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Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

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1 declarations, 0 contain sorry, 0 declared as axioms

The fundamental identity in the local case

Ideal.sum_ramification_inertia_eq_finrank says that if S is a finite flat algebra over a domain R and p is a prime of R, then ∑ q, e q * f q = finrank R S, the sum being over the primes q of S above p. If S is local then there is exactly one such q, namely the maximal ideal of S, and the identity becomes e * f = finrank R S.

Imports (2)

Imported by 2 files

theorem Ideal.ramificationIdx_mul_inertiaDeg_eq_finrank_of_isLocalRing The fundamental identity e * f = [S : R] for a local ring S, finite over a Dedekind domain R, and p a nonzero maximal ideal of R line 28 source
Source documentation
The fundamental identity e * f = [S : R] for a local ring S, finite over a Dedekind domain R, and p a nonzero maximal ideal of R.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDedekindDomain, IsLocalRing, Algebra, FaithfulSMul, Module.Finite, Ideal, Ideal.IsMaximal, Ne, Bot.bot, Eq, Nat, HMul.hMul, Ideal.ramificationIdx, IsLocalRing.maximalIdeal, Ideal.inertiaDeg, Module.finrank
Proof uses: IsDomain, IsDomain.of_faithfulSMul, Membership.mem, Set, Ideal.primesOver, Eq.mpr, Singleton.singleton, id, congrArg, IsLocalRing.primesOver_eq, Eq.refl, Ideal.LiesOver, And.right, Ideal.IsPrime, IsLocalRing.eq_maximalIdeal, Ideal.IsPrime.isMaximal, And.left, Subtype.property, Ideal.ne_bot_of_mem_primesOver, Subsingleton and 10 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem ramificationIdx_mul_inertiaDeg_eq_finrank_of_isLocalRing
    {R : Type*} [CommRing R] [IsDedekindDomain R]
    (S : Type*) [CommRing S] [IsDedekindDomain S] [IsLocalRing S] [Algebra R S] [FaithfulSMul R S]
    [Module.Finite R S] {p : Ideal R} [p.IsMaximal] (hp0 : p ≠ ⊥) :
    (IsLocalRing.maximalIdeal S).ramificationIdx R *
      (IsLocalRing.maximalIdeal S).inertiaDeg R = Module.finrank R S := by
…
```