Theorems · Theorem · commutative algebra
Module.lift_rank_of_isLocalizedModule_of_free
∀ {R : Type u_1} {M : Type uM} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (Rₛ : Type uR')
{Mₛ : Type uM'} [inst_3 : AddCommGroup Mₛ] [inst_4 : Module R Mₛ] [inst_5 : CommRing Rₛ] [inst_6 : Algebra R Rₛ]
[inst_7 : Module Rₛ Mₛ] [IsScalarTower R Rₛ Mₛ] (S : Submonoid R) (f : M →ₗ[R] Mₛ) [IsLocalization S Rₛ]
[IsLocalizedModule S f] [Module.Free R M] [Nontrivial Rₛ],
Cardinal.lift.{uM, uM'} (Module.rank Rₛ Mₛ) = Cardinal.lift.{uM', uM} (Module.rank R M)Also see IsLocalizedModule.lift_rank_eq for a version for non-free modules,
but requires S to not contain any zero-divisors.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
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Cites28
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- 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
- Algebrastatement and proof · cited by 11,388
- LinearMapstatement and proof · cited by 10,215
- Algebra.algebraMapproof · cited by 4,706
- IsScalarTowerstatement and proof · cited by 3,896
- Submonoidstatement and proof · cited by 3,086
- one_mulproof · cited by 2,841
- Cardinalstatement and proof · cited by 2,598
- TensorProductproof · cited by 2,545
Cited by1
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- Module.finrank_of_isLocalizedModule_of_freeproof · cited by 2