feat(RingTheory): category of finite étale algebras over a separably closed field

[chrisflav]

We define the category of finite étale R-algebras for a ring R and show it is equivalent to FintypeCat if R is a separably closed field.

From Pi1.

---

• depends on: feat(RingTheory): PrimeSpectrum.sigmaToPi is an open embedding #38016

[github-actions]

PR summary 8ec19128ad

Import changes exceeding 2%

%	File
+21.54%	Mathlib.RingTheory.TotallySplit

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.RingTheory.TotallySplit	1787	2172	+385 (+21.54%)

Import changes for all files

Files	Import difference
Mathlib.RingTheory.TotallySplit	385
Mathlib.RingTheory.Etale.Finite (new file)	2209

---

Declarations diff

+ FiniteEtale
+ FiniteEtale.baseChange
+ FiniteEtale.baseChangeSelfIso
+ FiniteEtale.equivOfIsSepClosed
+ FiniteEtale.fiber
+ FiniteEtale.fiberIsoBaseChangeFiber
+ FiniteEtale.fiberIsoComp
+ FiniteEtale.fiberIsoFiniteSpec
+ FiniteEtale.finiteSpec
+ FiniteEtale.isoMk
+ FiniteEtale.of
+ FiniteEtale.ofHom
+ algHomEquivPrimeSpectrum
+ algebraMap_bijective
+ bijective_algebraMap_quotient
+ coe_evalAlgHom
+ comap_evalRingHom_basicOpen
+ equivPiOfIsSepClosed
+ equivPiOfIsSepClosed_comap
+ equivPiOfIsSepClosed_self_apply
+ equivPi_apply
+ etale
+ finite
+ finiteEtale
+ instance (R : FiniteEtale k) : IsArtinianRing R
+ instance (R : Type*) [CommRing R] (K : Type*) [Field K] [Algebra R K]
+ instance (S : FiniteEtale.{v} R) : Algebra.Etale R S
+ instance (S : FiniteEtale.{v} R) : Module.Finite R S
+ instance (Ω : Type u) [Field Ω] [IsSepClosed Ω] : (FiniteEtale.fiber.{u} Ω Ω).IsEquivalence
+ instance (Ω : Type u) [Field Ω] [IsSepClosed Ω] : (FiniteEtale.finiteSpec.{u} Ω).IsEquivalence
+ instance : CoeSort (FiniteEtale.{v} R) (Type v) := ⟨fun R ↦ R.obj⟩
+ instance : Etale R R
+ instance : FormallyEtale R R := of_formallyUnramified_and_formallySmooth
+ instance [EssFiniteType K A] [FormallyEtale K A] (p : Ideal A) [p.IsPrime] :
+ instance [IsSepClosed k] [EssFiniteType k R] [FormallyEtale k R] : IsFiniteSplit k R := by
+ isOpenEmbedding_sigmaToPi
+ liftEquivRight
+ sigmaHomeoPi
+ sigmaHomeoPi_apply
+ sigmaToPi_apply
+ sigmaToPi_mk_basicOpen

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

Increase in tech debt: (relative, absolute) = (3.00, 0.00)

Current number	Change	Type
6488	3	backward.isDefEq.respectTransparency

Current commit cef8ea14c7
Reference commit 8ec19128ad

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[mathlib-dependent-issues]

This PR/issue depends on:

• feat(RingTheory): PrimeSpectrum.sigmaToPi is an open embedding #38016
  By Dependent Issues (🤖). Happy coding!