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82da87f | 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 | 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
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