Localization (commutative algebra)

Localization is commonly done with respect to a multiplicatively closed set S (also called a multiplicative set or a multiplicative system) of elements of a ring R, that is a subset of R that is closed under multiplication, and contains 1.

The requirement that S must be a multiplicative set is natural, since it implies that all denominators introduced by the localization belong to S. The localization by a set U that is not multiplicatively closed can also be defined, by taking as possible denominators all products of elements of U. However, the same localization is obtained by using the multiplicatively closed set S of all products of elements of U. As this often makes reasoning and notation simpler, it is standard practice to consider only localizations by multiplicative sets.

For example, the localization by a single element s introduces fractions of the form ${\displaystyle {\tfrac {a}{s}},}$ but also products of such fractions, such as ${\displaystyle {\tfrac {ab}{s^{2}}}.}$ So, the denominators will belong to the multiplicative set ${\displaystyle \{1,s,s^{2},s^{3},\ldots \}}$ of the powers of s. Therefore, one generally talks of "the localization by the powers of an element" rather than of "the localization by an element".

The localization of a ring R by a multiplicative set S is generally denoted ${\displaystyle S^{-1}R,}$ but other notations are commonly used in some special cases: if ${\displaystyle S=\{1,t,t^{2},\ldots \}}$ consists of the powers of a single element, ${\displaystyle S^{-1}R}$ is often denoted ${\displaystyle R_{t};}$ if ${\displaystyle S=R\setminus {\mathfrak {p}}}$ is the complement of a prime ideal ${\displaystyle {\mathfrak {p}}}$, then ${\displaystyle S^{-1}R}$ is denoted ${\displaystyle R_{\mathfrak {p}}.}$

In the remainder of this article, only localizations by a multiplicative set are considered.

When the ring R is an integral domain and S does not contain 0, the ring ${\displaystyle S^{-1}R}$ is a subring of the field of fractions of R. As such, the localization of a domain is a domain.

More precisely, it is the subring of the field of fractions of R, that consists of the fractions ${\displaystyle {\tfrac {a}{s}}}$ such that ${\displaystyle s\in S.}$ This is a subring since the sum ${\displaystyle {\tfrac {a}{s}}+{\tfrac {b}{t}}={\tfrac {at+bs}{st}},}$ and the product ${\displaystyle {\tfrac {a}{s}}\,{\tfrac {b}{t}}={\tfrac {ab}{st}}}$ of two elements of ${\displaystyle S^{-1}R}$ are in ${\displaystyle S^{-1}R.}$ This results from the defining property of a multiplicative set, which implies also that ${\displaystyle 1={\tfrac {1}{1}}\in S^{-1}R.}$ In this case, R is a subring of ${\displaystyle S^{-1}R.}$ It is shown below that this is no longer true in general, typically when S contains zero divisors.

For example, the decimal fractions are the localization of the ring of integers by the multiplicative set of the powers of ten. In this case, ${\displaystyle S^{-1}R}$ consists of the rational numbers that can be written as ${\displaystyle {\tfrac {n}{10^{k}}},}$ where n is an integer, and k is a nonnegative integer.

In the general case, a problem arises with zero divisors. Let S be a multiplicative set in a commutative ring R. Suppose that ${\displaystyle s\in S,}$ and ${\displaystyle 0\neq a\in R}$ is a zero divisor with ${\displaystyle as=0.}$ Then ${\displaystyle {\tfrac {a}{1}}}$ is the image in ${\displaystyle S^{-1}R}$ of ${\displaystyle a\in R,}$ and one has ${\displaystyle {\tfrac {a}{1}}={\tfrac {as}{s}}={\tfrac {0}{s}}={\tfrac {0}{1}}.}$ Thus some nonzero elements of R must be zero in ${\displaystyle S^{-1}R.}$ The construction that follows is designed for taking this into account.

Given R and S as above, one considers the equivalence relation on ${\displaystyle R\times S}$ that is defined by ${\displaystyle (r_{1},s_{1})\sim (r_{2},s_{2})}$ if there exists a ${\displaystyle t\in S}$ such that ${\displaystyle t(s_{1}r_{2}-s_{2}r_{1})=0.}$

The localization ${\displaystyle S^{-1}R}$ is defined as the set of the equivalence classes for this relation. The class of (r, s) is denoted as ${\displaystyle {\frac {r}{s}},}$ ${\displaystyle r/s,}$ or ${\displaystyle s^{-1}r.}$ So, one has ${\displaystyle {\tfrac {r_{1}}{s_{1}}}={\tfrac {r_{2}}{s_{2}}}}$ if and only if there is a ${\displaystyle t\in S}$ such that ${\displaystyle t(s_{1}r_{2}-s_{2}r_{1})=0.}$ The reason for the ${\displaystyle t}$ is to handle cases such as the above ${\displaystyle {\tfrac {a}{1}}={\tfrac {0}{1}},}$ where ${\displaystyle s_{1}r_{2}-s_{2}r_{1}}$ is nonzero even though the fractions should be regarded as equal.

The localization ${\displaystyle S^{-1}R}$ is a commutative ring with addition

${\displaystyle {\frac {r_{1}}{s_{1}}}+{\frac {r_{2}}{s_{2}}}={\frac {r_{1}s_{2}+r_{2}s_{1}}{s_{1}s_{2}}},}$

multiplication

${\displaystyle {\frac {r_{1}}{s_{1}}}\,{\frac {r_{2}}{s_{2}}}={\frac {r_{1}r_{2}}{s_{1}s_{2}}},}$

additive identity ${\displaystyle {\tfrac {0}{1}},}$ and multiplicative identity ${\displaystyle {\tfrac {1}{1}}.}$

The function

${\displaystyle r\mapsto {\frac {r}{1}}}$

defines a ring homomorphism from ${\displaystyle R}$ into ${\displaystyle S^{-1}R,}$ which is injective if and only if S does not contain any zero divisors.

If ${\displaystyle 0\in S,}$ then ${\displaystyle S^{-1}R}$ is the zero ring that has 0 as unique element.

If S is the set of all regular elements of R (that is the elements that are not zero divisors), ${\displaystyle S^{-1}R}$ is called the total ring of fractions of R.

The (above defined) ring homomorphism ${\displaystyle j\colon R\to S^{-1}R}$ satisfies a universal property that is described below. This characterizes ${\displaystyle S^{-1}R}$ up to an isomorphism. So all properties of localizations can be deduced from the universal property, independently from the way they have been constructed. Moreover, many important properties of localization are easily deduced from the general properties of universal properties, while their direct proof may be together technical, straightforward and boring.

The universal property satisfied by ${\displaystyle j\colon R\to S^{-1}R}$ is the following:

If ${\displaystyle f\colon R\to T}$ is a ring homomorphism that maps every element of S to a unit (invertible element) in T, there exists a unique ring homomorphism ${\displaystyle g\colon S^{-1}R\to T}$ such that ${\displaystyle f=g\circ j.}$

Using category theory, this can be expressed by saying that localization is a functor that is left adjoint to a forgetful functor. More precisely, let ${\displaystyle {\mathcal {C}}}$ and ${\displaystyle {\mathcal {D}}}$ be the categories whose objects are pairs of a commutative ring and a submonoid of, respectively, the multiplicative monoid or the group of units of the ring. The morphisms of these categories are the ring homomorphisms that map the submonoid of the first object into the submonoid of the second one. Finally, let ${\displaystyle {\mathcal {F}}\colon {\mathcal {D}}\to {\mathcal {C}}}$ be the forgetful functor that forgets that the elements of the second element of the pair are invertible.

Then the factorization ${\displaystyle f=g\circ j}$ of the universal property defines a bijection

${\displaystyle \hom _{\mathcal {C}}((R,S),{\mathcal {F}}(T,U))\to \hom _{\mathcal {D}}((S^{-1}R,j(S)),(T,U)).}$

This may seem a rather tricky way of expressing the universal property, but it is useful for showing easily many properties, by using the fact that the composition of two left adjoint functors is a left adjoint functor.

• If ${\displaystyle R=\mathbb {Z} }$ is the ring of integers, and ${\displaystyle S=\mathbb {Z} \setminus \{0\},}$ then ${\displaystyle S^{-1}R}$ is the field ${\displaystyle \mathbb {Q} }$ of the rational numbers.
• If R is an integral domain, and ${\displaystyle S=R\setminus \{0\},}$ then ${\displaystyle S^{-1}R}$ is the field of fractions of R. The preceding example is a special case of this one.
• If R is a commutative ring, and if S is the subset of its elements that are not zero divisors, then ${\displaystyle S^{-1}R}$ is the total ring of fractions of R. In this case, S is the largest multiplicative set such that the homomorphism ${\displaystyle R\to S^{-1}R}$ is injective. The preceding example is a special case of this one.
• If x is an element of a commutative ring R and ${\displaystyle S=\{1,x,x^{2},\ldots \},}$ then ${\displaystyle S^{-1}R}$ can be identified (is canonically isomorphic to) ${\displaystyle R[x^{-1}]=R[s]/(xs-1).}$ (The proof consists of showing that this ring satisfies the above universal property.) This sort of localization plays a fundamental role in the definition of an affine scheme.
• If ${\displaystyle {\mathfrak {p}}}$ is a prime ideal of a commutative ring R, the set complement ${\displaystyle S=R\setminus {\mathfrak {p}}}$ of ${\displaystyle {\mathfrak {p}}}$ in R is a multiplicative set (by the definition of a prime ideal). The ring ${\displaystyle S^{-1}R}$ is a local ring that is generally denoted ${\displaystyle R_{\mathfrak {p}},}$ and called the local ring of R at ${\displaystyle {\mathfrak {p}}.}$ This sort of localization is fundamental in commutative algebra, because many properties of a commutative ring can be read on its local rings. Such a property is often called a local property. For example, a ring is regular if and only if all its local rings are regular.

Localization is a rich construction that has many useful properties. In this section, only the properties relative to rings and to a single localization are considered. Properties concerning ideals, modules, or several multiplicative sets are considered in other sections.

• ${\displaystyle S^{-1}R=0}$ if and only if S contains 0.
• The ring homomorphism ${\displaystyle R\to S^{-1}R}$ is injective if and only if S does not contain any zero divisors.
• The ring homomorphism ${\displaystyle R\to S^{-1}R}$ is an epimorphism in the category of rings, that is not surjective in general.
• The ring ${\displaystyle S^{-1}R}$ is a flat R-module (see § Localization of a module for details).
• If ${\displaystyle S=R\setminus {\mathfrak {p}}}$ is the complement of a prime ideal ${\displaystyle {\mathfrak {p}}}$, then ${\displaystyle S^{-1}R,}$ denoted ${\displaystyle R_{\mathfrak {p}},}$ is a local ring; that is, it has only one maximal ideal.

Properties to be moved in another section

• Localization commutes with formations of finite sums, products, intersections and radicals;[1] e.g., if ${\displaystyle {\sqrt {I}}}$ denote the radical of an ideal I in R, then

${\displaystyle {\sqrt {I}}\cdot S^{-1}R={\sqrt {I\cdot S^{-1}R}}\,.}$

In particular, R is reduced if and only if its total ring of fractions is reduced.[2]

• Let R be an integral domain with the field of fractions K. Then its localization ${\displaystyle R_{\mathfrak {p}}}$ at a prime ideal ${\displaystyle {\mathfrak {p}}}$ can be viewed as a subring of K. Moreover,

${\displaystyle R=\bigcap _{\mathfrak {p}}R_{\mathfrak {p}}=\bigcap _{\mathfrak {m}}R_{\mathfrak {m}}}$

where the first intersection is over all prime ideals and the second over the maximal ideals.[3]

• There is a bijection between the set of prime ideals of S⁻¹R and the set of prime ideals of R that do not intersect S. This bijection is induced by the given homomorphism R → S ⁻¹R.

Let ${\displaystyle S\subseteq R}$ be a multiplicative set. The saturation ${\displaystyle {\hat {S}}}$ of ${\displaystyle S}$ is the set

${\displaystyle {\hat {S}}=\{r\in R\colon \exists s\in R,rs\in S\}.}$

The multiplicative set S is saturated if it equals its saturation, that is, if ${\displaystyle {\hat {S}}=S}$, or equivalently, if ${\displaystyle rs\in S}$ implies that r and s are in S.

If S is not saturated, and ${\displaystyle rs\in S,}$ then ${\displaystyle {\frac {s}{rs}}}$ is a multiplicative inverse of the image of r in ${\displaystyle S^{-1}R.}$ So, the images of the elements of ${\displaystyle {\hat {S}}}$ are all invertible in ${\displaystyle S^{-1}R,}$ and the universal property implies that ${\displaystyle S^{-1}R}$ and ${\displaystyle {\hat {S}}{}^{-1}R}$ are canonically isomorphic, that is, there is a unique isomorphism between them that fixes the images of the elements of R.

If S and T are two multiplicative sets, then ${\displaystyle S^{-1}R}$ and ${\displaystyle T^{-1}R}$ are isomorphic if and only if they have the same saturation, or, equivalently, if s belongs to one of the multiplicative set, then there exists ${\displaystyle t\in R}$ such that st belongs to the other.

Saturated multiplicative sets are not widely used explicitly, since, for verifying that a set is saturated, one must know all units of the ring.

If M is a submodule of an R-module N, and S is a multiplicative set in R, one has ${\displaystyle S^{-1}M\subseteq S^{-1}N.}$ This implies that, if ${\displaystyle f\colon M\to N}$ is an injective module homomorphism, then

${\displaystyle S^{-1}R\otimes _{R}f:\quad S^{-1}R\otimes _{R}M\to S^{-1}R\otimes _{R}N}$

is also an injective homomorphism.

Since the tensor product is a right exact functor, this implies that localization by S maps exact sequences of R-modules to exact sequences of ${\displaystyle S^{-1}R}$-modules. In other words, localization is an exact functor, and ${\displaystyle S^{-1}R}$ is a flat R-module.

This flatness and the fact that localization solves a universal property make that localization preserves many properties of modules and rings, and is compatible with solutions of other universal properties. For example, the natural map

${\displaystyle S^{-1}(M\otimes _{R}N)\to S^{-1}M\otimes _{S^{-1}R}S^{-1}N}$

is an isomorphism. If ${\displaystyle M}$ is a finitely presented module, the natural map

${\displaystyle S^{-1}\operatorname {Hom} _{R}(M,N)\to \operatorname {Hom} _{S^{-1}R}(S^{-1}M,S^{-1}N)}$

is also an isomorphism.[4]

If a module M is a finitely generated over R, one has

${\displaystyle S^{-1}(\operatorname {Ann} _{R}(M))=\operatorname {Ann} _{S^{-1}R}(S^{-1}M),}$

where ${\displaystyle \operatorname {Ann} }$ denotes annihilator, that is the ideal of the elements of the ring that map to zero all elements of the module.[5] In particular,

${\displaystyle S^{-1}M=0\quad \iff \quad S\cap \operatorname {Ann} _{R}(M)\neq \emptyset ,}$

that is, if ${\displaystyle tM=0}$ for some ${\displaystyle t\in S.}$[6]

A property P of an R-module M is a local property if the following conditions are equivalent:

• P holds for M.
• P holds for all ${\displaystyle M_{\mathfrak {p}},}$ where ${\displaystyle {\mathfrak {p}}}$ is a prime ideal of R.
• P holds for all ${\displaystyle M_{\mathfrak {m}},}$ where ${\displaystyle {\mathfrak {m}}}$ is a maximal ideal of R.

The following are local properties:

• M is zero.
• M is torsion-free (in the case where R is a commutative domain).
• M is a flat module.
• M is an invertible module (in the case where R is a commutative domain, and M is a submodule of the field of fractions of R).
• ${\displaystyle f\colon M\to N}$ is injective (resp. surjective), where N is another R-module.

On the other hand, some properties are not local properties. For example, an infinite direct product of fields is not an integral domain nor a Noetherian ring, while all its local rings are fields, and therefore Noetherian integral domains.