Theorems · Theorem · commutative algebra

IsLocalRing.subsingleton_tensorProduct

∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [inst_3 : IsLocalRing R] [Module.Finite R M],
  Subsingleton (TensorProduct R (IsLocalRing.ResidueField R) M) ↔ Subsingleton M

Defined in: Mathlib.RingTheory.LocalRing.Module
Cited by: 4 results in Mathlib
Foundations: Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module IsLocalRing Module.Finite

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Cites16

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

DFunLike.coeproof · cited by 62,936
Modulestatement and proof · cited by 20,661
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
Top.topproof · cited by 9,680
Submoduleproof · cited by 7,192
Bot.botproof · cited by 4,720
TensorProductstatement and proof · cited by 2,545
Module.Finitestatement and proof · cited by 1,032
IsLocalRingstatement and proof · cited by 339
IsLocalRing.ResidueFieldstatement and proof · cited by 156
TensorProduct.mkproof · cited by 129

Cited by4

Results whose statement or proof uses this declaration.

Algebra.FormallyUnramified.of_map_maximalIdealproof · cited by 1
IsLocalRing.split_injective_iff_lTensor_residueField_injectiveproof · cited by 1
Module.exists_basis_of_basis_baseChangeproof · cited by 1
Module.mem_support_iff_nontrivial_residueField_tensorProductproof · cited by 0