Theorems · Theorem · nonassociative algebras

RootPairing.baseOf_pairwise_pairing_le_zero

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [Finite ι] [inst : AddCommGroup M]
  [inst_1 : AddCommGroup N] [inst_2 : CommRing R] [inst_3 : Module R M] [inst_4 : Module R N] (P : RootPairing ι R M N)
  {S : Type u_5} [inst_5 : LinearOrder S] [inst_6 : AddCommGroup S] [IsOrderedAddMonoid S] (f : M →+ S) [CharZero R]
  [IsDomain R] [inst_10 : P.IsCrystallographic],
  (∀ (i : ι), f (P.root i) ≠ 0) → (IsAddIndecomposable.baseOf (⇑P.root) f).Pairwise fun i j => P.pairingIn ℤ i j ≤ 0

This is [serre1965](Ch. V, §9, Lemma 3).

Defined in

Mathlib.LinearAlgebra.RootSystem.BaseExists

Cited by

1 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FiniteAddCommGroupAddCommGroupCommRingModuleModuleLinearOrderAddCommGroupIsOrderedAddMonoidCharZeroIsDomainRootPairing.IsCrystallographic