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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 | import OAI.AlgebraicGeometry.StableCoordinate.DeterminantRigidity
import OAI.AlgebraicGeometry.StableCoordinate.DeterminantGradedObstruction
namespace OAI
/-!
# The final elementary step of determinant rigidity
A derivation which fixes the three polynomial base generators is determined
by a single coefficient. Preservation of the actual fiber grading makes that
coefficient have weight `-2`. If it lies in the polynomial base, whose weights
are nonnegative, it vanishes; so does the derivation. The coefficient's
polynomiality is supplied separately by the locally nilpotent chart argument.
-/
namespace StableCoordinate.DeterminantRing
variable (k : Type*) [CommRing k]
/-- The actual weight obstruction annihilates every vertical derivation whose
coefficient is polynomial in the base variables. -/
theorem derivation_eq_zero_of_fixed_base_and_coefficient_mem
(D : Derivation k (C k) (C k))
(ha : D (a k) = 0) (hd : D (d k) = 0) (hu : D (u k) = 0)
(hgraded : ∀ n : ℤ, ∀ x ∈ Obstruction.fiberPiece k n,
D x ∈ Obstruction.fiberPiece k n)
(hq : rigidityCoefficient D (b k) (c k) ∈ (baseMap (k := k)).range) :
D = 0 := by
have hb : b k ∈ Obstruction.fiberPiece k (-1) := by
change Obstruction.cX k 2 ∈ Obstruction.fiberPiece k (-1)
simpa [Obstruction.fiberWeight] using Obstruction.cX_mem_fiberPiece k 2
have hc : c k ∈ Obstruction.fiberPiece k (-1) := by
change Obstruction.cX k 3 ∈ Obstruction.fiberPiece k (-1)
simpa [Obstruction.fiberWeight] using Obstruction.cX_mem_fiberPiece k 3
have hzero : rigidityCoefficient D (b k) (c k) = 0 :=
DeterminantGradedObstruction.coefficient_eq_zero_of_mem_baseRange k
(hgraded (-1) (b k) hb) (hgraded (-1) (c k) hc) hq
obtain ⟨hb', hc'⟩ := derivation_rigidityCoefficient D (a k) (d k) (b k) (c k)
(determinant_relation k) ha hd
rw [hzero, mul_zero] at hb' hc'
exact derivation_eq_zero k D ha hd hb' hc' hu
end StableCoordinate.DeterminantRing
end OAI
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