Theorems · Theorem · linear algebra

TensorProduct.vanishesTrivially_iff_sum_tmul_eq_zero_of_rTensor_injective

∀ (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},
  Function.Injective ⇑(LinearMap.rTensor N (Submodule.span R (Set.range m)).subtype) →
    (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], generalization. Assume that the submodule $M' \subseteq M$ generated by the $m_i$ satisfies the property that the map $M' \otimes N \to M \otimes N$ is injective. 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 100 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleFintype