feat(Mathlib/RingTheory/LocalRing): extended maximal ideal lies in Jacobson radical for finite extensions#108
Open
chrisflav wants to merge 2 commits into
Conversation
…cobson radical for finite extensions For a local ring `A` and a finite `A`-algebra `B`, the image of the maximal ideal under `algebraMap A B` lies in the Jacobson radical of `B`, units mod `m·B` lift, and (Noetherian) Krull intersection `⨅ (m·B)^n = ⊥` holds. Extracted from archon's `Proetale/Mathlib/RingTheory/Etale/HenselianPair.lean` and generalised by dropping the henselian hypothesis (these results only need `IsLocalRing A + Module.Finite A B`). Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
chrisflav
added
archon
…al extensions Address review on PR chrisflav#108: * Rename file from `LocalRing/Finite.lean` to `LocalRing/Integral.lean` to reflect the actual hypothesis used. * Weaken `[Module.Finite A B]` to `[Algebra.IsIntegral A B]` in `map_maximalIdeal_le_jacobson_bot` and `isUnit_of_isUnit_quotient_mk_map_maximalIdeal`. * State `iInf_pow_map_maximalIdeal_eq_bot` with `[Algebra.IsIntegral A B]` and the natural `[IsNoetherianRing B]` hypothesis (instead of `[IsNoetherianRing A] [Module.Finite A B]`). * Add the primitive `comap_eq_maximalIdeal_of_isMaximal`: every maximal ideal of `B` contracts to the maximal ideal of `A`. Derive `map_maximalIdeal_le_jacobson_bot` as a one-line corollary. * Expose `isLocalHom_quotient_mk_map_maximalIdeal` as a named lemma; the `IsUnit`-flavoured lemma becomes a one-line corollary. * Update the module docstring accordingly. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
chrisflav
removed
the
awaiting-auto-review
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
1 participant
Add this suggestion to a batch that can be applied as a single commit.This suggestion is invalid because no changes were made to the code.Suggestions cannot be applied while the pull request is closed.Suggestions cannot be applied while viewing a subset of changes.Only one suggestion per line can be applied in a batch.Add this suggestion to a batch that can be applied as a single commit.Applying suggestions on deleted lines is not supported.You must change the existing code in this line in order to create a valid suggestion.Outdated suggestions cannot be applied.This suggestion has been applied or marked resolved.Suggestions cannot be applied from pending reviews.Suggestions cannot be applied on multi-line comments.Suggestions cannot be applied while the pull request is queued to merge.Suggestion cannot be applied right now. Please check back later.
For a local ring
Aand a finiteA-algebraB, the image of the maximal ideal ofAunderalgebraMap A Blies in the Jacobson radical ofB. Three consequences:IsLocalRing.map_maximalIdeal_le_jacobson_bot:(maximalIdeal A).map (algebraMap A B) ≤ Ideal.jacobson ⊥.IsLocalRing.isUnit_of_isUnit_quotient_mk_map_maximalIdeal:an element of
Bwhose image inB ⧸ m·Bis a unit is itself a unit inB.IsLocalRing.iInf_pow_map_maximalIdeal_eq_bot(NoetherianA):⨅ n, (m·B)^n = ⊥— Krull's intersection theorem for the extended ideal.Proof goes via integral going-up (
Ideal.isMaximal_comap_of_isIntegral_of_isMaximal) for the Jacobson containment, the standard local-hom criterion (isLocalHom_of_le_jacobson_bot) for unit lifting, andIdeal.iInf_pow_smul_eq_bot_of_le_jacobsonfor Krull intersection.Extracted from archon's
Proetale/Mathlib/RingTheory/Etale/HenselianPair.leanand generalised by dropping the henselian / étale hypotheses (these results only needIsLocalRing A + Module.Finite A B).