Theorems · Theorem · commutative algebra
CommRingCat.isPushout_of_isLocalization
∀ {R S Rₘ Sₘ : Type u} [inst : CommRing R] [inst_1 : CommRing Rₘ] [inst_2 : Algebra R Rₘ] [inst_3 : CommRing S]
[inst_4 : CommRing Sₘ] [inst_5 : Algebra S Sₘ] (f : R →+* S) (fₘ : Rₘ →+* Sₘ),
fₘ.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp f →
∀ (M : Submonoid R) [IsLocalization M Rₘ] [IsLocalization (Submonoid.map f M) Sₘ],
CategoryTheory.IsPushout (CommRingCat.ofHom f) (CommRingCat.ofHom (algebraMap R Rₘ))
(CommRingCat.ofHom (algebraMap S Sₘ)) (CommRingCat.ofHom fₘ)- Cited by
- 1 results in Mathlib
- Foundations
- Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
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
- Algebrastatement and proof · cited by 11,388
- RingHomstatement and proof · cited by 10,189
- Algebra.algebraMapstatement and proof · cited by 4,706
- IsScalarTowerproof · cited by 3,896
- Submonoidstatement and proof · cited by 3,086
- CommRingCatstatement · cited by 2,333
- RingHom.compstatement and proof · cited by 899
- IsLocalizationstatement and proof · cited by 636
- RingHom.toAlgebraproof · cited by 337
- CommRingCat.ofHomstatement · cited by 259
- CategoryTheory.IsPushoutstatement · cited by 219
Cited by1
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.Scheme.isPullback_toSpecΓ_toSpecΓproof · cited by 1