feat: port RingTheory.RingHom.FiniteType (#5075)

Author: xroblot

Mathlib.lean

@@ -2544,6 +2544,7 @@ import Mathlib.RingTheory.QuotientNilpotent
 import Mathlib.RingTheory.QuotientNoetherian
 import Mathlib.RingTheory.ReesAlgebra
 import Mathlib.RingTheory.RingHom.Finite
+import Mathlib.RingTheory.RingHom.FiniteType
 import Mathlib.RingTheory.RingHom.Integral
 import Mathlib.RingTheory.RingHom.Surjective
 import Mathlib.RingTheory.RingHomProperties

Mathlib/RingTheory/RingHom/FiniteType.lean

@@ -0,0 +1,106 @@
+/-
+Copyright (c) 2021 Andrew Yang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Andrew Yang
+
+! This file was ported from Lean 3 source module ring_theory.ring_hom.finite_type
+! leanprover-community/mathlib commit 64fc7238fb41b1a4f12ff05e3d5edfa360dd768c
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.RingTheory.LocalProperties
+import Mathlib.RingTheory.Localization.InvSubmonoid
+
+/-!
+
+# The meta properties of finite-type ring homomorphisms.
+
+The main result is `RingHom.finiteType_is_local`.
+
+-/
+
+
+namespace RingHom
+
+open scoped Pointwise
+
+theorem finiteType_stableUnderComposition : StableUnderComposition @FiniteType := by
+  introv R hf hg
+  exact hg.comp hf
+#align ring_hom.finite_type_stable_under_composition RingHom.finiteType_stableUnderComposition
+
+theorem finiteType_holdsForLocalizationAway : HoldsForLocalizationAway @FiniteType := by
+  introv R _
+  suffices Algebra.FiniteType R S by
+    rw [RingHom.FiniteType]
+    convert this; ext;
+    rw [Algebra.smul_def]; rfl
+  exact IsLocalization.finiteType_of_monoid_fg (Submonoid.powers r) S
+#align ring_hom.finite_type_holds_for_localization_away RingHom.finiteType_holdsForLocalizationAway
+
+theorem finiteType_ofLocalizationSpanTarget : OfLocalizationSpanTarget @FiniteType := by
+  -- Setup algebra intances.
+  rw [ofLocalizationSpanTarget_iff_finite]
+  introv R hs H
+  classical
+  letI := f.toAlgebra
+  replace H : ∀ r : s, Algebra.FiniteType R (Localization.Away (r : S))
+  · intro r; simp_rw [RingHom.FiniteType] at H; convert H r; ext; simp_rw [Algebra.smul_def]; rfl
+  replace H := fun r => (H r).1
+  constructor
+  -- Suppose `s : Finset S` spans `S`, and each `Sᵣ` is finitely generated as an `R`-algebra.
+  -- Say `t r : Finset Sᵣ` generates `Sᵣ`. By assumption, we may find `lᵢ` such that
+  -- `∑ lᵢ * sᵢ = 1`. I claim that all `s` and `l` and the numerators of `t` and generates `S`.
+  choose t ht using H
+  obtain ⟨l, hl⟩ :=
+    (Finsupp.mem_span_iff_total S (s : Set S) 1).mp
+      (show (1 : S) ∈ Ideal.span (s : Set S) by rw [hs]; trivial)
+  let sf := fun x : s => IsLocalization.finsetIntegerMultiple (Submonoid.powers (x : S)) (t x)
+  use s.attach.biUnion sf ∪ s ∪ l.support.image l
+  rw [eq_top_iff]
+  -- We need to show that every `x` falls in the subalgebra generated by those elements.
+  -- Since all `s` and `l` are in the subalgebra, it suffices to check that `sᵢ ^ nᵢ • x` falls in
+  -- the algebra for each `sᵢ` and some `nᵢ`.
+  rintro x -
+  apply Subalgebra.mem_of_span_eq_top_of_smul_pow_mem _ (s : Set S) l hl _ _ x _
+  · intro x hx
+    apply Algebra.subset_adjoin
+    rw [Finset.coe_union, Finset.coe_union]
+    exact Or.inl (Or.inr hx)
+  · intro i
+    by_cases h : l i = 0; · rw [h]; exact zero_mem _
+    apply Algebra.subset_adjoin
+    rw [Finset.coe_union, Finset.coe_image]
+    exact Or.inr (Set.mem_image_of_mem _ (Finsupp.mem_support_iff.mpr h))
+  · intro r
+    rw [Finset.coe_union, Finset.coe_union, Finset.coe_biUnion]
+    -- Since all `sᵢ` and numerators of `t r` are in the algebra, it suffices to show that the
+    -- image of `x` in `Sᵣ` falls in the `R`-adjoin of `t r`, which is of course true.
+    -- Porting note: The following `obtain` fails because Lean wants to know right away what the
+    -- placeholders are, so we need to provide a little more guidance
+    -- obtain ⟨⟨_, n₂, rfl⟩, hn₂⟩ := IsLocalization.exists_smul_mem_of_mem_adjoin
+    --   (Submonoid.powers (r : S)) x (t r) (Algebra.adjoin R _) _ _ _
+    rw [show ∀ A : Set S, (∃ n, (r : S) ^ n • x ∈ Algebra.adjoin R A) ↔
+      (∃ m : (Submonoid.powers (r : S)), (m : S) • x ∈ Algebra.adjoin R A) by
+      { exact fun _ => by simp [Submonoid.mem_powers_iff] }]
+    refine IsLocalization.exists_smul_mem_of_mem_adjoin
+      (Submonoid.powers (r : S)) x (t r) (Algebra.adjoin R _) ?_ ?_ ?_
+    · intro x hx
+      apply Algebra.subset_adjoin
+      exact Or.inl (Or.inl ⟨_, ⟨r, rfl⟩, _, ⟨s.mem_attach r, rfl⟩, hx⟩)
+    · rw [Submonoid.powers_eq_closure, Submonoid.closure_le, Set.singleton_subset_iff]
+      apply Algebra.subset_adjoin
+      exact Or.inl (Or.inr r.2)
+    · rw [ht]; trivial
+#align ring_hom.finite_type_of_localization_span_target RingHom.finiteType_ofLocalizationSpanTarget
+
+theorem finiteType_is_local : PropertyIsLocal @FiniteType :=
+  ⟨localization_finiteType, finiteType_ofLocalizationSpanTarget, finiteType_stableUnderComposition,
+    finiteType_holdsForLocalizationAway⟩
+#align ring_hom.finite_type_is_local RingHom.finiteType_is_local
+
+theorem finiteType_respectsIso : RingHom.RespectsIso @RingHom.FiniteType :=
+  RingHom.finiteType_is_local.respectsIso
+#align ring_hom.finite_type_respects_iso RingHom.finiteType_respectsIso
+
+end RingHom