Theorems · Theorem · commutative algebra

Algebra.Generators.cotangentSpaceBasis_repr_tmul

∀ {R : Type u} {S : Type v} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] {ι : Type w}
  (P : Algebra.Generators R S ι) (r : S) (x : P.Ring) (i : ι),
  (P.cotangentSpaceBasis.repr (r ⊗ₜ[P.Ring] (KaehlerDifferential.D R P.Ring) x)) i =
    r * (MvPolynomial.aeval P.val) ((MvPolynomial.pderiv i) x)

Defined in: Mathlib.RingTheory.Extension.Cotangent.Basic
Cited by: 4 results in Mathlib
Foundations: Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing CommRing Algebra

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Cites28

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
RingHom.idstatement · cited by 18,349
CommRingstatement and proof · cited by 17,173
Algebrastatement and proof · cited by 11,388
Finsuppstatement · cited by 5,255
LinearEquivstatement · cited by 3,317
AlgHomstatement · cited by 3,236
mul_commproof · cited by 2,262
MvPolynomialstatement · cited by 2,140
TensorProduct.tmulstatement · cited by 1,182
Module.Basis.reprstatement · cited by 498
MvPolynomial.aevalstatement and proof · cited by 298

Cited by4

Results whose statement or proof uses this declaration.

Algebra.PreSubmersivePresentation.cotangentComplexAux_applyproof · cited by 3
Algebra.Generators.cotangentRestrict_mkproof · cited by 1
Algebra.Presentation.differentials.comm₁₂_singleproof · cited by 1
Algebra.Generators.cotangentSpaceBasis_repr_one_tmulproof · cited by 1