Theorems · Theorem · commutative algebra
Module.Flat.trans
∀ (R : Type u) (S : Type v) (M : Type w) [inst : CommSemiring R] [inst_1 : CommSemiring S] [inst_2 : Algebra R S] [inst_3 : AddCommMonoid M] [inst_4 : Module R M] [inst_5 : Module S M] [IsScalarTower R S M] [Module.Flat R S] [Module.Flat S M], Module.Flat R M
If S is a flat R-algebra, then any flat S-Module is also R-flat.
- Defined in
- Mathlib.RingTheory.Flat.Stability
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 105 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites24
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
- RingHom.idproof · cited by 18,349
- AddCommMonoidstatement and proof · cited by 12,281
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- LinearMapproof · cited by 10,215
- Submoduleproof · cited by 7,192
- IsScalarTowerstatement and proof · cited by 3,896
- TensorProductproof · cited by 2,545
- LinearEquiv.toLinearMapproof · cited by 1,171
- Submodule.subtypeproof · cited by 480
Cited by9
Results whose statement or proof uses this declaration.
- RingHom.Flat.compproof · cited by 8
- Module.flat_iff_of_isLocalizationproof · cited by 2
- Module.Flat.of_flat_tensorProductproof · cited by 2
- Module.FaithfullyFlat.transproof · cited by 2
- Module.Flat.ulift_left_iffproof · cited by 1
- RingHom.Flat.localRingHomproof · cited by 1
- RingHom.Flat.ofLocalizationPrimeproof · cited by 1
- Algebra.FormallySmooth.flat_of_algHom_of_isNoetherianRingproof · cited by 1
- Algebra.WeaklyEtale.transproof · cited by 0