Theorems · Theorem · algebraic geometry
Module.mem_freeLocus_of_isLocalization
∀ {R : Type uR} {M : Type uM} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (p : PrimeSpectrum R)
(Rₚ : Type u_1) (Mₚ : Type u_2) [inst_3 : CommRing Rₚ] [inst_4 : Algebra R Rₚ] [IsLocalization.AtPrime Rₚ p.asIdeal]
[inst_6 : AddCommGroup Mₚ] [inst_7 : Module R Mₚ] (f : M →ₗ[R] Mₚ) [IsLocalizedModule p.asIdeal.primeCompl f]
[inst_9 : Module Rₚ Mₚ] [IsScalarTower R Rₚ Mₚ], p ∈ Module.freeLocus R M ↔ Module.Free Rₚ Mₚ- Cited by
- 3 results in Mathlib
- Foundations
- Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites41
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement · cited by 53,352
- 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
- LinearEquivproof · cited by 3,317
- LinearEquiv.toLinearMapproof · cited by 1,171
Cited by3
Results whose statement or proof uses this declaration.
- Module.mem_freeLocus_iff_tensorproof · cited by 1
- Module.freeLocus_localizationproof · cited by 1
- Module.freeLocus_congrproof · cited by 0