Theorems · Theorem · ring theory

TensorProduct.gradedMul_algebraMap

∀ {R : Type u_1} {ι : Type u_2} [inst : CommSemiring ι] [inst_1 : Module ι (Additive ℤˣ)] [inst_2 : DecidableEq ι]
  (𝒜 : ι → Type u_3) (ℬ : ι → Type u_4) [inst_3 : CommRing R] [inst_4 : (i : ι) → AddCommGroup (𝒜 i)]
  [inst_5 : (i : ι) → AddCommGroup (ℬ i)] [inst_6 : (i : ι) → Module R (𝒜 i)] [inst_7 : (i : ι) → Module R (ℬ i)]
  [inst_8 : DirectSum.GRing 𝒜] [inst_9 : DirectSum.GRing ℬ] [inst_10 : DirectSum.GAlgebra R 𝒜]
  [inst_11 : DirectSum.GAlgebra R ℬ] (x : TensorProduct R (DirectSum ι fun i => 𝒜 i) (DirectSum ι fun i => ℬ i))
  (r : R), ((TensorProduct.gradedMul R 𝒜 ℬ) x) ((algebraMap R (DirectSum ι 𝒜)) r ⊗ₜ[R] 1) = r • x

Defined in

Mathlib.LinearAlgebra.TensorProduct.Graded.External

Cited by

1 results in Mathlib

Foundations

Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringModuleDecidableEqCommRingAddCommGroupAddCommGroupModuleModuleDirectSum.GRingDirectSum.GRingDirectSum.GAlgebraDirectSum.GAlgebra