Theorems · Theorem · commutative algebra
Algebra.FormallyUnramified.iff_exists_tensorProduct
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
[Algebra.EssFiniteType R S],
Algebra.FormallyUnramified R S ↔
∃ t, (∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * t = 0) ∧ (Algebra.TensorProduct.lmul' R) t = 1Proposition I.2.3 + I.2.6 of [iversen]
A finite-type R-algebra S is (formally) unramified iff there exists a t : S ⊗[R] S satisfying
1. t annihilates every 1 ⊗ s - s ⊗ 1.
2. the image of t is 1 under the map S ⊗[R] S → S.
- Defined in
- Mathlib.RingTheory.Unramified.Finite
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 91 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.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Idealproof · cited by 4,748
- Set.rangeproof · cited by 4,705
- mul_oneproof · cited by 3,885
- AlgHomstatement · cited by 3,236
- one_mulproof · cited by 2,841
- TensorProductstatement and proof · cited by 2,545
- Nat.cast_oneproof · cited by 2,501
- mul_commproof · cited by 2,262
- le_antisymmproof · cited by 2,068
Cited by2
Results whose statement or proof uses this declaration.
- Algebra.FormallyUnramified.lmul_elemproof · cited by 2
- Algebra.FormallyUnramified.one_tmul_sub_tmul_one_mul_elemproof · cited by 1