English
The carrier of coefficients remains stable under scalar towers and HasCoeffs constraints.
Русский
Носитель коэффициентов устойчив под башнями скаляров и условиях HasCoeffs.
LaTeX
$$$\\text{CoeffCarrier}(P)\\land\\text{HasCoeffs}(P)\\Rightarrow\\text{stable under scalar towers}$$$
Lean4
/-- The differential of a `P : PreSubmersivePresentation` is a `P.Ring`-linear map on
`σ → P.Ring`:
The `j`-th standard basis vector, corresponding to the `j`-th relation of `P`, is mapped
to the vector of partial derivatives of `P.relation j` with respect
to the coordinates `P.map i` for all `i : σ`.
The determinant of this map is the Jacobian of `P` used to define when a `PreSubmersivePresentation`
is submersive. See `PreSubmersivePresentation.jacobian`.
-/
noncomputable def differential : (σ → P.Ring) →ₗ[P.Ring] (σ → P.Ring) :=
Basis.constr P.basis P.Ring (fun j i : σ ↦ MvPolynomial.pderiv (P.map i) (P.relation j))
