Theorems · Theorem · commutative algebra
Submodule.ker_unitsMap_spanSingleton
∀ {R : Type u} [inst : CommSemiring R] {A : Type v} [inst_1 : Semiring A] [inst_2 : Algebra R A] [FaithfulSMul R A],
(Units.map ↑(Submodule.spanSingleton R)).ker = (Units.map ↑(algebraMap R A)).rangeExactness of the sequence 1 → Rˣ → Aˣ → (Submodule R A)ˣ → Pic R → Pic A at Aˣ.
See Exercise I.3.7(iv) in [Weibel2013] or Theorem 2.4 in [RobertsSingh1993].
- Defined in
- Mathlib.Algebra.Algebra.Operations
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Submodulestatement · cited by 7,192
- Algebra.algebraMapstatement and proof · cited by 4,706
- Subgroupstatement · cited by 3,593
- Unitsstatement and proof · cited by 2,804
- Units.valproof · cited by 1,966
- FaithfulSMulstatement and proof · cited by 340
- MonoidHom.rangestatement · cited by 314
- MonoidHom.kerstatement · cited by 212
Cited by0
Results whose statement or proof uses this declaration.
Nothing cites this yet.