theorem corollary23_jacobian_rank {k : Type u_1} [Field k] {n m : ℕ} (f : Fin m → MvPolynomial (Fin n) k) (x : Fin n → k) (hx : ∀ (i : Fin m), (MvPolynomial.eval x) (f i) = 0) (hI : Ideal.span (Set.range f) ≤ maxIdealOfPoint x) (hdim : ringKrullDim (localRingAtPoint (Ideal.span (Set.range f)) x hI) = ↑(n - m)) : (jacobianMatrix f x).rank = m ↔ IsRegularLocalRing (localRingAtPoint (Ideal.span (Set.range f)) x hI)

Corollary 23 / Proposition 31 (Lecture 19), restated. For a complete intersection V(f_1,…,f_m) of expected dimension n - m, the local ring at x is regular iff the Jacobian matrix has full rank m at x.

theorem corollary23_jacobian_hypersurface {k : Type u_1} [Field k] {n : ℕ} (P : MvPolynomial (Fin n) k) (x : Fin n → k) (hx : (MvPolynomial.eval x) P = 0) (hI : Ideal.span (Set.range fun (x : Fin 1) => P) ≤ maxIdealOfPoint x) (hdim : ringKrullDim (localRingAtPoint (Ideal.span (Set.range fun (x : Fin 1) => P)) x hI) = ↑(n - 1)) : (∃ (i : Fin n), (MvPolynomial.eval x) ((MvPolynomial.pderiv i) P) ≠ 0) ↔ IsRegularLocalRing (localRingAtPoint (Ideal.span (Set.range fun (x : Fin 1) => P)) x hI)

Hypersurface case of the Jacobian criterion: for V(P) ⊂ 𝔸ⁿ of dimension n-1, the point x is smooth iff some partial derivative ∂P/∂x_i(x) is nonzero.