Localizations of commutative rings #

We characterize the localization of a commutative ring R at a submonoid M up to isomorphism; that is, a commutative ring S is the localization of R at M iff we can find a ring homomorphism f : R →+* S satisfying 3 properties:

For all y ∈ M, f y is a unit;
For all z : S, there exists (x, y) : R × M such that z * f y = f x;
For all x, y : R, f x = f y iff there exists c ∈ M such that x * c = y * c.

In the following, let R, P be commutative rings, S, Q be R- and P-algebras and M, T be submonoids of R and P respectively, e.g.:

variable (R S P Q : Type*) [CommRing R] [CommRing S] [CommRing P] [CommRing Q]
variable [Algebra R S] [Algebra P Q] (M : Submonoid R) (T : Submonoid P)

Main definitions #

IsLocalization (M : Submonoid R) (S : Type*) is a typeclass expressing that S is a localization of R at M, i.e. the canonical map algebraMap R S : R →+* S is a localization map (satisfying the above properties).
IsLocalization.mk' S is a surjection sending (x, y) : R × M to f x * (f y)⁻¹
IsLocalization.lift is the ring homomorphism from S induced by a homomorphism from R which maps elements of M to invertible elements of the codomain.
IsLocalization.map S Q is the ring homomorphism from S to Q which maps elements of M to elements of T
IsLocalization.ringEquivOfRingEquiv: if R and P are isomorphic by an isomorphism sending M to T, then S and Q are isomorphic
IsLocalization.algEquiv: if Q is another localization of R at M, then S and Q are isomorphic as R-algebras

Main results #

Localization M S, a construction of the localization as a quotient type, defined in GroupTheory.MonoidLocalization, has CommRing, Algebra R and IsLocalization M instances if R is a ring. Localization.Away, Localization.AtPrime and FractionRing are abbreviations for Localizations and have their corresponding IsLocalization instances

Implementation notes #

In maths it is natural to reason up to isomorphism, but in Lean we cannot naturally rewrite one structure with an isomorphic one; one way around this is to isolate a predicate characterizing a structure up to isomorphism, and reason about things that satisfy the predicate.

A previous version of this file used a fully bundled type of ring localization maps, then used a type synonym f.codomain for f : LocalizationMap M S to instantiate the R-algebra structure on S. This results in defining ad-hoc copies for everything already defined on S. By making IsLocalization a predicate on the algebraMap R S, we can ensure the localization map commutes nicely with other algebraMaps.

To prove most lemmas about a localization map algebraMap R S in this file we invoke the corresponding proof for the underlying CommMonoid localization map IsLocalization.toLocalizationMap M S, which can be found in GroupTheory.MonoidLocalization and the namespace Submonoid.LocalizationMap.

To reason about the localization as a quotient type, use mk_eq_of_mk' and associated lemmas. These show the quotient map mk : R → M → Localization M equals the surjection LocalizationMap.mk' induced by the map algebraMap : R →+* Localization M. The lemma mk_eq_of_mk' hence gives you access to the results in the rest of the file, which are about the LocalizationMap.mk' induced by any localization map.

The proof that "a CommRing K which is the localization of an integral domain R at R \ {0} is a field" is a def rather than an instance, so if you want to reason about a field of fractions K, assume [Field K] instead of just [CommRing K].

Tags #

localization, ring localization, commutative ring localization, characteristic predicate, commutative ring, field of fractions

Localizations of commutative rings #

Main definitions #

Main results #

Implementation notes #

Tags #

Constructing a localization at a given submonoid #