algebra.category.Ring.instances
⟷
Mathlib.Algebra.Category.Ring.Instances
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,9 +3,9 @@ Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
-import Mathbin.Algebra.Category.Ring.Basic
-import Mathbin.RingTheory.Localization.Away.Basic
-import Mathbin.RingTheory.Ideal.LocalRing
+import Algebra.Category.Ring.Basic
+import RingTheory.Localization.Away.Basic
+import RingTheory.Ideal.LocalRing
#align_import algebra.category.Ring.instances from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,16 +2,13 @@
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 algebra.category.Ring.instances
-! leanprover-community/mathlib commit 38df578a6450a8c5142b3727e3ae894c2300cae0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Algebra.Category.Ring.Basic
import Mathbin.RingTheory.Localization.Away.Basic
import Mathbin.RingTheory.Ideal.LocalRing
+#align_import algebra.category.Ring.instances from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"
+
/-!
# Ring-theoretic results in terms of categorical languages
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -22,15 +22,19 @@ import Mathbin.RingTheory.Ideal.LocalRing
open CategoryTheory
+#print localization_unit_isIso /-
instance localization_unit_isIso (R : CommRingCat) :
IsIso (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) :=
IsIso.of_iso (IsLocalization.atOne R (Localization.Away (1 : R))).toRingEquiv.toCommRingCatIso
#align localization_unit_is_iso localization_unit_isIso
+-/
+#print localization_unit_isIso' /-
instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) := by
cases R; exact localization_unit_isIso _
#align localization_unit_is_iso' localization_unit_isIso'
+-/
#print IsLocalization.epi /-
theorem IsLocalization.epi {R : Type _} [CommRing R] (M : Submonoid R) (S : Type _) [CommRing S]
@@ -39,15 +43,19 @@ theorem IsLocalization.epi {R : Type _} [CommRing R] (M : Submonoid R) (S : Type
#align is_localization.epi IsLocalization.epi
-/
+#print Localization.epi /-
instance Localization.epi {R : Type _} [CommRing R] (M : Submonoid R) :
Epi (CommRingCat.ofHom <| algebraMap R <| Localization M) :=
IsLocalization.epi M _
#align localization.epi Localization.epi
+-/
+#print Localization.epi' /-
instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) := by cases R;
exact IsLocalization.epi M _
#align localization.epi' Localization.epi'
+-/
#print CommRingCat.isLocalRingHom_comp /-
instance CommRingCat.isLocalRingHom_comp {R S T : CommRingCat} (f : R ⟶ S) (g : S ⟶ T)
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -22,17 +22,11 @@ import Mathbin.RingTheory.Ideal.LocalRing
open CategoryTheory
-/- warning: localization_unit_is_iso -> localization_unit_isIso is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align localization_unit_is_iso localization_unit_isIsoₓ'. -/
instance localization_unit_isIso (R : CommRingCat) :
IsIso (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) :=
IsIso.of_iso (IsLocalization.atOne R (Localization.Away (1 : R))).toRingEquiv.toCommRingCatIso
#align localization_unit_is_iso localization_unit_isIso
-/- warning: localization_unit_is_iso' -> localization_unit_isIso' is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align localization_unit_is_iso' localization_unit_isIso'ₓ'. -/
instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) := by
cases R; exact localization_unit_isIso _
@@ -45,23 +39,11 @@ theorem IsLocalization.epi {R : Type _} [CommRing R] (M : Submonoid R) (S : Type
#align is_localization.epi IsLocalization.epi
-/
-/- warning: localization.epi -> Localization.epi is a dubious translation:
-lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))), CategoryTheory.Epi.{u1, succ u1} CommRingCat.{u1} CommRingCat.largeCategory.{u1} (CommRingCat.of.{u1} R _inst_1) (CommRingCat.of.{u1} (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) (Localization.commRing.{u1} R _inst_1 M)) (CommRingCat.ofHom.{u1} R (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) _inst_1 (Localization.commRing.{u1} R _inst_1 M) (algebraMap.{u1, u1} R (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) (CommRing.toRing.{u1} (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) (Localization.commRing.{u1} R _inst_1 M))) (Localization.algebra.{u1, u1} R (CommRing.toCommSemiring.{u1} R _inst_1) M R (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))
-but is expected to have type
- forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))), CategoryTheory.Epi.{u1, succ u1} CommRingCat.{u1} instCommRingCatLargeCategory.{u1} (CommRingCat.of.{u1} R _inst_1) (CommRingCat.of.{u1} (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) (Localization.instCommRingLocalizationToCommMonoid.{u1} R _inst_1 M)) (CommRingCat.ofHom.{u1} R (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) _inst_1 (Localization.instCommRingLocalizationToCommMonoid.{u1} R _inst_1 M) (algebraMap.{u1, u1} R (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) M)) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} R (CommRing.toCommSemiring.{u1} R _inst_1) M R (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))
-Case conversion may be inaccurate. Consider using '#align localization.epi Localization.epiₓ'. -/
instance Localization.epi {R : Type _} [CommRing R] (M : Submonoid R) :
Epi (CommRingCat.ofHom <| algebraMap R <| Localization M) :=
IsLocalization.epi M _
#align localization.epi Localization.epi
-/- warning: localization.epi' -> Localization.epi' is a dubious translation:
-lean 3 declaration is
- forall {R : CommRingCat.{u1}} (M : Submonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (MulZeroOneClass.toMulOneClass.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (NonAssocSemiring.toMulZeroOneClass.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (NonAssocRing.toNonAssocSemiring.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (Ring.toNonAssocRing.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toRing.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R))))))), CategoryTheory.Epi.{u1, succ u1} CommRingCat.{u1} CommRingCat.largeCategory.{u1} R (CommRingCat.of.{u1} (Localization.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommMonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) M) (Localization.commRing.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R) M)) (CommRingCat.ofHom.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (Localization.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommMonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) M) (CommRingCat.commRing.{u1} R) (Localization.commRing.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R) M) (algebraMap.{u1, u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (Localization.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommMonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) M) (CommRing.toCommSemiring.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) (Ring.toSemiring.{u1} (Localization.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommMonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) M) (CommRing.toRing.{u1} (Localization.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommMonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) M) (Localization.commRing.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R) M))) (Localization.algebra.{u1, u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommSemiring.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) M (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommSemiring.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) (Algebra.id.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommSemiring.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R))))))
-but is expected to have type
- forall {R : CommRingCat.{u1}} (M : Submonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (MulZeroOneClass.toMulOneClass.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (NonAssocSemiring.toMulZeroOneClass.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toNonAssocSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))), CategoryTheory.Epi.{u1, succ u1} CommRingCat.{u1} instCommRingCatLargeCategory.{u1} R (CommRingCat.of.{u1} (Localization.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M) (Localization.instCommRingLocalizationToCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R) M)) (CommRingCat.ofHom.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Localization.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M) (CommRingCat.instCommRingα_1.{u1} R) (Localization.instCommRingLocalizationToCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R) M) (algebraMap.{u1, u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Localization.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (CommSemiring.toSemiring.{u1} (Localization.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M)) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (Algebra.id.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))
-Case conversion may be inaccurate. Consider using '#align localization.epi' Localization.epi'ₓ'. -/
instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) := by cases R;
exact IsLocalization.epi M _
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -34,10 +34,8 @@ instance localization_unit_isIso (R : CommRingCat) :
<too large>
Case conversion may be inaccurate. Consider using '#align localization_unit_is_iso' localization_unit_isIso'ₓ'. -/
instance localization_unit_isIso' (R : CommRingCat) :
- @IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) :=
- by
- cases R
- exact localization_unit_isIso _
+ @IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) := by
+ cases R; exact localization_unit_isIso _
#align localization_unit_is_iso' localization_unit_isIso'
#print IsLocalization.epi /-
@@ -65,9 +63,7 @@ but is expected to have type
forall {R : CommRingCat.{u1}} (M : Submonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (MulZeroOneClass.toMulOneClass.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (NonAssocSemiring.toMulZeroOneClass.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toNonAssocSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))), CategoryTheory.Epi.{u1, succ u1} CommRingCat.{u1} instCommRingCatLargeCategory.{u1} R (CommRingCat.of.{u1} (Localization.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M) (Localization.instCommRingLocalizationToCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R) M)) (CommRingCat.ofHom.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Localization.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M) (CommRingCat.instCommRingα_1.{u1} R) (Localization.instCommRingLocalizationToCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R) M) (algebraMap.{u1, u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Localization.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (CommSemiring.toSemiring.{u1} (Localization.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M)) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (Algebra.id.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))
Case conversion may be inaccurate. Consider using '#align localization.epi' Localization.epi'ₓ'. -/
instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
- @Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) :=
- by
- cases R
+ @Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) := by cases R;
exact IsLocalization.epi M _
#align localization.epi' Localization.epi'
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -23,28 +23,22 @@ import Mathbin.RingTheory.Ideal.LocalRing
open CategoryTheory
/- warning: localization_unit_is_iso -> localization_unit_isIso is a dubious translation:
-lean 3 declaration is
- forall (R : CommRingCat.{u1}), CategoryTheory.IsIso.{u1, succ u1} CommRingCat.{u1} CommRingCat.largeCategory.{u1} (CommRingCat.of.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) (CommRingCat.of.{u1} (Localization.Away.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommMonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) (OfNat.ofNat.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) 1 (OfNat.mk.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) 1 (One.one.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (AddMonoidWithOne.toOne.{u1} (coeSort.{succ (succ u1), succ (succ 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Case conversion may be inaccurate. Consider using '#align localization_unit_is_iso localization_unit_isIsoₓ'. -/
instance localization_unit_isIso (R : CommRingCat) :
IsIso (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) :=
IsIso.of_iso (IsLocalization.atOne R (Localization.Away (1 : R))).toRingEquiv.toCommRingCatIso
#align localization_unit_is_iso localization_unit_isIso
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(CommMonoid.toMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))) (OfNat.ofNat.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) 1 (One.toOfNat1.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toOne.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))))) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (Submonoid.powers.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommMonoid.toMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))) (OfNat.ofNat.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) 1 (One.toOfNat1.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toOne.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))) (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (Algebra.id.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))
-Case conversion may be inaccurate. Consider using '#align localization_unit_is_iso' localization_unit_is_iso'ₓ'. -/
-instance localization_unit_is_iso' (R : CommRingCat) :
+/- warning: localization_unit_is_iso' -> localization_unit_isIso' is a dubious translation:
+<too large>
+Case conversion may be inaccurate. Consider using '#align localization_unit_is_iso' localization_unit_isIso'ₓ'. -/
+instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) :=
by
cases R
exact localization_unit_isIso _
-#align localization_unit_is_iso' localization_unit_is_iso'
+#align localization_unit_is_iso' localization_unit_isIso'
#print IsLocalization.epi /-
theorem IsLocalization.epi {R : Type _} [CommRing R] (M : Submonoid R) (S : Type _) [CommRing S]
mathlib commit https://github.com/leanprover-community/mathlib/commit/8d33f09cd7089ecf074b4791907588245aec5d1b
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
! This file was ported from Lean 3 source module algebra.category.Ring.instances
-! leanprover-community/mathlib commit a7c017d750512a352b623b1824d75da5998457d0
+! leanprover-community/mathlib commit 38df578a6450a8c5142b3727e3ae894c2300cae0
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -14,6 +14,9 @@ import Mathbin.RingTheory.Ideal.LocalRing
/-!
# Ring-theoretic results in terms of categorical languages
+
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828
@@ -19,11 +19,23 @@ import Mathbin.RingTheory.Ideal.LocalRing
open CategoryTheory
+/- warning: localization_unit_is_iso -> localization_unit_isIso is a dubious translation:
+lean 3 declaration is
+ forall (R : CommRingCat.{u1}), CategoryTheory.IsIso.{u1, succ u1} CommRingCat.{u1} CommRingCat.largeCategory.{u1} (CommRingCat.of.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) (CommRingCat.of.{u1} (Localization.Away.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommMonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) (OfNat.ofNat.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) 1 (OfNat.mk.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) 1 (One.one.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (AddMonoidWithOne.toOne.{u1} (coeSort.{succ (succ u1), succ (succ 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R) (Ring.toAddCommGroupWithOne.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toRing.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)))))))))))) (CommRingCat.ofHom.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (Localization.Away.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommMonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) (OfNat.ofNat.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) 1 (OfNat.mk.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) 1 (One.one.{u1} (coeSort.{succ (succ u1), 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+but is expected to have type
+ forall (R : CommRingCat.{u1}), CategoryTheory.IsIso.{u1, succ u1} CommRingCat.{u1} instCommRingCatLargeCategory.{u1} (CommRingCat.of.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (CommRingCat.of.{u1} (Localization.Away.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (OfNat.ofNat.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) 1 (One.toOfNat1.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toOne.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))) (Localization.instCommRingLocalizationToCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R) (Submonoid.powers.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommMonoid.toMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))) (OfNat.ofNat.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) 1 (One.toOfNat1.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toOne.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))))) (CommRingCat.ofHom.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Localization.Away.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (OfNat.ofNat.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) 1 (One.toOfNat1.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toOne.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))) (CommRingCat.instCommRingα_1.{u1} R) (Localization.instCommRingLocalizationToCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R) (Submonoid.powers.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommMonoid.toMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))) (OfNat.ofNat.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) 1 (One.toOfNat1.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toOne.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)))))))) (algebraMap.{u1, u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Localization.Away.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (OfNat.ofNat.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) 1 (One.toOfNat1.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toOne.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (CommSemiring.toSemiring.{u1} (Localization.Away.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (OfNat.ofNat.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) 1 (One.toOfNat1.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toOne.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (Submonoid.powers.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommMonoid.toMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))) (OfNat.ofNat.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) 1 (One.toOfNat1.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toOne.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))))) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (Submonoid.powers.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommMonoid.toMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))) (OfNat.ofNat.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) 1 (One.toOfNat1.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toOne.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))) (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (Algebra.id.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))
+Case conversion may be inaccurate. Consider using '#align localization_unit_is_iso localization_unit_isIsoₓ'. -/
instance localization_unit_isIso (R : CommRingCat) :
IsIso (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) :=
IsIso.of_iso (IsLocalization.atOne R (Localization.Away (1 : R))).toRingEquiv.toCommRingCatIso
#align localization_unit_is_iso localization_unit_isIso
+/- warning: localization_unit_is_iso' -> localization_unit_is_iso' is a dubious translation:
+lean 3 declaration is
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u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)))))))) (algebraMap.{u1, u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Localization.Away.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (OfNat.ofNat.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) 1 (One.toOfNat1.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toOne.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (CommSemiring.toSemiring.{u1} (Localization.Away.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (OfNat.ofNat.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) 1 (One.toOfNat1.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toOne.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (Submonoid.powers.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommMonoid.toMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))) (OfNat.ofNat.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) 1 (One.toOfNat1.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toOne.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))))) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (Submonoid.powers.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommMonoid.toMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))) (OfNat.ofNat.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) 1 (One.toOfNat1.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toOne.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))) (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (Algebra.id.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))
+Case conversion may be inaccurate. Consider using '#align localization_unit_is_iso' localization_unit_is_iso'ₓ'. -/
instance localization_unit_is_iso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) :=
by
@@ -31,16 +43,30 @@ instance localization_unit_is_iso' (R : CommRingCat) :
exact localization_unit_isIso _
#align localization_unit_is_iso' localization_unit_is_iso'
+#print IsLocalization.epi /-
theorem IsLocalization.epi {R : Type _} [CommRing R] (M : Submonoid R) (S : Type _) [CommRing S]
[Algebra R S] [IsLocalization M S] : Epi (CommRingCat.ofHom <| algebraMap R S) :=
⟨fun T f₁ f₂ => @IsLocalization.ringHom_ext R _ M S _ _ T _ _ _ _⟩
#align is_localization.epi IsLocalization.epi
+-/
+/- warning: localization.epi -> Localization.epi is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))), CategoryTheory.Epi.{u1, succ u1} CommRingCat.{u1} CommRingCat.largeCategory.{u1} (CommRingCat.of.{u1} R _inst_1) (CommRingCat.of.{u1} (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) (Localization.commRing.{u1} R _inst_1 M)) (CommRingCat.ofHom.{u1} R (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) _inst_1 (Localization.commRing.{u1} R _inst_1 M) (algebraMap.{u1, u1} R (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) (CommRing.toRing.{u1} (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) (Localization.commRing.{u1} R _inst_1 M))) (Localization.algebra.{u1, u1} R (CommRing.toCommSemiring.{u1} R _inst_1) M R (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))), CategoryTheory.Epi.{u1, succ u1} CommRingCat.{u1} instCommRingCatLargeCategory.{u1} (CommRingCat.of.{u1} R _inst_1) (CommRingCat.of.{u1} (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) (Localization.instCommRingLocalizationToCommMonoid.{u1} R _inst_1 M)) (CommRingCat.ofHom.{u1} R (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) _inst_1 (Localization.instCommRingLocalizationToCommMonoid.{u1} R _inst_1 M) (algebraMap.{u1, u1} R (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (Localization.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) M)) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} R (CommRing.toCommSemiring.{u1} R _inst_1) M R (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))
+Case conversion may be inaccurate. Consider using '#align localization.epi Localization.epiₓ'. -/
instance Localization.epi {R : Type _} [CommRing R] (M : Submonoid R) :
Epi (CommRingCat.ofHom <| algebraMap R <| Localization M) :=
IsLocalization.epi M _
#align localization.epi Localization.epi
+/- warning: localization.epi' -> Localization.epi' is a dubious translation:
+lean 3 declaration is
+ forall {R : CommRingCat.{u1}} (M : Submonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (MulZeroOneClass.toMulOneClass.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (NonAssocSemiring.toMulZeroOneClass.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (NonAssocRing.toNonAssocSemiring.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (Ring.toNonAssocRing.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toRing.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R))))))), CategoryTheory.Epi.{u1, succ u1} CommRingCat.{u1} CommRingCat.largeCategory.{u1} R (CommRingCat.of.{u1} (Localization.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommMonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) M) (Localization.commRing.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R) M)) (CommRingCat.ofHom.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (Localization.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommMonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) M) (CommRingCat.commRing.{u1} R) (Localization.commRing.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R) M) (algebraMap.{u1, u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (Localization.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommMonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) M) (CommRing.toCommSemiring.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) (Ring.toSemiring.{u1} (Localization.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommMonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) M) (CommRing.toRing.{u1} (Localization.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommMonoid.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) M) (Localization.commRing.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R) M))) (Localization.algebra.{u1, u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommSemiring.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) M (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommSemiring.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R)) (Algebra.id.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRing.toCommSemiring.{u1} (coeSort.{succ (succ u1), succ (succ u1)} CommRingCat.{u1} Type.{u1} CommRingCat.hasCoeToSort.{u1} R) (CommRingCat.commRing.{u1} R))))))
+but is expected to have type
+ forall {R : CommRingCat.{u1}} (M : Submonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (MulZeroOneClass.toMulOneClass.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (NonAssocSemiring.toMulZeroOneClass.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Semiring.toNonAssocSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommSemiring.toSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))), CategoryTheory.Epi.{u1, succ u1} CommRingCat.{u1} instCommRingCatLargeCategory.{u1} R (CommRingCat.of.{u1} (Localization.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M) (Localization.instCommRingLocalizationToCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R) M)) (CommRingCat.ofHom.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Localization.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M) (CommRingCat.instCommRingα_1.{u1} R) (Localization.instCommRingLocalizationToCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R) M) (algebraMap.{u1, u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (Localization.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (CommSemiring.toSemiring.{u1} (Localization.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M)) (Localization.instAlgebraLocalizationToCommMonoidToSemiringInstCommSemiringLocalizationToCommMonoid.{u1, u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) M (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R)) (Algebra.id.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRing.toCommSemiring.{u1} (CategoryTheory.Bundled.α.{u1, u1} CommRing.{u1} R) (CommRingCat.instCommRingα_1.{u1} R))))))
+Case conversion may be inaccurate. Consider using '#align localization.epi' Localization.epi'ₓ'. -/
instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) :=
by
@@ -48,21 +74,27 @@ instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
exact IsLocalization.epi M _
#align localization.epi' Localization.epi'
+#print CommRingCat.isLocalRingHom_comp /-
instance CommRingCat.isLocalRingHom_comp {R S T : CommRingCat} (f : R ⟶ S) (g : S ⟶ T)
[IsLocalRingHom g] [IsLocalRingHom f] : IsLocalRingHom (f ≫ g) :=
isLocalRingHom_comp _ _
#align CommRing.is_local_ring_hom_comp CommRingCat.isLocalRingHom_comp
+-/
+#print isLocalRingHom_of_iso /-
theorem isLocalRingHom_of_iso {R S : CommRingCat} (f : R ≅ S) : IsLocalRingHom f.Hom :=
{
map_nonunit := fun a ha => by
convert f.inv.is_unit_map ha
rw [CategoryTheory.Iso.hom_inv_id_apply] }
#align is_local_ring_hom_of_iso isLocalRingHom_of_iso
+-/
+#print isLocalRingHom_of_isIso /-
-- see Note [lower instance priority]
instance (priority := 100) isLocalRingHom_of_isIso {R S : CommRingCat} (f : R ⟶ S) [IsIso f] :
IsLocalRingHom f :=
isLocalRingHom_of_iso (asIso f)
#align is_local_ring_hom_of_is_iso isLocalRingHom_of_isIso
+-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/f51de8769c34652d82d1c8e5f8f18f8374782bed
@@ -4,12 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
! This file was ported from Lean 3 source module algebra.category.Ring.instances
-! leanprover-community/mathlib commit c1918ac18519732d5bdfa2b6917c016c4d6a8148
+! leanprover-community/mathlib commit a7c017d750512a352b623b1824d75da5998457d0
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Algebra.Category.Ring.Basic
-import Mathbin.RingTheory.Localization.Away
+import Mathbin.RingTheory.Localization.Away.Basic
import Mathbin.RingTheory.Ideal.LocalRing
/-!
mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee
@@ -21,7 +21,7 @@ open CategoryTheory
instance localization_unit_isIso (R : CommRingCat) :
IsIso (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) :=
- IsIso.of_iso (IsLocalization.atOne R (Localization.Away (1 : R))).toRingEquiv.toCommRingIso
+ IsIso.of_iso (IsLocalization.atOne R (Localization.Away (1 : R))).toRingEquiv.toCommRingCatIso
#align localization_unit_is_iso localization_unit_isIso
instance localization_unit_is_iso' (R : CommRingCat) :
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
Type _
and Sort _
(#6499)
We remove all possible occurences of Type _
and Sort _
in favor of Type*
and Sort*
.
This has nice performance benefits.
@@ -27,12 +27,12 @@ instance localization_unit_isIso' (R : CommRingCat) :
exact localization_unit_isIso _
#align localization_unit_is_iso' localization_unit_isIso'
-theorem IsLocalization.epi {R : Type _} [CommRing R] (M : Submonoid R) (S : Type _) [CommRing S]
+theorem IsLocalization.epi {R : Type*} [CommRing R] (M : Submonoid R) (S : Type _) [CommRing S]
[Algebra R S] [IsLocalization M S] : Epi (CommRingCat.ofHom <| algebraMap R S) :=
⟨fun {T} _ _ => @IsLocalization.ringHom_ext R _ M S _ _ T _ _ _ _⟩
#align is_localization.epi IsLocalization.epi
-instance Localization.epi {R : Type _} [CommRing R] (M : Submonoid R) :
+instance Localization.epi {R : Type*} [CommRing R] (M : Submonoid R) :
Epi (CommRingCat.ofHom <| algebraMap R <| Localization M) :=
IsLocalization.epi M _
#align localization.epi Localization.epi
@@ -2,16 +2,13 @@
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 algebra.category.Ring.instances
-! leanprover-community/mathlib commit a7c017d750512a352b623b1824d75da5998457d0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.LocalRing
+#align_import algebra.category.Ring.instances from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
+
/-!
# Ring-theoretic results in terms of categorical languages
-/
@@ -24,11 +24,11 @@ instance localization_unit_isIso (R : CommRingCat) :
IsIso.of_iso (IsLocalization.atOne R (Localization.Away (1 : R))).toRingEquiv.toCommRingCatIso
#align localization_unit_is_iso localization_unit_isIso
-instance localization_unit_is_iso' (R : CommRingCat) :
+instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) := by
cases R
exact localization_unit_isIso _
-#align localization_unit_is_iso' localization_unit_is_iso'
+#align localization_unit_is_iso' localization_unit_isIso'
theorem IsLocalization.epi {R : Type _} [CommRing R] (M : Submonoid R) (S : Type _) [CommRing S]
[Algebra R S] [IsLocalization M S] : Epi (CommRingCat.ofHom <| algebraMap R S) :=
The unported dependencies are