Theorems · Definition · commutative algebra
RingHom.Flat
{R : Type u} → {S : Type v} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → PropA ring homomorphism f : R →+* S is flat if S is flat as an R module.
- Defined in
- Mathlib.RingTheory.RingHom.Flat
- Cited by
- 60 results in Mathlib
- Foundations
- Depth 19 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- RingHomstatement and proof · cited by 10,189
- Module.Flatproof · cited by 279
Cited by65
Results whose statement or proof uses this declaration.
- RingHom.Flat.compstatement and proof · cited by 8
- RingHom.Flat.of_bijectivestatement · cited by 8
- AlgebraicGeometry.Scheme.Hom.finrank_SpecMap_eq_finrankstatement and proof · cited by 6
- RingHom.FaithfullyFlat.iff_flat_and_comap_surjectivestatement and proof · cited by 5
- RingHom.flat_algebraMap_iffstatement · cited by 4
- AlgebraicGeometry.Scheme.Hom.flat_appLEstatement · cited by 3
- AlgebraicGeometry.isIso_pushoutSection_of_iSup_eqstatement and proof · cited by 2
- RingHom.Flat.comp_iff_of_bijective_rightstatement and proof · cited by 2
- RingHom.Flat.idstatement · cited by 2
- AlgebraicGeometry.mono_pushoutSection_of_iSup_eqstatement and proof · cited by 2
- RingHom.Flat.stableUnderCompositionstatement and proof · cited by 2
- AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_left_of_ringHomFlatstatement and proof · cited by 2