Theorems · Theorem · commutative algebra

Algebra.norm_norm

∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Module.Free R S]
  {A : Type u_6} [inst_4 : Ring A] [inst_5 : Algebra R A] [inst_6 : Algebra S A] [IsScalarTower R S A] [Module.Free S A]
  {a : A}, (Algebra.norm R) ((Algebra.norm S) a) = (Algebra.norm R) a

Let A/S/R be a tower of finite free tower of rings (with R and S commutative). Then $\text{Norm}_{A/R} = \text{Norm}_{A/S} \circ \text{Norm}_{S/R}$.

Defined in

Mathlib.RingTheory.Norm.Transitivity

Cited by

5 results in Mathlib

Foundations

Depth 132 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraModule.FreeRingAlgebraAlgebraIsScalarTowerModule.Free