Theorems · Definition · commutative algebra

Algebra.PreSubmersivePresentation.differential

{R : Type u} →
  {S : Type v} →
    {ι : Type w} →
      {σ : Type t} →
        [inst : CommRing R] →
          [inst_1 : CommRing S] →
            [inst_2 : Algebra R S] →
              (P : Algebra.PreSubmersivePresentation R S ι σ) → [Finite σ] → (σ → P.Ring) →ₗ[P.Ring] σ → P.Ring

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.

Defined in

Mathlib.RingTheory.Extension.Presentation.Submersive

Cited by

2 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraFinite