Theorems · Theorem · commutative algebra
Algebra.FormallyUnramified.exists_algEquiv_prod
∀ (R S : Type u) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Algebra.EssFiniteType R S] [Algebra.FormallyUnramified R S], ∃ T x x_1, Nonempty (TensorProduct R S S ≃ₐ[S] S × T)
If S is an unramified R-algebra, S ⊗[R] S splits as S × T for some R-algebra T.
In particular, the diagonal is an open and closed immersion.
- Defined in
- Mathlib.RingTheory.Unramified.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites25
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
- mul_oneproof · cited by 3,885
- TensorProductstatement and proof · cited by 2,545
- HasQuotient.Quotientproof · cited by 2,301
- AlgEquivstatement and proof · cited by 1,681
- Submodule.spanproof · cited by 1,504
- TensorProduct.tmulproof · cited by 1,182
- Ideal.spanproof · cited by 948
- IsIdempotentElemproof · cited by 217
- add_sub_cancelproof · cited by 195
- AlgEquiv.transproof · cited by 108
Cited by1
Results whose statement or proof uses this declaration.
- Algebra.IsFiniteSplit.exists_tensorProduct_of_etaleproof · cited by 0