Theorems · Theorem · nonassociative algebras

RootPairing.rootFormIn_self_smul_coroot

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : AddCommGroup M]
  [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] (P : RootPairing ι R M N) (S : Type u_5)
  [inst_5 : CommRing S] [inst_6 : Algebra S R] [inst_7 : FaithfulSMul S R] [inst_8 : Module S M]
  [inst_9 : IsScalarTower S R M] [inst_10 : Module S N] [IsScalarTower S R N] [inst_12 : P.IsValuedIn S]
  [inst_13 : Fintype ι] (i : ι),
  ((P.RootFormIn S) (P.rootSpanMem S i)) (P.rootSpanMem S i) • P.coroot i = 2 • (P.PolarizationIn S) (P.rootSpanMem S i)

A version of SGA3 XXI Lemma 1.2.1 (10), adapted to change of rings.

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear

Cited by

2 results in Mathlib

Foundations

Depth 60 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleCommRingAlgebraFaithfulSMulModuleIsScalarTowerModuleIsScalarTowerRootPairing.IsValuedInFintype