_injective k) | |
| (chartMapA_baseMap k) (chartMapD_baseMap k) hA hD | |
| /-- Exact membership characterization, with both implications retained. -/ | |
| theorem mem_range_baseMap_iff_constant_charts (q : C k) : | |
| q ∈ Set.range (baseMap (k := k)) ↔ | |
| (∃ v : BaseAway k 0, chartMapA k q = Polynomial.C v) ∧ | |
| (∃ w : BaseAway k 1, chartMapD k q = Polynomial.C w) := by | |
| constructor | |
| · rintro ⟨r, rfl⟩ | |
| exact ⟨⟨_, chartMapA_baseMap k r⟩, ⟨_, chartMapD_baseMap k r⟩⟩ | |
| · rintro ⟨⟨v, hA⟩, ⟨w, hD⟩⟩ | |
| exact exists_base_of_constant_charts k hA hD | |
| end StableCoordinate.DeterminantRing | |
| end OAI | |