Theorems · Theorem · commutative algebra
Ideal.finprod_heightOneSpectrum_factorization_coe
∀ {R : Type u_1} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K]
[inst_3 : IsFractionRing R K] [inst_4 : IsDedekindDomain R] {I : Ideal R},
I ≠ 0 →
∏ᶠ (v : IsDedekindDomain.HeightOneSpectrum R),
↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk I).factors) =
↑IThe ideal I equals the finprod ∏_v v^(val_v(I)), when both sides are regarded as fractional
ideals of R.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 154 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
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
- Fieldstatement and proof · cited by 7,404
- Idealstatement and proof · cited by 4,748
- le_reflproof · cited by 2,061
- nonZeroDivisorsstatement and proof · cited by 895
- IsFractionRingstatement and proof · cited by 738
- IsDedekindDomainstatement and proof · cited by 668
- FractionalIdealstatement and proof · cited by 423
- IsDedekindDomain.HeightOneSpectrumstatement and proof · cited by 338
- zpow_natCastproof · cited by 271
- finprodstatement and proof · cited by 257
Cited by1
Results whose statement or proof uses this declaration.
- FractionalIdeal.finprod_heightOneSpectrum_factorizationproof · cited by 2