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Fill in v_adicCompletionComapAlgHom #245

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Fill in v_adicCompletionComapAlgHom #245

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37 changes: 28 additions & 9 deletions FLT/DedekindDomain/FiniteAdeleRing/BaseChange.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -63,8 +63,6 @@ def comap (w : HeightOneSpectrum B) : HeightOneSpectrum A where
isPrime := Ideal.comap_isPrime (algebraMap A B) w.asIdeal
ne_bot := mt Ideal.eq_bot_of_comap_eq_bot w.ne_bot

open scoped algebraMap
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I feel like this only causes trouble (e.g. the show (r : adicCompletion K (comap A w)) = @UniformSpace.Completion.coe' K this r from rfl below) and we don't really use it anyway.


lemma mk_count_factors_map
(hAB : Function.Injective (algebraMap A B))
(w : HeightOneSpectrum B) (I : Ideal A) [DecidableEq (Associates (Ideal A))]
Expand Down Expand Up @@ -219,22 +217,43 @@ noncomputable def adicCompletionComapAlgHom
subst hvw
simp only [RingHom.toMonoidHom_eq_coe, OneHom.toFun_eq_coe, MonoidHom.toOneHom_coe,
MonoidHom.coe_coe]
have : (adicCompletionComapRingHom A K _ w rfl) (r : adicCompletion K (comap A w)) =
have : (adicCompletionComapRingHom A K _ w rfl) (algebraMap _ _ r) =
(algebraMap L (adicCompletion L w)) (algebraMap K L r) := by
letI : UniformSpace L := w.adicValued.toUniformSpace
letI : UniformSpace K := (comap A w).adicValued.toUniformSpace
rw [adicCompletionComapRingHom, UniformSpace.Completion.mapRingHom]
rw [show (r : adicCompletion K (comap A w)) = @UniformSpace.Completion.coe' K this r from rfl]
apply UniformSpace.Completion.extensionHom_coe
rw [this, ← IsScalarTower.algebraMap_apply K L]
cont := sorry -- #235
cont :=
letI : UniformSpace K := v.adicValued.toUniformSpace;
letI : UniformSpace L := w.adicValued.toUniformSpace;
UniformSpace.Completion.continuous_extension

omit [IsIntegralClosure B A L] [FiniteDimensional K L] [Algebra.IsSeparable K L] in
lemma adicCompletionComapAlgHom_coe
(v : HeightOneSpectrum A) (w : HeightOneSpectrum B) (hvw : v = comap A w) (x : K) :
adicCompletionComapAlgHom A K L B v w hvw x = algebraMap K L x :=
(adicCompletionComapAlgHom A K L B v w hvw).commutes _

-- this name is surely wrong
open WithZeroTopology in
lemma v_adicCompletionComapAlgHom
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This is the right name according to the naming convention.
We could probably rename adicCompletionComapAlgHom to adicCompletionComap.

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@kbuzzard kbuzzard Nov 30, 2024 •
edited
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I am totally confused by your comment. How can w(x)=v(x)^e be called v_adicCompletionComapAlgHom? Where's the AlgHom in the statement?

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Your statement does not typecheck because $w$ is a valuation on $\hat{L}$ while $v$ is a valuation on $\hat{K}$.
So the statement is $w(f(x)) = v(x)^e$ where $f : \hat{K} \to \hat{L}$ is the map you called adicCompletionComap

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Fair enough!

(v : HeightOneSpectrum A) (w : HeightOneSpectrum B) (hvw : v = comap A w) (x) :
Valued.v (adicCompletionComapAlgHom A K L B v w hvw x) ^
(Ideal.ramificationIdx (algebraMap A B) (comap A w).asIdeal w.asIdeal) = Valued.v x := sorry
-- #234
Valued.v (adicCompletionComapAlgHom A K L B v w hvw x) = Valued.v x ^
Ideal.ramificationIdx (algebraMap A B) (comap A w).asIdeal w.asIdeal := by
revert x
apply funext_iff.mp
symm
letI : UniformSpace K := v.adicValued.toUniformSpace
letI : UniformSpace L := w.adicValued.toUniformSpace
apply UniformSpace.Completion.ext
· exact Valued.continuous_valuation.pow _
· exact Valued.continuous_valuation.comp (adicCompletionComapAlgHom ..).cont
intro a
simp only [Valued.valuedCompletion_apply, adicCompletionComapAlgHom_coe]
show v.valuation a ^ _ = (w.valuation _)
subst hvw
rw [← valuation_comap A K L B w a]

noncomputable def adicCompletionComapAlgHom' (v : HeightOneSpectrum A) :
(HeightOneSpectrum.adicCompletion K v) →ₐ[K]
Expand All @@ -245,7 +264,7 @@ noncomputable def adicCompletionContinuousComapAlgHom (v : HeightOneSpectrum A)
(HeightOneSpectrum.adicCompletion K v) →A[K]
(∀ w : {w : HeightOneSpectrum B // v = comap A w}, HeightOneSpectrum.adicCompletion L w.1) where
__ := adicCompletionComapAlgHom' A K L B v
cont := sorry -- #236
cont := continuous_pi fun w ↦ (adicCompletionComapAlgHom A K L B v _ w.2).cont

open scoped TensorProduct -- ⊗ notation for tensor product

Expand Down