Theorems · Theorem · commutative algebra

Module.FaithfullyFlat.lTensor_exact_iff_exact

∀ (R : Type u) (M : Type v) [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {N1 : Type u_1}
  [inst_3 : AddCommGroup N1] [inst_4 : Module R N1] {N2 : Type u_2} [inst_5 : AddCommGroup N2] [inst_6 : Module R N2]
  {N3 : Type u_3} [inst_7 : AddCommGroup N3] [inst_8 : Module R N3] (l12 : N1 →ₗ[R] N2) (l23 : N2 →ₗ[R] N3)
  [Module.FaithfullyFlat R M],
  Function.Exact ⇑(LinearMap.lTensor M l12) ⇑(LinearMap.lTensor M l23) ↔ Function.Exact ⇑l12 ⇑l23

Defined in: Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
Cited by: 2 results in Mathlib
Foundations: Depth 107 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module AddCommGroup Module AddCommGroup Module AddCommGroup Module Module.FaithfullyFlat

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Cites12

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Modulestatement and proof · cited by 20,661
RingHom.idstatement and proof · cited by 18,349
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
LinearMapstatement and proof · cited by 10,215
TensorProductstatement · cited by 2,545
LinearMap.lTensorstatement and proof · cited by 203
Function.Exactstatement and proof · cited by 182
Module.FaithfullyFlatstatement and proof · cited by 72
Module.Flat.lTensor_exactproof · cited by 7
Module.FaithfullyFlat.lTensor_reflects_exactproof · cited by 3

Cited by2

Results whose statement or proof uses this declaration.

Module.FaithfullyFlat.lTensor_injective_iff_injectiveproof · cited by 4
Module.FaithfullyFlat.lTensor_surjective_iff_surjectiveproof · cited by 3