Theorems · Theorem · linear algebra

TensorProduct.vanishesTrivially_iff_sum_tmul_eq_zero

∀ (R : Type u_1) [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {N : Type u_3}
  [inst_3 : AddCommGroup N] [inst_4 : Module R N] {ι : Type u_4} [inst_5 : Fintype ι] {m : ι → M} {n : ι → N},
  Submodule.span R (Set.range m) = ⊤ → (TensorProduct.VanishesTrivially R m n ↔ ∑ i, m i ⊗ₜ[R] n i = 0)

Equational criterion for vanishing [A. Altman and S. Kleiman, A term of commutative algebra (Lemma 8.16)][altman2021term]. Assume that the $m_i$ generate $M$. Then the expression $\sum_i m_i \otimes n_i$ vanishes trivially if and only if it vanishes.

Defined in

Mathlib.LinearAlgebra.TensorProduct.Vanishing

Cited by

0 results in Mathlib

Foundations

Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleFintype