ring_theory.dedekind_domain.integral_closure ⟷ Mathlib.RingTheory.DedekindDomain.IntegralClosure

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -123,7 +123,7 @@ theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type _
     conv_lhs => rw [← Classical.choose_spec (hc' i)]
     rw [← IsScalarTower.algebraMap_smul K (Classical.choose (hc' i)) (db i)]
   refine' ⟨fun i => db.repr (algebraMap C L x) i, fun i => _, (db.sum_repr _).symm⟩
-  rw [BilinForm.dualBasis_repr_apply]
+  rw [LinearMap.BilinForm.dualBasis_repr_apply]
   exact is_integral_trace (IsIntegral.mul hx (hb_int i))
 #align is_integral_closure.range_le_span_dual_basis IsIntegralClosure.range_le_span_dualBasis
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -3,7 +3,7 @@ Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
 -/
-import LinearAlgebra.FreeModule.Pid
+import LinearAlgebra.FreeModule.PID
 import RingTheory.DedekindDomain.Basic
 import RingTheory.Localization.Module
 import RingTheory.Trace
@@ -142,7 +142,7 @@ theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type _} [Finty
 
 variable (A) (K)
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (y «expr ≠ » (0 : A)) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (y «expr ≠ » (0 : A)) -/
 #print exists_integral_multiples /-
 /-- Send a set of `x`'es in a finite extension `L` of the fraction field of `R`
 to `(y : R) • x ∈ integral_closure R L`. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -203,7 +203,7 @@ theorem FiniteDimensional.exists_is_basis_integral :
   · intro x; simp only [mul_inv_cancel_left₀ hy']
   · rintro ⟨x', hx'⟩
     simp only [Algebra.smul_def, Finset.mem_image, exists_prop, Finset.mem_univ, true_and_iff] at
-      his' 
+      his'
     simp only [Basis.map_apply, LinearEquiv.coe_mk]
     exact his' _ ⟨_, rfl⟩
 #align finite_dimensional.exists_is_basis_integral FiniteDimensional.exists_is_basis_integral

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -66,10 +66,10 @@ variable [Algebra K L] [Algebra A L] [IsScalarTower A K L]
 
 variable [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L]
 
-#print IsIntegralClosure.isLocalization /-
+#print IsIntegralClosure.isLocalization_of_isSeparable /-
 /- If `L` is a separable extension of `K = Frac(A)` and `L` has no zero smul divisors by `A`,
 then `L` is the localization of the integral closure `C` of `A` in `L` at `A⁰`. -/
-theorem IsIntegralClosure.isLocalization [IsSeparable K L] [NoZeroSMulDivisors A L] :
+theorem IsIntegralClosure.isLocalization_of_isSeparable [IsSeparable K L] [NoZeroSMulDivisors A L] :
     IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L :=
   by
   haveI : IsDomain C :=
@@ -91,7 +91,7 @@ theorem IsIntegralClosure.isLocalization [IsSeparable K L] [NoZeroSMulDivisors A
     refine' congr_arg (algebraMap C L) ((mul_right_inj' _).mp h)
     rw [Subtype.coe_mk, map_ne_zero_iff _ (NoZeroSMulDivisors.algebraMap_injective A C)]
     exact mem_non_zero_divisors_iff_ne_zero.mp hm
-#align is_integral_closure.is_localization IsIntegralClosure.isLocalization
+#align is_integral_closure.is_localization IsIntegralClosure.isLocalization_of_isSeparable
 -/
 
 variable [FiniteDimensional K L]
@@ -263,7 +263,7 @@ theorem IsIntegralClosure.rank [IsPrincipalIdealRing A] [NoZeroSMulDivisors A L]
   haveI : Module.Free A C := IsIntegralClosure.module_free A K L C
   haveI : IsNoetherian A C := IsIntegralClosure.isNoetherian A K L C
   haveI : IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L :=
-    IsIntegralClosure.isLocalization A K L C
+    IsIntegralClosure.isLocalization_of_isSeparable A K L C
   let b := Basis.localizationLocalization K A⁰ L (Module.Free.chooseBasis A C)
   rw [FiniteDimensional.finrank_eq_card_chooseBasisIndex, FiniteDimensional.finrank_eq_card_basis b]
 #align is_integral_closure.rank IsIntegralClosure.rank

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -124,7 +124,7 @@ theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type _
     rw [← IsScalarTower.algebraMap_smul K (Classical.choose (hc' i)) (db i)]
   refine' ⟨fun i => db.repr (algebraMap C L x) i, fun i => _, (db.sum_repr _).symm⟩
   rw [BilinForm.dualBasis_repr_apply]
-  exact is_integral_trace (isIntegral_mul hx (hb_int i))
+  exact is_integral_trace (IsIntegral.mul hx (hb_int i))
 #align is_integral_closure.range_le_span_dual_basis IsIntegralClosure.range_le_span_dualBasis
 -/
 
@@ -161,9 +161,9 @@ theorem exists_integral_multiples (s : Finset L) :
     refine' ⟨y * y', mul_ne_zero hy hy', fun x'' hx'' => _⟩
     rcases finset.mem_insert.mp hx'' with (rfl | hx'')
     · rw [mul_smul, Algebra.smul_def, Algebra.smul_def, mul_comm _ x'', hx']
-      exact isIntegral_mul isIntegral_algebraMap x'.2
+      exact IsIntegral.mul isIntegral_algebraMap x'.2
     · rw [mul_comm, mul_smul, Algebra.smul_def]
-      exact isIntegral_mul isIntegral_algebraMap (hs _ hx'')
+      exact IsIntegral.mul isIntegral_algebraMap (hs _ hx'')
     · rw [IsScalarTower.algebraMap_eq A K L]
       apply (algebraMap K L).Injective.comp
       exact IsFractionRing.injective _ _
@@ -222,10 +222,10 @@ theorem IsIntegralClosure.isNoetherian [IsIntegrallyClosed A] [IsNoetherianRing
   let b' := (trace_form K L).dualBasis (traceForm_nondegenerate K L) b
   letI := isNoetherian_span_of_finite A (Set.finite_range b')
   let f : C →ₗ[A] Submodule.span A (Set.range b') :=
-    (Submodule.ofLe (IsIntegralClosure.range_le_span_dualBasis C b hb_int)).comp
+    (Submodule.inclusion (IsIntegralClosure.range_le_span_dualBasis C b hb_int)).comp
       ((Algebra.linearMap C L).restrictScalars A).range_restrict
   refine' isNoetherian_of_ker_bot f _
-  rw [LinearMap.ker_comp, Submodule.ker_ofLe, Submodule.comap_bot, LinearMap.ker_codRestrict]
+  rw [LinearMap.ker_comp, Submodule.ker_inclusion, Submodule.comap_bot, LinearMap.ker_codRestrict]
   exact LinearMap.ker_eq_bot_of_injective (IsIntegralClosure.algebraMap_injective C A L)
 #align is_integral_closure.is_noetherian IsIntegralClosure.isNoetherian
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,10 +3,10 @@ Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
 -/
-import Mathbin.LinearAlgebra.FreeModule.Pid
-import Mathbin.RingTheory.DedekindDomain.Basic
-import Mathbin.RingTheory.Localization.Module
-import Mathbin.RingTheory.Trace
+import LinearAlgebra.FreeModule.Pid
+import RingTheory.DedekindDomain.Basic
+import RingTheory.Localization.Module
+import RingTheory.Trace
 
 #align_import ring_theory.dedekind_domain.integral_closure from "leanprover-community/mathlib"@"e8e130de9dba4ed6897183c3193c752ffadbcc77"
 
@@ -142,7 +142,7 @@ theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type _} [Finty
 
 variable (A) (K)
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (y «expr ≠ » (0 : A)) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (y «expr ≠ » (0 : A)) -/
 #print exists_integral_multiples /-
 /-- Send a set of `x`'es in a finite extension `L` of the fraction field of `R`
 to `(y : R) • x ∈ integral_closure R L`. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,17 +2,14 @@
 Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-
-! This file was ported from Lean 3 source module ring_theory.dedekind_domain.integral_closure
-! leanprover-community/mathlib commit e8e130de9dba4ed6897183c3193c752ffadbcc77
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.LinearAlgebra.FreeModule.Pid
 import Mathbin.RingTheory.DedekindDomain.Basic
 import Mathbin.RingTheory.Localization.Module
 import Mathbin.RingTheory.Trace
 
+#align_import ring_theory.dedekind_domain.integral_closure from "leanprover-community/mathlib"@"e8e130de9dba4ed6897183c3193c752ffadbcc77"
+
 /-!
 # Integral closure of Dedekind domains
 
@@ -145,7 +142,7 @@ theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type _} [Finty
 
 variable (A) (K)
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (y «expr ≠ » (0 : A)) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (y «expr ≠ » (0 : A)) -/
 #print exists_integral_multiples /-
 /-- Send a set of `x`'es in a finite extension `L` of the fraction field of `R`
 to `(y : R) • x ∈ integral_closure R L`. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/6285167a053ad0990fc88e56c48ccd9fae6550eb

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
 
 ! This file was ported from Lean 3 source module ring_theory.dedekind_domain.integral_closure
-! leanprover-community/mathlib commit 4cf7ca0e69e048b006674cf4499e5c7d296a89e0
+! leanprover-community/mathlib commit e8e130de9dba4ed6897183c3193c752ffadbcc77
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -16,6 +16,9 @@ import Mathbin.RingTheory.Trace
 /-!
 # Integral closure of Dedekind domains
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file shows the integral closure of a Dedekind domain (in particular, the ring of integers
 of a number field) is a Dedekind domain.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/f2ad3645af9effcdb587637dc28a6074edc813f9

@@ -66,6 +66,7 @@ variable [Algebra K L] [Algebra A L] [IsScalarTower A K L]
 
 variable [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L]
 
+#print IsIntegralClosure.isLocalization /-
 /- If `L` is a separable extension of `K = Frac(A)` and `L` has no zero smul divisors by `A`,
 then `L` is the localization of the integral closure `C` of `A` in `L` at `A⁰`. -/
 theorem IsIntegralClosure.isLocalization [IsSeparable K L] [NoZeroSMulDivisors A L] :
@@ -91,11 +92,13 @@ theorem IsIntegralClosure.isLocalization [IsSeparable K L] [NoZeroSMulDivisors A
     rw [Subtype.coe_mk, map_ne_zero_iff _ (NoZeroSMulDivisors.algebraMap_injective A C)]
     exact mem_non_zero_divisors_iff_ne_zero.mp hm
 #align is_integral_closure.is_localization IsIntegralClosure.isLocalization
+-/
 
 variable [FiniteDimensional K L]
 
 variable {A K L}
 
+#print IsIntegralClosure.range_le_span_dualBasis /-
 theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type _} [Fintype ι]
     [DecidableEq ι] (b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
     ((Algebra.linearMap C L).restrictScalars A).range ≤
@@ -123,7 +126,9 @@ theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type _
   rw [BilinForm.dualBasis_repr_apply]
   exact is_integral_trace (isIntegral_mul hx (hb_int i))
 #align is_integral_closure.range_le_span_dual_basis IsIntegralClosure.range_le_span_dualBasis
+-/
 
+#print integralClosure_le_span_dualBasis /-
 theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type _} [Fintype ι] [DecidableEq ι]
     (b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
     (integralClosure A L).toSubmodule ≤
@@ -133,10 +138,12 @@ theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type _} [Finty
   intro x hx
   exact ⟨⟨x, hx⟩, rfl⟩
 #align integral_closure_le_span_dual_basis integralClosure_le_span_dualBasis
+-/
 
 variable (A) (K)
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (y «expr ≠ » (0 : A)) -/
+#print exists_integral_multiples /-
 /-- Send a set of `x`'es in a finite extension `L` of the fraction field of `R`
 to `(y : R) • x ∈ integral_closure R L`. -/
 theorem exists_integral_multiples (s : Finset L) :
@@ -161,9 +168,11 @@ theorem exists_integral_multiples (s : Finset L) :
       apply (algebraMap K L).Injective.comp
       exact IsFractionRing.injective _ _
 #align exists_integral_multiples exists_integral_multiples
+-/
 
 variable (L)
 
+#print FiniteDimensional.exists_is_basis_integral /-
 /-- If `L` is a finite extension of `K = Frac(A)`,
 then `L` has a basis over `A` consisting of integral elements. -/
 theorem FiniteDimensional.exists_is_basis_integral :
@@ -198,9 +207,11 @@ theorem FiniteDimensional.exists_is_basis_integral :
     simp only [Basis.map_apply, LinearEquiv.coe_mk]
     exact his' _ ⟨_, rfl⟩
 #align finite_dimensional.exists_is_basis_integral FiniteDimensional.exists_is_basis_integral
+-/
 
 variable (A K L) [IsSeparable K L]
 
+#print IsIntegralClosure.isNoetherian /-
 /- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is
 integrally closed and Noetherian, the integral closure `C` of `A` in `L` is
 Noetherian over `A`. -/
@@ -217,7 +228,9 @@ theorem IsIntegralClosure.isNoetherian [IsIntegrallyClosed A] [IsNoetherianRing
   rw [LinearMap.ker_comp, Submodule.ker_ofLe, Submodule.comap_bot, LinearMap.ker_codRestrict]
   exact LinearMap.ker_eq_bot_of_injective (IsIntegralClosure.algebraMap_injective C A L)
 #align is_integral_closure.is_noetherian IsIntegralClosure.isNoetherian
+-/
 
+#print IsIntegralClosure.isNoetherianRing /-
 /- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is
 integrally closed and Noetherian, the integral closure `C` of `A` in `L` is
 Noetherian. -/
@@ -225,7 +238,9 @@ theorem IsIntegralClosure.isNoetherianRing [IsIntegrallyClosed A] [IsNoetherianR
     IsNoetherianRing C :=
   isNoetherianRing_iff.mpr <| isNoetherian_of_tower A (IsIntegralClosure.isNoetherian A K L C)
 #align is_integral_closure.is_noetherian_ring IsIntegralClosure.isNoetherianRing
+-/
 
+#print IsIntegralClosure.module_free /-
 /- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a principal ring
 and `L` has no zero smul divisors by `A`, the integral closure `C` of `A` in `L` is
 a free `A`-module. -/
@@ -236,7 +251,9 @@ theorem IsIntegralClosure.module_free [NoZeroSMulDivisors A L] [IsPrincipalIdeal
   haveI : IsNoetherian A C := IsIntegralClosure.isNoetherian A K L _
   exact Module.free_of_finite_type_torsion_free'
 #align is_integral_closure.module_free IsIntegralClosure.module_free
+-/
 
+#print IsIntegralClosure.rank /-
 /- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a principal ring
 and `L` has no zero smul divisors by `A`, the `A`-rank of the integral closure `C` of `A` in `L`
 is equal to the `K`-rank of `L`. -/
@@ -250,9 +267,11 @@ theorem IsIntegralClosure.rank [IsPrincipalIdealRing A] [NoZeroSMulDivisors A L]
   let b := Basis.localizationLocalization K A⁰ L (Module.Free.chooseBasis A C)
   rw [FiniteDimensional.finrank_eq_card_chooseBasisIndex, FiniteDimensional.finrank_eq_card_basis b]
 #align is_integral_closure.rank IsIntegralClosure.rank
+-/
 
 variable {A K}
 
+#print integralClosure.isNoetherianRing /-
 /- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is
 integrally closed and Noetherian, the integral closure of `A` in `L` is
 Noetherian. -/
@@ -260,9 +279,11 @@ theorem integralClosure.isNoetherianRing [IsIntegrallyClosed A] [IsNoetherianRin
     IsNoetherianRing (integralClosure A L) :=
   IsIntegralClosure.isNoetherianRing A K L (integralClosure A L)
 #align integral_closure.is_noetherian_ring integralClosure.isNoetherianRing
+-/
 
 variable (A K) [IsDomain C]
 
+#print IsIntegralClosure.isDedekindDomain /-
 /- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain,
 the integral closure `C` of `A` in `L` is a Dedekind domain.
 
@@ -277,7 +298,9 @@ theorem IsIntegralClosure.isDedekindDomain [h : IsDedekindDomain A] : IsDedekind
       ⟨IsIntegralClosure.mk' C x (isIntegral_trans (IsIntegralClosure.isIntegral_algebra A L) _ hx),
         IsIntegralClosure.algebraMap_mk' _ _ _⟩⟩
 #align is_integral_closure.is_dedekind_domain IsIntegralClosure.isDedekindDomain
+-/
 
+#print integralClosure.isDedekindDomain /-
 /- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain,
 the integral closure of `A` in `L` is a Dedekind domain.
 
@@ -288,11 +311,13 @@ theorem integralClosure.isDedekindDomain [h : IsDedekindDomain A] :
     IsDedekindDomain (integralClosure A L) :=
   IsIntegralClosure.isDedekindDomain A K L (integralClosure A L)
 #align integral_closure.is_dedekind_domain integralClosure.isDedekindDomain
+-/
 
 variable [Algebra (FractionRing A) L] [IsScalarTower A (FractionRing A) L]
 
 variable [FiniteDimensional (FractionRing A) L] [IsSeparable (FractionRing A) L]
 
+#print integralClosure.isDedekindDomain_fractionRing /-
 /- If `L` is a finite separable extension of `Frac(A)`, where `A` is a Dedekind domain,
 the integral closure of `A` in `L` is a Dedekind domain.
 
@@ -303,6 +328,7 @@ instance integralClosure.isDedekindDomain_fractionRing [IsDedekindDomain A] :
     IsDedekindDomain (integralClosure A L) :=
   integralClosure.isDedekindDomain A (FractionRing A) L
 #align integral_closure.is_dedekind_domain_fraction_ring integralClosure.isDedekindDomain_fractionRing
+-/
 
 end IsIntegralClosure
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -136,8 +136,6 @@ theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type _} [Finty
 
 variable (A) (K)
 
-include K
-
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (y «expr ≠ » (0 : A)) -/
 /-- Send a set of `x`'es in a finite extension `L` of the fraction field of `R`
 to `(y : R) • x ∈ integral_closure R L`. -/
@@ -203,8 +201,6 @@ theorem FiniteDimensional.exists_is_basis_integral :
 
 variable (A K L) [IsSeparable K L]
 
-include L
-
 /- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is
 integrally closed and Noetherian, the integral closure `C` of `A` in `L` is
 Noetherian over `A`. -/
@@ -293,8 +289,6 @@ theorem integralClosure.isDedekindDomain [h : IsDedekindDomain A] :
   IsIntegralClosure.isDedekindDomain A K L (integralClosure A L)
 #align integral_closure.is_dedekind_domain integralClosure.isDedekindDomain
 
-omit K
-
 variable [Algebra (FractionRing A) L] [IsScalarTower A (FractionRing A) L]
 
 variable [FiniteDimensional (FractionRing A) L] [IsSeparable (FractionRing A) L]

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -138,7 +138,7 @@ variable (A) (K)
 
 include K
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (y «expr ≠ » (0 : A)) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (y «expr ≠ » (0 : A)) -/
 /-- Send a set of `x`'es in a finite extension `L` of the fraction field of `R`
 to `(y : R) • x ∈ integral_closure R L`. -/
 theorem exists_integral_multiples (s : Finset L) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -305,10 +305,10 @@ the integral closure of `A` in `L` is a Dedekind domain.
 See also the lemma `integral_closure.is_dedekind_domain` where you can choose
 the field of fractions yourself.
 -/
-instance integralClosure.isDedekindDomainFractionRing [IsDedekindDomain A] :
+instance integralClosure.isDedekindDomain_fractionRing [IsDedekindDomain A] :
     IsDedekindDomain (integralClosure A L) :=
   integralClosure.isDedekindDomain A (FractionRing A) L
-#align integral_closure.is_dedekind_domain_fraction_ring integralClosure.isDedekindDomainFractionRing
+#align integral_closure.is_dedekind_domain_fraction_ring integralClosure.isDedekindDomain_fractionRing
 
 end IsIntegralClosure
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -142,7 +142,7 @@ include K
 /-- Send a set of `x`'es in a finite extension `L` of the fraction field of `R`
 to `(y : R) • x ∈ integral_closure R L`. -/
 theorem exists_integral_multiples (s : Finset L) :
-    ∃ (y : _)(_ : y ≠ (0 : A)), ∀ x ∈ s, IsIntegral A (y • x) :=
+    ∃ (y : _) (_ : y ≠ (0 : A)), ∀ x ∈ s, IsIntegral A (y • x) :=
   by
   haveI := Classical.decEq L
   refine' s.induction _ _
@@ -169,7 +169,7 @@ variable (L)
 /-- If `L` is a finite extension of `K = Frac(A)`,
 then `L` has a basis over `A` consisting of integral elements. -/
 theorem FiniteDimensional.exists_is_basis_integral :
-    ∃ (s : Finset L)(b : Basis s K L), ∀ x, IsIntegral A (b x) :=
+    ∃ (s : Finset L) (b : Basis s K L), ∀ x, IsIntegral A (b x) :=
   by
   letI := Classical.decEq L
   letI : IsNoetherian K L := IsNoetherian.iff_fg.2 inferInstance
@@ -196,7 +196,7 @@ theorem FiniteDimensional.exists_is_basis_integral :
   · intro x; simp only [mul_inv_cancel_left₀ hy']
   · rintro ⟨x', hx'⟩
     simp only [Algebra.smul_def, Finset.mem_image, exists_prop, Finset.mem_univ, true_and_iff] at
-      his'
+      his' 
     simp only [Basis.map_apply, LinearEquiv.coe_mk]
     exact his' _ ⟨_, rfl⟩
 #align finite_dimensional.exists_is_basis_integral FiniteDimensional.exists_is_basis_integral

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -41,7 +41,7 @@ dedekind domain, dedekind ring
 
 variable (R A K : Type _) [CommRing R] [CommRing A] [Field K]
 
-open nonZeroDivisors Polynomial
+open scoped nonZeroDivisors Polynomial
 
 variable [IsDomain A]
 
@@ -56,7 +56,7 @@ of a number field is a Dedekind domain. -/
 
 open Algebra
 
-open BigOperators
+open scoped BigOperators
 
 variable (A K) [Algebra A K] [IsFractionRing A K]
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -192,10 +192,8 @@ theorem FiniteDimensional.exists_is_basis_integral :
           left_inv := _
           right_inv := _ },
       _⟩
-  · intro x
-    simp only [inv_mul_cancel_left₀ hy']
-  · intro x
-    simp only [mul_inv_cancel_left₀ hy']
+  · intro x; simp only [inv_mul_cancel_left₀ hy']
+  · intro x; simp only [mul_inv_cancel_left₀ hy']
   · rintro ⟨x', hx'⟩
     simp only [Algebra.smul_def, Finset.mem_image, exists_prop, Finset.mem_univ, true_and_iff] at
       his'

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -99,9 +99,9 @@ variable {A K L}
 theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type _} [Fintype ι]
     [DecidableEq ι] (b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
     ((Algebra.linearMap C L).restrictScalars A).range ≤
-      Submodule.span A (Set.range <| (traceForm K L).dualBasis (traceFormNondegenerate K L) b) :=
+      Submodule.span A (Set.range <| (traceForm K L).dualBasis (traceForm_nondegenerate K L) b) :=
   by
-  let db := (trace_form K L).dualBasis (traceFormNondegenerate K L) b
+  let db := (trace_form K L).dualBasis (traceForm_nondegenerate K L) b
   rintro _ ⟨x, rfl⟩
   simp only [LinearMap.coe_restrictScalars, Algebra.linearMap_apply]
   have hx : IsIntegral A (algebraMap C L x) := (IsIntegralClosure.isIntegral A L x).algebraMap
@@ -127,7 +127,7 @@ theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type _
 theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type _} [Fintype ι] [DecidableEq ι]
     (b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
     (integralClosure A L).toSubmodule ≤
-      Submodule.span A (Set.range <| (traceForm K L).dualBasis (traceFormNondegenerate K L) b) :=
+      Submodule.span A (Set.range <| (traceForm K L).dualBasis (traceForm_nondegenerate K L) b) :=
   by
   refine' le_trans _ (IsIntegralClosure.range_le_span_dualBasis (integralClosure A L) b hb_int)
   intro x hx
@@ -214,7 +214,7 @@ theorem IsIntegralClosure.isNoetherian [IsIntegrallyClosed A] [IsNoetherianRing
     IsNoetherian A C := by
   haveI := Classical.decEq L
   obtain ⟨s, b, hb_int⟩ := FiniteDimensional.exists_is_basis_integral A K L
-  let b' := (trace_form K L).dualBasis (traceFormNondegenerate K L) b
+  let b' := (trace_form K L).dualBasis (traceForm_nondegenerate K L) b
   letI := isNoetherian_span_of_finite A (Set.finite_range b')
   let f : C →ₗ[A] Submodule.span A (Set.range b') :=
     (Submodule.ofLe (IsIntegralClosure.range_le_span_dualBasis C b hb_int)).comp

mathlib commit https://github.com/leanprover-community/mathlib/commit/2651125b48fc5c170ab1111afd0817c903b1fc6c

@@ -185,7 +185,7 @@ theorem FiniteDimensional.exists_is_basis_integral :
     ⟨s',
       bs'.map
         {
-          LinearMap.Algebra.lmul _ _
+          Algebra.lmul _ _
             (algebraMap A L y) with
           toFun := fun x => algebraMap A L y * x
           invFun := fun x => (algebraMap A L y)⁻¹ * x

mathlib commit https://github.com/leanprover-community/mathlib/commit/1a4df69ca1a9a0e5e26bfe12e2b92814216016d0

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
 
 ! This file was ported from Lean 3 source module ring_theory.dedekind_domain.integral_closure
-! leanprover-community/mathlib commit 641b6a82006416ec431b2987b354af9311fed4f2
+! leanprover-community/mathlib commit 4cf7ca0e69e048b006674cf4499e5c7d296a89e0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -254,7 +254,7 @@ theorem IsIntegralClosure.rank [IsPrincipalIdealRing A] [NoZeroSMulDivisors A L]
   haveI : IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L :=
     IsIntegralClosure.isLocalization A K L C
   let b := Basis.localizationLocalization K A⁰ L (Module.Free.chooseBasis A C)
-  rw [Module.Free.finrank_eq_card_chooseBasisIndex, FiniteDimensional.finrank_eq_card_basis b]
+  rw [FiniteDimensional.finrank_eq_card_chooseBasisIndex, FiniteDimensional.finrank_eq_card_basis b]
 #align is_integral_closure.rank IsIntegralClosure.rank
 
 variable {A K}

mathlib commit https://github.com/leanprover-community/mathlib/commit/641b6a82006416ec431b2987b354af9311fed4f2

@@ -4,11 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
 
 ! This file was ported from Lean 3 source module ring_theory.dedekind_domain.integral_closure
-! leanprover-community/mathlib commit 6155d4351090a6fad236e3d2e4e0e4e7342668e8
+! leanprover-community/mathlib commit 641b6a82006416ec431b2987b354af9311fed4f2
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
+import Mathbin.LinearAlgebra.FreeModule.Pid
 import Mathbin.RingTheory.DedekindDomain.Basic
+import Mathbin.RingTheory.Localization.Module
 import Mathbin.RingTheory.Trace
 
 /-!
@@ -56,14 +58,44 @@ open Algebra
 
 open BigOperators
 
-variable {A K} [Algebra A K] [IsFractionRing A K]
+variable (A K) [Algebra A K] [IsFractionRing A K]
 
-variable {L : Type _} [Field L] (C : Type _) [CommRing C]
+variable (L : Type _) [Field L] (C : Type _) [CommRing C]
 
-variable [Algebra K L] [FiniteDimensional K L] [Algebra A L] [IsScalarTower A K L]
+variable [Algebra K L] [Algebra A L] [IsScalarTower A K L]
 
 variable [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L]
 
+/- If `L` is a separable extension of `K = Frac(A)` and `L` has no zero smul divisors by `A`,
+then `L` is the localization of the integral closure `C` of `A` in `L` at `A⁰`. -/
+theorem IsIntegralClosure.isLocalization [IsSeparable K L] [NoZeroSMulDivisors A L] :
+    IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L :=
+  by
+  haveI : IsDomain C :=
+    (IsIntegralClosure.equiv A C L (integralClosure A L)).toRingEquiv.IsDomain (integralClosure A L)
+  haveI : NoZeroSMulDivisors A C := IsIntegralClosure.noZeroSMulDivisors A L
+  refine' ⟨_, fun z => _, fun x y => ⟨fun h => ⟨1, _⟩, _⟩⟩
+  · rintro ⟨_, x, hx, rfl⟩
+    rw [isUnit_iff_ne_zero, map_ne_zero_iff _ (IsIntegralClosure.algebraMap_injective C A L),
+      Subtype.coe_mk, map_ne_zero_iff _ (NoZeroSMulDivisors.algebraMap_injective A C)]
+    exact mem_non_zero_divisors_iff_ne_zero.mp hx
+  · obtain ⟨m, hm⟩ :=
+      IsIntegral.exists_multiple_integral_of_isLocalization A⁰ z (IsSeparable.isIntegral K z)
+    obtain ⟨x, hx⟩ : ∃ x, algebraMap C L x = m • z := is_integral_closure.is_integral_iff.mp hm
+    refine' ⟨⟨x, algebraMap A C m, m, SetLike.coe_mem m, rfl⟩, _⟩
+    rw [Subtype.coe_mk, ← IsScalarTower.algebraMap_apply, hx, mul_comm, Submonoid.smul_def,
+      smul_def]
+  · simp only [IsIntegralClosure.algebraMap_injective C A L h]
+  · rintro ⟨⟨_, m, hm, rfl⟩, h⟩
+    refine' congr_arg (algebraMap C L) ((mul_right_inj' _).mp h)
+    rw [Subtype.coe_mk, map_ne_zero_iff _ (NoZeroSMulDivisors.algebraMap_injective A C)]
+    exact mem_non_zero_divisors_iff_ne_zero.mp hm
+#align is_integral_closure.is_localization IsIntegralClosure.isLocalization
+
+variable [FiniteDimensional K L]
+
+variable {A K L}
+
 theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type _} [Fintype ι]
     [DecidableEq ι] (b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
     ((Algebra.linearMap C L).restrictScalars A).range ≤
@@ -200,6 +232,31 @@ theorem IsIntegralClosure.isNoetherianRing [IsIntegrallyClosed A] [IsNoetherianR
   isNoetherianRing_iff.mpr <| isNoetherian_of_tower A (IsIntegralClosure.isNoetherian A K L C)
 #align is_integral_closure.is_noetherian_ring IsIntegralClosure.isNoetherianRing
 
+/- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a principal ring
+and `L` has no zero smul divisors by `A`, the integral closure `C` of `A` in `L` is
+a free `A`-module. -/
+theorem IsIntegralClosure.module_free [NoZeroSMulDivisors A L] [IsPrincipalIdealRing A] :
+    Module.Free A C :=
+  by
+  haveI : NoZeroSMulDivisors A C := IsIntegralClosure.noZeroSMulDivisors A L
+  haveI : IsNoetherian A C := IsIntegralClosure.isNoetherian A K L _
+  exact Module.free_of_finite_type_torsion_free'
+#align is_integral_closure.module_free IsIntegralClosure.module_free
+
+/- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a principal ring
+and `L` has no zero smul divisors by `A`, the `A`-rank of the integral closure `C` of `A` in `L`
+is equal to the `K`-rank of `L`. -/
+theorem IsIntegralClosure.rank [IsPrincipalIdealRing A] [NoZeroSMulDivisors A L] :
+    FiniteDimensional.finrank A C = FiniteDimensional.finrank K L :=
+  by
+  haveI : Module.Free A C := IsIntegralClosure.module_free A K L C
+  haveI : IsNoetherian A C := IsIntegralClosure.isNoetherian A K L C
+  haveI : IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L :=
+    IsIntegralClosure.isLocalization A K L C
+  let b := Basis.localizationLocalization K A⁰ L (Module.Free.chooseBasis A C)
+  rw [Module.Free.finrank_eq_card_chooseBasisIndex, FiniteDimensional.finrank_eq_card_basis b]
+#align is_integral_closure.rank IsIntegralClosure.rank
+
 variable {A K}
 
 /- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

@@ -106,7 +106,7 @@ variable (A) (K)
 
 include K
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (y «expr ≠ » (0 : A)) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (y «expr ≠ » (0 : A)) -/
 /-- Send a set of `x`'es in a finite extension `L` of the fraction field of `R`
 to `(y : R) • x ∈ integral_closure R L`. -/
 theorem exists_integral_multiples (s : Finset L) :
@@ -153,7 +153,7 @@ theorem FiniteDimensional.exists_is_basis_integral :
     ⟨s',
       bs'.map
         {
-          Algebra.lmul _ _
+          LinearMap.Algebra.lmul _ _
             (algebraMap A L y) with
           toFun := fun x => algebraMap A L y * x
           invFun := fun x => (algebraMap A L y)⁻¹ * x

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

A PR analogous to #12338: reformatting proofs following the multiple goals linter of #12339.

@@ -127,13 +127,13 @@ theorem exists_integral_multiples (s : Finset L) :
     have := exists_integral_multiple
       ((IsFractionRing.isAlgebraic_iff A K L).mpr (.of_finite _ x))
       ((injective_iff_map_eq_zero (algebraMap A L)).mp ?_)
-    rcases this with ⟨x', y', hy', hx'⟩
-    refine' ⟨y * y', mul_ne_zero hy hy', fun x'' hx'' => _⟩
-    rcases Finset.mem_insert.mp hx'' with (rfl | hx'')
-    · rw [mul_smul, Algebra.smul_def, Algebra.smul_def, mul_comm _ x'', hx']
-      exact isIntegral_algebraMap.mul x'.2
-    · rw [mul_comm, mul_smul, Algebra.smul_def]
-      exact isIntegral_algebraMap.mul (hs _ hx'')
+    · rcases this with ⟨x', y', hy', hx'⟩
+      refine' ⟨y * y', mul_ne_zero hy hy', fun x'' hx'' => _⟩
+      rcases Finset.mem_insert.mp hx'' with (rfl | hx'')
+      · rw [mul_smul, Algebra.smul_def, Algebra.smul_def, mul_comm _ x'', hx']
+        exact isIntegral_algebraMap.mul x'.2
+      · rw [mul_comm, mul_smul, Algebra.smul_def]
+        exact isIntegral_algebraMap.mul (hs _ hx'')
     · rw [IsScalarTower.algebraMap_eq A K L]
       apply (algebraMap K L).injective.comp
       exact IsFractionRing.injective _ _

All of these changes appear to be oversights to me.

@@ -62,7 +62,7 @@ variable (L : Type*) [Field L] (C : Type*) [CommRing C]
 variable [Algebra K L] [Algebra A L] [IsScalarTower A K L]
 variable [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L]
 
-/- If `L` is an algebraic extension of `K = Frac(A)` and `L` has no zero smul divisors by `A`,
+/-- If `L` is an algebraic extension of `K = Frac(A)` and `L` has no zero smul divisors by `A`,
 then `L` is the localization of the integral closure `C` of `A` in `L` at `A⁰`. -/
 theorem IsIntegralClosure.isLocalization (hKL : Algebra.IsAlgebraic K L) :
     IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L := by
@@ -170,7 +170,7 @@ theorem FiniteDimensional.exists_is_basis_integral :
 
 variable [IsSeparable K L]
 
-/- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is
+/-- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is
 integrally closed and Noetherian, the integral closure `C` of `A` in `L` is
 Noetherian over `A`. -/
 theorem IsIntegralClosure.isNoetherian [IsIntegrallyClosed A] [IsNoetherianRing A] :
@@ -187,7 +187,7 @@ theorem IsIntegralClosure.isNoetherian [IsIntegrallyClosed A] [IsNoetherianRing
   exact LinearMap.ker_eq_bot_of_injective (IsIntegralClosure.algebraMap_injective C A L)
 #align is_integral_closure.is_noetherian IsIntegralClosure.isNoetherian
 
-/- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is
+/-- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is
 integrally closed and Noetherian, the integral closure `C` of `A` in `L` is
 Noetherian. -/
 theorem IsIntegralClosure.isNoetherianRing [IsIntegrallyClosed A] [IsNoetherianRing A] :
@@ -195,7 +195,7 @@ theorem IsIntegralClosure.isNoetherianRing [IsIntegrallyClosed A] [IsNoetherianR
   isNoetherianRing_iff.mpr <| isNoetherian_of_tower A (IsIntegralClosure.isNoetherian A K L C)
 #align is_integral_closure.is_noetherian_ring IsIntegralClosure.isNoetherianRing
 
-/- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a principal ring
+/-- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a principal ring
 and `L` has no zero smul divisors by `A`, the integral closure `C` of `A` in `L` is
 a free `A`-module. -/
 theorem IsIntegralClosure.module_free [NoZeroSMulDivisors A L] [IsPrincipalIdealRing A] :
@@ -205,7 +205,7 @@ theorem IsIntegralClosure.module_free [NoZeroSMulDivisors A L] [IsPrincipalIdeal
   inferInstance
 #align is_integral_closure.module_free IsIntegralClosure.module_free
 
-/- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a principal ring
+/-- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a principal ring
 and `L` has no zero smul divisors by `A`, the `A`-rank of the integral closure `C` of `A` in `L`
 is equal to the `K`-rank of `L`. -/
 theorem IsIntegralClosure.rank [IsPrincipalIdealRing A] [NoZeroSMulDivisors A L] :
@@ -220,7 +220,7 @@ theorem IsIntegralClosure.rank [IsPrincipalIdealRing A] [NoZeroSMulDivisors A L]
 
 variable {A K}
 
-/- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is
+/-- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is
 integrally closed and Noetherian, the integral closure of `A` in `L` is
 Noetherian. -/
 theorem integralClosure.isNoetherianRing [IsIntegrallyClosed A] [IsNoetherianRing A] :
@@ -230,13 +230,12 @@ theorem integralClosure.isNoetherianRing [IsIntegrallyClosed A] [IsNoetherianRin
 
 variable (A K) [IsDomain C]
 
-/- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain,
+/-- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain,
 the integral closure `C` of `A` in `L` is a Dedekind domain.
 
-Can't be an instance since `A`, `K` or `L` can't be inferred. See also the instance
+This cannot be an instance since `A`, `K` or `L` can't be inferred. See also the instance
 `integralClosure.isDedekindDomain_fractionRing` where `K := FractionRing A`
-and `C := integralClosure A L`.
--/
+and `C := integralClosure A L`. -/
 theorem IsIntegralClosure.isDedekindDomain [IsDedekindDomain A] : IsDedekindDomain C :=
   have : IsFractionRing C L := IsIntegralClosure.isFractionRing_of_finite_extension A K L C
   { IsIntegralClosure.isNoetherianRing A K L C,
@@ -246,12 +245,11 @@ theorem IsIntegralClosure.isDedekindDomain [IsDedekindDomain A] : IsDedekindDoma
         IsIntegralClosure.algebraMap_mk' _ _ _⟩ with : IsDedekindDomain C }
 #align is_integral_closure.is_dedekind_domain IsIntegralClosure.isDedekindDomain
 
-/- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain,
+/-- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain,
 the integral closure of `A` in `L` is a Dedekind domain.
 
-Can't be an instance since `K` can't be inferred. See also the instance
-`integralClosure.isDedekindDomain_fractionRing` where `K := FractionRing A`.
--/
+This cannot be an instance since `K` can't be inferred. See also the instance
+`integralClosure.isDedekindDomain_fractionRing` where `K := FractionRing A`. -/
 theorem integralClosure.isDedekindDomain [IsDedekindDomain A] :
     IsDedekindDomain (integralClosure A L) :=
   IsIntegralClosure.isDedekindDomain A K L (integralClosure A L)
@@ -260,12 +258,11 @@ theorem integralClosure.isDedekindDomain [IsDedekindDomain A] :
 variable [Algebra (FractionRing A) L] [IsScalarTower A (FractionRing A) L]
 variable [FiniteDimensional (FractionRing A) L] [IsSeparable (FractionRing A) L]
 
-/- If `L` is a finite separable extension of `Frac(A)`, where `A` is a Dedekind domain,
+/-- If `L` is a finite separable extension of `Frac(A)`, where `A` is a Dedekind domain,
 the integral closure of `A` in `L` is a Dedekind domain.
 
 See also the lemma `integralClosure.isDedekindDomain` where you can choose
-the field of fractions yourself.
--/
+the field of fractions yourself. -/
 instance integralClosure.isDedekindDomain_fractionRing [IsDedekindDomain A] :
     IsDedekindDomain (integralClosure A L) :=
   integralClosure.isDedekindDomain A (FractionRing A) L

Remove structure BilinForm from LinearAlgebra/BilinearForm/Basic and migrate all of _root_.BilinForm to LinearMap.BilinForm

Closes: #10553

This isn't the end of the story, as there's still a lot of overlap between LinearAlgebra/BilinearForm and LinearAlgebra/SesquilinearForm but that can be sorted out in subsequent PRs.

Supersedes:

• #11057
• #11032
• #10432
• #10422

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Christopher Hoskin <christopher.hoskin@overleaf.com> Co-authored-by: Christopher Hoskin <mans0954@users.noreply.github.com> Co-authored-by: Vierkantor <vierkantor@vierkantor.com>

@@ -95,10 +95,11 @@ theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type*}
     [DecidableEq ι] (b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
     LinearMap.range ((Algebra.linearMap C L).restrictScalars A) ≤
     Submodule.span A (Set.range <| (traceForm K L).dualBasis (traceForm_nondegenerate K L) b) := by
-  rw [← BilinForm.dualSubmodule_span_of_basis, ← BilinForm.le_flip_dualSubmodule,
-    Submodule.span_le]
+  rw [← LinearMap.BilinForm.dualSubmodule_span_of_basis,
+    ← LinearMap.BilinForm.le_flip_dualSubmodule, Submodule.span_le]
   rintro _ ⟨i, rfl⟩ _ ⟨y, rfl⟩
-  simp only [LinearMap.coe_restrictScalars, linearMap_apply, BilinForm.flip_apply, traceForm_apply]
+  simp only [LinearMap.coe_restrictScalars, linearMap_apply, LinearMap.BilinForm.flip_apply,
+    traceForm_apply]
   refine IsIntegrallyClosed.isIntegral_iff.mp ?_
   exact isIntegral_trace ((IsIntegralClosure.isIntegral A L y).algebraMap.mul (hb_int i))
 #align is_integral_closure.range_le_span_dual_basis IsIntegralClosure.range_le_span_dualBasis

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -58,11 +58,8 @@ open Algebra
 open scoped BigOperators
 
 variable [Algebra A K] [IsFractionRing A K]
-
 variable (L : Type*) [Field L] (C : Type*) [CommRing C]
-
 variable [Algebra K L] [Algebra A L] [IsScalarTower A K L]
-
 variable [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L]
 
 /- If `L` is an algebraic extension of `K = Frac(A)` and `L` has no zero smul divisors by `A`,
@@ -92,7 +89,6 @@ theorem IsIntegralClosure.isLocalization_of_isSeparable [IsSeparable K L] :
 #align is_integral_closure.is_localization IsIntegralClosure.isLocalization_of_isSeparable
 
 variable [FiniteDimensional K L]
-
 variable {A K L}
 
 theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type*} [Fintype ι]
@@ -261,7 +257,6 @@ theorem integralClosure.isDedekindDomain [IsDedekindDomain A] :
 #align integral_closure.is_dedekind_domain integralClosure.isDedekindDomain
 
 variable [Algebra (FractionRing A) L] [IsScalarTower A (FractionRing A) L]
-
 variable [FiniteDimensional (FractionRing A) L] [IsSeparable (FractionRing A) L]
 
 /- If `L` is a finite separable extension of `Frac(A)`, where `A` is a Dedekind domain,

This uses the improved shake script from #9772 to reduce imports across mathlib. The corresponding noshake.json file has been added to #9772.

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
 -/
 import Mathlib.LinearAlgebra.FreeModule.PID
+import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
 import Mathlib.LinearAlgebra.BilinearForm.DualLattice
 import Mathlib.RingTheory.DedekindDomain.Basic
 import Mathlib.RingTheory.Localization.Module

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

@@ -201,10 +201,10 @@ theorem IsIntegralClosure.isNoetherianRing [IsIntegrallyClosed A] [IsNoetherianR
 and `L` has no zero smul divisors by `A`, the integral closure `C` of `A` in `L` is
 a free `A`-module. -/
 theorem IsIntegralClosure.module_free [NoZeroSMulDivisors A L] [IsPrincipalIdealRing A] :
-    Module.Free A C := by
+    Module.Free A C :=
   haveI : NoZeroSMulDivisors A C := IsIntegralClosure.noZeroSMulDivisors A L
   haveI : IsNoetherian A C := IsIntegralClosure.isNoetherian A K L _
-  exact Module.free_of_finite_type_torsion_free'
+  inferInstance
 #align is_integral_closure.module_free IsIntegralClosure.module_free
 
 /- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a principal ring

Follow-up #9184

@@ -120,7 +120,7 @@ variable (A K)
 /-- Send a set of `x`s in a finite extension `L` of the fraction field of `R`
 to `(y : R) • x ∈ integralClosure R L`. -/
 theorem exists_integral_multiples (s : Finset L) :
-    ∃ (y : _) (_ : y ≠ (0 : A)), ∀ x ∈ s, IsIntegral A (y • x) := by
+    ∃ y ≠ (0 : A), ∀ x ∈ s, IsIntegral A (y • x) := by
   haveI := Classical.decEq L
   refine' s.induction _ _
   · use 1, one_ne_zero

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

@@ -64,12 +64,13 @@ variable [Algebra K L] [Algebra A L] [IsScalarTower A K L]
 
 variable [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L]
 
-/- If `L` is a separable extension of `K = Frac(A)` and `L` has no zero smul divisors by `A`,
+/- If `L` is an algebraic extension of `K = Frac(A)` and `L` has no zero smul divisors by `A`,
 then `L` is the localization of the integral closure `C` of `A` in `L` at `A⁰`. -/
-theorem IsIntegralClosure.isLocalization [IsSeparable K L] [NoZeroSMulDivisors A L] :
+theorem IsIntegralClosure.isLocalization (hKL : Algebra.IsAlgebraic K L) :
     IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L := by
   haveI : IsDomain C :=
     (IsIntegralClosure.equiv A C L (integralClosure A L)).toMulEquiv.isDomain (integralClosure A L)
+  haveI : NoZeroSMulDivisors A L := NoZeroSMulDivisors.trans A K L
   haveI : NoZeroSMulDivisors A C := IsIntegralClosure.noZeroSMulDivisors A L
   refine' ⟨_, fun z => _, fun {x y} h => ⟨1, _⟩⟩
   · rintro ⟨_, x, hx, rfl⟩
@@ -77,13 +78,17 @@ theorem IsIntegralClosure.isLocalization [IsSeparable K L] [NoZeroSMulDivisors A
       Subtype.coe_mk, map_ne_zero_iff _ (NoZeroSMulDivisors.algebraMap_injective A C)]
     exact mem_nonZeroDivisors_iff_ne_zero.mp hx
   · obtain ⟨m, hm⟩ :=
-      IsIntegral.exists_multiple_integral_of_isLocalization A⁰ z (IsSeparable.isIntegral K z)
+      IsIntegral.exists_multiple_integral_of_isLocalization A⁰ z (hKL z).isIntegral
     obtain ⟨x, hx⟩ : ∃ x, algebraMap C L x = m • z := IsIntegralClosure.isIntegral_iff.mp hm
     refine' ⟨⟨x, algebraMap A C m, m, SetLike.coe_mem m, rfl⟩, _⟩
     rw [Subtype.coe_mk, ← IsScalarTower.algebraMap_apply, hx, mul_comm, Submonoid.smul_def,
       smul_def]
   · simp only [IsIntegralClosure.algebraMap_injective C A L h]
-#align is_integral_closure.is_localization IsIntegralClosure.isLocalization
+
+theorem IsIntegralClosure.isLocalization_of_isSeparable [IsSeparable K L] :
+    IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L :=
+  IsIntegralClosure.isLocalization A K L C (IsSeparable.isAlgebraic _ _)
+#align is_integral_closure.is_localization IsIntegralClosure.isLocalization_of_isSeparable
 
 variable [FiniteDimensional K L]
 
@@ -210,7 +215,7 @@ theorem IsIntegralClosure.rank [IsPrincipalIdealRing A] [NoZeroSMulDivisors A L]
   haveI : Module.Free A C := IsIntegralClosure.module_free A K L C
   haveI : IsNoetherian A C := IsIntegralClosure.isNoetherian A K L C
   haveI : IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L :=
-    IsIntegralClosure.isLocalization A K L C
+    IsIntegralClosure.isLocalization A K L C (Algebra.IsIntegral.of_finite _ _).isAlgebraic
   let b := Basis.localizationLocalization K A⁰ L (Module.Free.chooseBasis A C)
   rw [FiniteDimensional.finrank_eq_card_chooseBasisIndex, FiniteDimensional.finrank_eq_card_basis b]
 #align is_integral_closure.rank IsIntegralClosure.rank

Also moved Algebra.discr_eq_discr_of_toMatrix_coeff_isIntegral to Mathlib/NumberTheory/NumberField/Discriminant.lean.

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
 -/
 import Mathlib.LinearAlgebra.FreeModule.PID
+import Mathlib.LinearAlgebra.BilinearForm.DualLattice
 import Mathlib.RingTheory.DedekindDomain.Basic
 import Mathlib.RingTheory.Localization.Module
 import Mathlib.RingTheory.Trace
@@ -92,26 +93,12 @@ theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type*}
     [DecidableEq ι] (b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
     LinearMap.range ((Algebra.linearMap C L).restrictScalars A) ≤
     Submodule.span A (Set.range <| (traceForm K L).dualBasis (traceForm_nondegenerate K L) b) := by
-  let db := (traceForm K L).dualBasis (traceForm_nondegenerate K L) b
-  rintro _ ⟨x, rfl⟩
-  simp only [LinearMap.coe_restrictScalars, Algebra.linearMap_apply]
-  have hx : IsIntegral A (algebraMap C L x) := (IsIntegralClosure.isIntegral A L x).algebraMap
-  rsuffices ⟨c, x_eq⟩ : ∃ c : ι → A, algebraMap C L x = ∑ i, c i • db i
-  · rw [x_eq]
-    refine' Submodule.sum_mem _ fun i _ => Submodule.smul_mem _ _ (Submodule.subset_span _)
-    rw [Set.mem_range]
-    exact ⟨i, rfl⟩
-  suffices ∃ c : ι → K, (∀ i, IsIntegral A (c i)) ∧ algebraMap C L x = ∑ i, c i • db i by
-    obtain ⟨c, hc, hx⟩ := this
-    have hc' : ∀ i, IsLocalization.IsInteger A (c i) := fun i =>
-      IsIntegrallyClosed.isIntegral_iff.mp (hc i)
-    use fun i => Classical.choose (hc' i)
-    refine' hx.trans (Finset.sum_congr rfl fun i _ => _)
-    conv_lhs => rw [← Classical.choose_spec (hc' i)]
-    rw [← IsScalarTower.algebraMap_smul K (Classical.choose (hc' i)) (db i)]
-  refine' ⟨fun i => db.repr (algebraMap C L x) i, fun i => _, (db.sum_repr _).symm⟩
-  simp_rw [BilinForm.dualBasis_repr_apply]
-  exact isIntegral_trace (hx.mul (hb_int i))
+  rw [← BilinForm.dualSubmodule_span_of_basis, ← BilinForm.le_flip_dualSubmodule,
+    Submodule.span_le]
+  rintro _ ⟨i, rfl⟩ _ ⟨y, rfl⟩
+  simp only [LinearMap.coe_restrictScalars, linearMap_apply, BilinForm.flip_apply, traceForm_apply]
+  refine IsIntegrallyClosed.isIntegral_iff.mp ?_
+  exact isIntegral_trace ((IsIntegralClosure.isIntegral A L y).algebraMap.mul (hb_int i))
 #align is_integral_closure.range_le_span_dual_basis IsIntegralClosure.range_le_span_dualBasis
 
 theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type*} [Fintype ι] [DecidableEq ι]

This matches Set.inclusion, Subring.inclusion, Subalgebra.inclusion, etc.

Also renames the homOfLe spellings in Algebra/Lie to match.

Note that we leave LieSubalgebra.ofLe, as this is a completely different statement!

As requested by @alreadydone.

@@ -190,10 +190,10 @@ theorem IsIntegralClosure.isNoetherian [IsIntegrallyClosed A] [IsNoetherianRing
   let b' := (traceForm K L).dualBasis (traceForm_nondegenerate K L) b
   letI := isNoetherian_span_of_finite A (Set.finite_range b')
   let f : C →ₗ[A] Submodule.span A (Set.range b') :=
-    (Submodule.ofLe (IsIntegralClosure.range_le_span_dualBasis C b hb_int)).comp
+    (Submodule.inclusion (IsIntegralClosure.range_le_span_dualBasis C b hb_int)).comp
       ((Algebra.linearMap C L).restrictScalars A).rangeRestrict
   refine' isNoetherian_of_ker_bot f _
-  rw [LinearMap.ker_comp, Submodule.ker_ofLe, Submodule.comap_bot, LinearMap.ker_codRestrict]
+  rw [LinearMap.ker_comp, Submodule.ker_inclusion, Submodule.comap_bot, LinearMap.ker_codRestrict]
   exact LinearMap.ker_eq_bot_of_injective (IsIntegralClosure.algebraMap_injective C A L)
 #align is_integral_closure.is_noetherian IsIntegralClosure.isNoetherian
 

Zulip

Initially I just wanted to add more dot notations for IsIntegral and IsAlgebraic (done in #8437); then I noticed near-duplicates Algebra.isIntegral_of_finite [Field R] [Ring A] and RingHom.IsIntegral.of_finite [CommRing R] [CommRing A] so I went on to generalize the latter to cover the former, and generalized everything in the IntegralClosure file to the noncommutative case whenever possible.

In the process I noticed more golfs, which result in this PR. Most notably, isIntegral_of_mem_of_FG is now proven using Cayley-Hamilton and doesn't depend on the Noetherian case isIntegral_of_noetherian; the latter is now proven using the former. In total the golfs makes mathlib 227 lines leaner (+487 -714).

The main changes are in the single file RingTheory/IntegralClosure:

• Change the definition of Algebra.IsIntegral which makes it unfold to IsIntegral rather than RingHom.IsIntegralElem because the former has much more APIs.

• Fix lemma names involving is_integral which are actually about IsIntegralElem: RingHom.is_integral_map → RingHom.isIntegralElem_map RingHom.is_integral_of_mem_closure → RingHom.IsIntegralElem.of_mem_closure RingHom.is_integral_zero/one → RingHom.isIntegralElem_zero/one RingHom.is_integral_add/neg/sub/mul/of_mul_unit → RingHom.IsIntegralElem.add/neg/sub/mul/of_mul_unit

• Add a lemma Algebra.IsIntegral.of_injective.

• Move isIntegral_of_(submodule_)noetherian down and golf them.

• Remove (Algebra.)isIntegral_of_finite that work only over fields, in favor of the more general (Algebra.)isIntegral.of_finite.

• Merge duplicate lemmas isIntegral_of_isScalarTower and isIntegral_tower_top_of_isIntegral into IsIntegral.tower_top.

• Golf IsIntegral.of_mem_of_fg by first proving IsIntegral.of_finite using Cayley-Hamilton.

• Add a docstring mentioning the Kurosh problem at Algebra.IsIntegral.finite. The negative solution to the problem means the theorem doesn't generalize to noncommutative algebras.

• Golf IsIntegral.tmul and isField_of_isIntegral_of_isField(').

• Combine isIntegral_trans_aux into isIntegral_trans and golf.

• Add Algebra namespace to isIntegral_sup.

• rename lemmas for dot notation: RingHom.isIntegral_trans → RingHom.IsIntegral.trans RingHom.isIntegral_quotient/tower_bot/top_of_isIntegral → RingHom.IsIntegral.quotient/tower_bot/top isIntegral_of_mem_closure' → IsIntegral.of_mem_closure' (and the '' version) isIntegral_of_surjective → Algebra.isIntegral_of_surjective

The next changed file is RingTheory/Algebraic:

• Rename: of_larger_base → tower_top (for consistency with IsIntegral) Algebra.isAlgebraic_of_finite → Algebra.IsAlgebraic.of_finite Algebra.isAlgebraic_trans → Algebra.IsAlgebraic.trans

• Add new lemmasAlgebra.IsIntegral.isAlgebraic, isAlgebraic_algHom_iff, and Algebra.IsAlgebraic.of_injective to streamline some proofs.

The generalization from CommRing to Ring requires an additional lemma scaleRoots_eval₂_mul_of_commute in Polynomial/ScaleRoots.

A lemma Algebra.lmul_injective is added to Algebra/Bilinear (in order to golf the proof of IsIntegral.of_mem_of_fg).

In all other files, I merely fix the changed names, or use newly available dot notations.

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -111,7 +111,7 @@ theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type*}
     rw [← IsScalarTower.algebraMap_smul K (Classical.choose (hc' i)) (db i)]
   refine' ⟨fun i => db.repr (algebraMap C L x) i, fun i => _, (db.sum_repr _).symm⟩
   simp_rw [BilinForm.dualBasis_repr_apply]
-  exact isIntegral_trace (IsIntegral.mul hx (hb_int i))
+  exact isIntegral_trace (hx.mul (hb_int i))
 #align is_integral_closure.range_le_span_dual_basis IsIntegralClosure.range_le_span_dualBasis
 
 theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type*} [Fintype ι] [DecidableEq ι]
@@ -135,15 +135,15 @@ theorem exists_integral_multiples (s : Finset L) :
     rintro x ⟨⟩
   · rintro x s hx ⟨y, hy, hs⟩
     have := exists_integral_multiple
-      ((IsFractionRing.isAlgebraic_iff A K L).mpr (isAlgebraic_of_finite _ _ x))
+      ((IsFractionRing.isAlgebraic_iff A K L).mpr (.of_finite _ x))
       ((injective_iff_map_eq_zero (algebraMap A L)).mp ?_)
     rcases this with ⟨x', y', hy', hx'⟩
     refine' ⟨y * y', mul_ne_zero hy hy', fun x'' hx'' => _⟩
     rcases Finset.mem_insert.mp hx'' with (rfl | hx'')
     · rw [mul_smul, Algebra.smul_def, Algebra.smul_def, mul_comm _ x'', hx']
-      exact IsIntegral.mul isIntegral_algebraMap x'.2
+      exact isIntegral_algebraMap.mul x'.2
     · rw [mul_comm, mul_smul, Algebra.smul_def]
-      exact IsIntegral.mul isIntegral_algebraMap (hs _ hx'')
+      exact isIntegral_algebraMap.mul (hs _ hx'')
     · rw [IsScalarTower.algebraMap_eq A K L]
       apply (algebraMap K L).injective.comp
       exact IsFractionRing.injective _ _

This PR tests a string-based tool for renaming declarations.

Inspired by this Zulip thread, I am trying to reduce the diff of #8406.

This PR makes the following renames:

| From | To |

@@ -111,7 +111,7 @@ theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type*}
     rw [← IsScalarTower.algebraMap_smul K (Classical.choose (hc' i)) (db i)]
   refine' ⟨fun i => db.repr (algebraMap C L x) i, fun i => _, (db.sum_repr _).symm⟩
   simp_rw [BilinForm.dualBasis_repr_apply]
-  exact isIntegral_trace (isIntegral_mul hx (hb_int i))
+  exact isIntegral_trace (IsIntegral.mul hx (hb_int i))
 #align is_integral_closure.range_le_span_dual_basis IsIntegralClosure.range_le_span_dualBasis
 
 theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type*} [Fintype ι] [DecidableEq ι]
@@ -141,9 +141,9 @@ theorem exists_integral_multiples (s : Finset L) :
     refine' ⟨y * y', mul_ne_zero hy hy', fun x'' hx'' => _⟩
     rcases Finset.mem_insert.mp hx'' with (rfl | hx'')
     · rw [mul_smul, Algebra.smul_def, Algebra.smul_def, mul_comm _ x'', hx']
-      exact isIntegral_mul isIntegral_algebraMap x'.2
+      exact IsIntegral.mul isIntegral_algebraMap x'.2
     · rw [mul_comm, mul_smul, Algebra.smul_def]
-      exact isIntegral_mul isIntegral_algebraMap (hs _ hx'')
+      exact IsIntegral.mul isIntegral_algebraMap (hs _ hx'')
     · rw [IsScalarTower.algebraMap_eq A K L]
       apply (algebraMap K L).injective.comp
       exact IsFractionRing.injective _ _

The other direction is a consequence of IsLocalization.map_units.

Also do the same for LocalizationMap and IsLocalizedModule.

This means we have one less fact to prove when constructing an IsLocalization (etc.) instance (thus many proofs are golfed), but once we construct it we still have access to the eq_iff_exists lemmas (without the prime) so the API doesn't get less powerful.

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -70,7 +70,7 @@ theorem IsIntegralClosure.isLocalization [IsSeparable K L] [NoZeroSMulDivisors A
   haveI : IsDomain C :=
     (IsIntegralClosure.equiv A C L (integralClosure A L)).toMulEquiv.isDomain (integralClosure A L)
   haveI : NoZeroSMulDivisors A C := IsIntegralClosure.noZeroSMulDivisors A L
-  refine' ⟨_, fun z => _, fun {x y} => ⟨fun h => ⟨1, _⟩, _⟩⟩
+  refine' ⟨_, fun z => _, fun {x y} h => ⟨1, _⟩⟩
   · rintro ⟨_, x, hx, rfl⟩
     rw [isUnit_iff_ne_zero, map_ne_zero_iff _ (IsIntegralClosure.algebraMap_injective C A L),
       Subtype.coe_mk, map_ne_zero_iff _ (NoZeroSMulDivisors.algebraMap_injective A C)]
@@ -82,10 +82,6 @@ theorem IsIntegralClosure.isLocalization [IsSeparable K L] [NoZeroSMulDivisors A
     rw [Subtype.coe_mk, ← IsScalarTower.algebraMap_apply, hx, mul_comm, Submonoid.smul_def,
       smul_def]
   · simp only [IsIntegralClosure.algebraMap_injective C A L h]
-  · rintro ⟨⟨_, m, hm, rfl⟩, h⟩
-    refine' congr_arg (algebraMap C L) ((mul_right_inj' _).mp h)
-    rw [Subtype.coe_mk, map_ne_zero_iff _ (NoZeroSMulDivisors.algebraMap_injective A C)]
-    exact mem_nonZeroDivisors_iff_ne_zero.mp hm
 #align is_integral_closure.is_localization IsIntegralClosure.isLocalization
 
 variable [FiniteDimensional K L]

Generalizes RingEquiv.noZeroDivisors and RingEquiv.isDomain to MulEquiv

Adds Function.Injective.isLeft/RightCancelMulZero, MulEquiv.toZeroHomClass, and MulEquiv.isField (the last one is useful for #6309)

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -68,7 +68,7 @@ then `L` is the localization of the integral closure `C` of `A` in `L` at `A⁰`
 theorem IsIntegralClosure.isLocalization [IsSeparable K L] [NoZeroSMulDivisors A L] :
     IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L := by
   haveI : IsDomain C :=
-    (IsIntegralClosure.equiv A C L (integralClosure A L)).toRingEquiv.isDomain (integralClosure A L)
+    (IsIntegralClosure.equiv A C L (integralClosure A L)).toMulEquiv.isDomain (integralClosure A L)
   haveI : NoZeroSMulDivisors A C := IsIntegralClosure.noZeroSMulDivisors A L
   refine' ⟨_, fun z => _, fun {x y} => ⟨fun h => ⟨1, _⟩, _⟩⟩
   · rintro ⟨_, x, hx, rfl⟩

This redefines the IsDedekindDomain class to be the conjunction through extends of the classes IsDomain, IsNoetherianRing, DimensionLEOne and IsIntegrallyClosed.

Zulip thread: https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Should.20.60IsDedekindDomain.60.20extend.20.60IsDomain.60.3F

@@ -251,12 +251,13 @@ Can't be an instance since `A`, `K` or `L` can't be inferred. See also the insta
 `integralClosure.isDedekindDomain_fractionRing` where `K := FractionRing A`
 and `C := integralClosure A L`.
 -/
-theorem IsIntegralClosure.isDedekindDomain [h : IsDedekindDomain A] : IsDedekindDomain C :=
-  haveI : IsFractionRing C L := IsIntegralClosure.isFractionRing_of_finite_extension A K L C
-  ⟨IsIntegralClosure.isNoetherianRing A K L C, h.dimensionLEOne.isIntegralClosure L _,
+theorem IsIntegralClosure.isDedekindDomain [IsDedekindDomain A] : IsDedekindDomain C :=
+  have : IsFractionRing C L := IsIntegralClosure.isFractionRing_of_finite_extension A K L C
+  { IsIntegralClosure.isNoetherianRing A K L C,
+    Ring.DimensionLEOne.isIntegralClosure A L C,
     (isIntegrallyClosed_iff L).mpr fun {x} hx =>
       ⟨IsIntegralClosure.mk' C x (isIntegral_trans (IsIntegralClosure.isIntegral_algebra A L) _ hx),
-        IsIntegralClosure.algebraMap_mk' _ _ _⟩⟩
+        IsIntegralClosure.algebraMap_mk' _ _ _⟩ with : IsDedekindDomain C }
 #align is_integral_closure.is_dedekind_domain IsIntegralClosure.isDedekindDomain
 
 /- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain,

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -36,7 +36,7 @@ dedekind domain, dedekind ring
 -/
 
 
-variable (R A K : Type _) [CommRing R] [CommRing A] [Field K]
+variable (R A K : Type*) [CommRing R] [CommRing A] [Field K]
 
 open scoped nonZeroDivisors Polynomial
 
@@ -57,7 +57,7 @@ open scoped BigOperators
 
 variable [Algebra A K] [IsFractionRing A K]
 
-variable (L : Type _) [Field L] (C : Type _) [CommRing C]
+variable (L : Type*) [Field L] (C : Type*) [CommRing C]
 
 variable [Algebra K L] [Algebra A L] [IsScalarTower A K L]
 
@@ -92,7 +92,7 @@ variable [FiniteDimensional K L]
 
 variable {A K L}
 
-theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type _} [Fintype ι]
+theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type*} [Fintype ι]
     [DecidableEq ι] (b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
     LinearMap.range ((Algebra.linearMap C L).restrictScalars A) ≤
     Submodule.span A (Set.range <| (traceForm K L).dualBasis (traceForm_nondegenerate K L) b) := by
@@ -118,7 +118,7 @@ theorem IsIntegralClosure.range_le_span_dualBasis [IsSeparable K L] {ι : Type _
   exact isIntegral_trace (isIntegral_mul hx (hb_int i))
 #align is_integral_closure.range_le_span_dual_basis IsIntegralClosure.range_le_span_dualBasis
 
-theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type _} [Fintype ι] [DecidableEq ι]
+theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type*} [Fintype ι] [DecidableEq ι]
     (b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
     Subalgebra.toSubmodule (integralClosure A L) ≤
     Submodule.span A (Set.range <| (traceForm K L).dualBasis (traceForm_nondegenerate K L) b) := by

The predicate that a ring has Krull dimension at most one was a regular def. I believe we should turn it into a class because:

• The property follows from the ring structure, e.g. because it is a PID or because it is an integral closure.
• We pass it around as a whole hypothesis, something instance synthesis can deal well with.
• It makes the definition of IsDedekindDomain the conjunction of a number of classes, so we could switch to extends for all its fields.

The main change in API is the addition of Ideal.IsPrime.isMaximal which is a restatement of the Krull dimension property with convenient dot notation: turn a prime ideal into a maximal ideal given the hypothesis that it's not zero.

Zulip thread: https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Should.20.60IsDedekindDomain.60.20extend.20.60IsDomain.60.3F/near/374515392

@@ -253,7 +253,7 @@ and `C := integralClosure A L`.
 -/
 theorem IsIntegralClosure.isDedekindDomain [h : IsDedekindDomain A] : IsDedekindDomain C :=
   haveI : IsFractionRing C L := IsIntegralClosure.isFractionRing_of_finite_extension A K L C
-  ⟨IsIntegralClosure.isNoetherianRing A K L C, h.dimensionLEOne.isIntegralClosure _ L _,
+  ⟨IsIntegralClosure.isNoetherianRing A K L C, h.dimensionLEOne.isIntegralClosure L _,
     (isIntegrallyClosed_iff L).mpr fun {x} hx =>
       ⟨IsIntegralClosure.mk' C x (isIntegral_trans (IsIntegralClosure.isIntegral_algebra A L) _ hx),
         IsIntegralClosure.algebraMap_mk' _ _ _⟩⟩

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,17 +2,14 @@
 Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-
-! This file was ported from Lean 3 source module ring_theory.dedekind_domain.integral_closure
-! leanprover-community/mathlib commit 4cf7ca0e69e048b006674cf4499e5c7d296a89e0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.LinearAlgebra.FreeModule.PID
 import Mathlib.RingTheory.DedekindDomain.Basic
 import Mathlib.RingTheory.Localization.Module
 import Mathlib.RingTheory.Trace
 
+#align_import ring_theory.dedekind_domain.integral_closure from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
+
 /-!
 # Integral closure of Dedekind domains