Theorems · Theorem · commutative algebra

Module.Flat.tfae_algebraMap_surjective

∀ (R : Type u_1) (S : Type u_2) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Module.Flat R S]
  [Module.Finite R S],
  [Function.Surjective ⇑(algebraMap R S), Function.Bijective ⇑(LinearMap.mul' R S),
      ∀ (p : PrimeSpectrum R), Module.rankAtStalk S p ≤ 1].TFAE

If S is a finite and flat R-algebra, R → S is surjective iff S ⊗[R] S → S is an isomorphism iff the rank of S is at most 1 at all primes.

Defined in: Mathlib.RingTheory.Flat.Rank
Cited by: 1 results in Mathlib
Foundations: Depth 123 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing CommRing Algebra Module.Flat Module.Finite

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Cites24

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
RingHom.idstatement · cited by 18,349
CommRingstatement and proof · cited by 17,173
Algebrastatement and proof · cited by 11,388
LinearMapstatement · cited by 10,215
RingHomstatement · cited by 10,189
Algebra.algebraMapstatement and proof · cited by 4,706
TensorProductstatement and proof · cited by 2,545
Module.Finitestatement and proof · cited by 1,032
Function.Bijectivestatement and proof · cited by 863
PrimeSpectrumstatement and proof · cited by 625
sqproof · cited by 280

Cited by1

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Module.rankAtStalk_le_one_iff_surjectiveproof · cited by 1