Theorems · Theorem · algebraic geometry

RingHom.SurjectiveOnStalks.exists_mul_eq_tmul

∀ {R : Type u_1} [inst : CommRing R] {S : Type u_2} [inst_1 : CommRing S] {T : Type u_3} [inst_2 : CommRing T]
  [inst_3 : Algebra R T] [inst_4 : Algebra R S],
  (algebraMap R T).SurjectiveOnStalks →
    ∀ (x : TensorProduct R S T) (J : Ideal T), J.IsPrime → ∃ t r a, r • t ∉ J ∧ 1 ⊗ₜ[R] (r • t) * x = a ⊗ₜ[R] t

If R → T is surjective on stalks, and J is some prime of T, then every element x in S ⊗[R] T satisfies (1 ⊗ r • t) * x = a ⊗ t for some r : R, a : S, and t : T such that r • t ∉ J.

Defined in

Mathlib.RingTheory.SurjectiveOnStalks

Cited by

4 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingCommRingAlgebraAlgebra