Localisation of a commutative ring

Intuition [Beginner]

Every fraction has a numerator and a denominator. The integers admit fractions like $3/4$ and $7/2$, but no integer $n$ satisfies $2\times n=1$. Localisation solves this by building a larger ring where chosen elements become invertible. Given a ring $A$ and a set $S$ of elements you want as denominators, the localisation $S^{-1}A$ is a new ring whose elements are fractions $a/s$ with $a$ in $A$ and $s$ in $S$, subject to cross-multiplication: $a/s$ equals $a^{\prime }/s^{\prime }$ when $s^{\prime }a=s{a}^{\prime }$.

Take $A=\mathbb{Z}$ and $S=\{1,2,4,8,16,\dots \}$, the powers of $2$. The localisation $S^{-1}\mathbb{Z}$ consists of all fractions $a/2^n$ with $a$ an integer and $n\ge 0$. These are the dyadic rationals: numbers like $3/4$, $-5/8$, $7/2$. The element $2$ is now invertible with inverse $1/2$, but $3$ and $5$ are not invertible because they were not placed in $S$.

Localisation zooms in on one part of a ring by making selected elements invertible while leaving others untouched. Different choices of $S$ illuminate different structural features. Why does this concept exist? Localisation provides the mechanism for studying a ring one prime at a time, which is the foundation of local algebra and the bridge from rings to geometry.

Visual [Beginner]

A large circle labelled $A$ with gold-highlighted elements forming the set $S$. A smaller circle to the right, labelled $S^{-1}A$, contains the original ring plus new inverse elements. An arrow connects the two, labelled "make $S$ invertible."

The picture shows that localisation enlarges the ring by adjoining inverses for elements of $S$ while preserving all the algebraic structure of $A$.

Worked example [Beginner]

Take $A=\mathbb{Z}$ and $S=\{1,3,9,27,81,\dots \}$, the powers of $3$. The localisation $S^{-1}\mathbb{Z}$ consists of fractions $a/3^n$.

Step 1. The integer $5$ maps to $5/1$ in $S^{-1}\mathbb{Z}$, which behaves exactly like $5$.

Step 2. The integer $3$ gains an inverse: $1/3$. This element did not exist in $\mathbb{Z}$; it is new. Similarly, $9$ has inverse $1/9$ and $27$ has inverse $1/27$.

Step 3. The fraction $6/9$ equals $2/3$ because $3\times 6=18=9\times 2$. Cross-multiplication confirms equality. Every fraction in this localisation can be written with a denominator that is a power of $3$.

What this tells us: localising $\mathbb{Z}$ at the powers of $3$ produces precisely the rational numbers whose denominators are powers of $3$, with the usual fraction arithmetic.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Throughout, $A$ is a commutative ring with identity ^{[Atiyah-Macdonald 1969]}.

Definition (Multiplicative set). A subset $S\subseteq A$ is a multiplicative set if $1\in S$ and $s,t\in S$ implies $st\in S$.

Definition (Localisation). The localisation of $A$ at $S$, written $S^{-1}A$, is the set of equivalence classes $a/s$ with $a\in A$, $s\in S$, where $$ \frac{a}{s} = \frac{a'}{s'} \iff \exists, t \in S:\ t(s' a - s a') = 0. $$ The ring operations are $$ \frac{a}{s} + \frac{a'}{s'} = \frac{s'a + sa'}{ss'}, \qquad \frac{a}{s} \cdot \frac{a'}{s'} = \frac{aa'}{ss'}. $$

The canonical map $\iota :A\to S^{-1}A$ sends $a\mapsto a/1$. This is a ring homomorphism, and every $s\in S$ becomes a unit in $S^{-1}A$: the inverse of $s/1$ is $1/s$.

Two important special cases:

1. Field of fractions. If $A$ is an integral domain and $S=A\setminus \{0\}$, then $S^{-1}A$ is the field of fractions of $A$. For $A=\mathbb{Z}$ this gives $\mathbb{Q}$.

2. Local ring at a prime. If $\mathfrak{p}$ is a prime ideal of $A$ and $S=A\setminus \mathfrak{p}$, then $S^{-1}A$ is denoted $A_{\mathfrak{p}}$. This is a local ring (a ring with exactly one maximal ideal) whose maximal ideal is $\mathfrak{p}{A}_{\mathfrak{p}}$.

Counterexamples to common slips [Intermediate+]

• The equivalence relation needs the witness $t$. When $A$ has no zero-divisors the condition simplifies to $s^{\prime }a=s{a}^{\prime }$. When zero-divisors are present, the witness $t\in S$ is required for transitivity of $\sim$.

• The canonical map $\iota$ need not be injective. If $s\in S$ satisfies $sa=0$, then $\iota (a)=a/1=0/1$ because $s(1\cdot a-0\cdot 1)=sa=0$. Elements annihilated by something in $S$ vanish in the localisation.

• If $0\in S$ then $S^{-1}A=0$. Every fraction $a/s$ equals $0/1$ witnessed by $t=0$. For this reason one always assumes $0\notin S$.

Key theorem with proof [Intermediate+]

Theorem (Universal property of localisation). Let $A$ be a commutative ring, $S$ a multiplicative set, and $B$ a commutative ring. If $f:A\to B$ is a ring homomorphism such that $f(s)$ is a unit in $B$ for every $s\in S$, then there exists a unique ring homomorphism $\bar{f}:S^{-1}A\to B$ satisfying $\bar{f}(a/1)=f(a)$ for all $a\in A$.

Proof. Existence. Define $\bar{f}(a/s)=f(a)f(s{)}^{-1}$. This is well-defined: if $a/s=a^{\prime }/s^{\prime }$, there exists $t\in S$ with $t(s^{\prime }a-s{a}^{\prime })=0$. Applying $f$ gives $f(t)(f(s^{\prime })f(a)-f(s)f(a^{\prime }))=0$ in $B$. Since $f(t)$ is a unit in $B$, multiply by $f(t{)}^{-1}$ to obtain $f(s^{\prime })f(a)=f(s)f(a^{\prime })$. Multiplying both sides by $f(s{)}^{-1}f(s^{\prime }{)}^{-1}$ gives $f(a)f(s{)}^{-1}=f(a^{\prime })f(s^{\prime }{)}^{-1}$.

The map $\bar{f}$ is a ring homomorphism: $$ \bar{f}!\left(\frac{a}{s} + \frac{a'}{s'}\right) = \bar{f}!\left(\frac{s'a + sa'}{ss'}\right) = \frac{f(s')f(a) + f(s)f(a')}{f(s)f(s')} = \frac{f(a)}{f(s)} + \frac{f(a')}{f(s')}. $$ Multiplicativity: $\bar{f}((a/s)(a^{\prime }/s^{\prime }))=f(a{a}^{\prime })f(s{s}^{\prime }{)}^{-1}=f(a)f(s{)}^{-1}f(a^{\prime })f(s^{\prime }{)}^{-1}=\bar{f}(a/s)\bar{f}(a^{\prime }/s^{\prime })$. Also $\bar{f}(1/1)=f(1)f(1{)}^{-1}=1$.

On elements $a/1$: $\bar{f}(a/1)=f(a)f(1{)}^{-1}=f(a)$, so $\bar{f}$ extends $f$.

Uniqueness. Suppose $g:S^{-1}A\to B$ is another ring homomorphism with $g(a/1)=f(a)$. For any $a/s\in S^{-1}A$: $$ g(a/s) = g\bigl((a/1)(1/s)\bigr) = g(a/1) \cdot g(1/s) = f(a) \cdot g(1/s). $$ Since $g(s/1)\cdot g(1/s)=g(1)=1$, we have $g(1/s)=g(s/1{)}^{-1}=f(s{)}^{-1}$. Therefore $g(a/s)=f(a)f(s{)}^{-1}=\bar{f}(a/s)$. $\square$

Bridge. The universal property builds toward 01.02.10 (tensor product of modules), where the localisation of a module $S^{-1}M$ is constructed as $S^{-1}A{\otimes }_{A}M$, and this is exactly the extension-of-scalars functor along $\iota :A\to S^{-1}A$. The foundational reason is that $S^{-1}A$ is the universal ring under $A$ that inverts $S$, and the central insight is that every homomorphism from $A$ to a ring where $S$ is inverted factors uniquely through $S^{-1}A$.

The bridge is between the ring-theoretic fraction construction and the module-theoretic tensor-product construction, and the pattern generalises to the localisation of quasi-coherent sheaves on a scheme ^{[Serre 1955]}.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Localisation of modules). Let $A$ be a commutative ring, $S$ a multiplicative set, and $M$ an $A$-module. The localisation $S^{-1}M$ is an $S^{-1}A$-module with scalar multiplication $(a/s)(m/t)=(am)/(st)$. There is a canonical isomorphism $S^{-1}M\cong S^{-1}A{\otimes }_{A}M$ of $S^{-1}A$-modules.

The isomorphism sends $m/s\mapsto (1/s)\otimes m$; its inverse sends $(a/s)\otimes m\mapsto (am)/s$ ^{[Atiyah-Macdonald 1969]}.

Theorem 2 (Exactness of localisation). The functor $S^{-1}(-):{\mathbf{Mod}}_A\to {\mathbf{Mod}}_{S^{-1}A}$ is exact. If $0\to M^{\prime }\stackrel{f}{\to }M\stackrel{g}{\to }{M}^{\prime \prime }\to 0$ is an exact sequence of $A$-modules, then $0\to S^{-1}{M}^{\prime }\stackrel{S^{-1}f}{\to }{S}^{-1}M\stackrel{S^{-1}g}{\to }{S}^{-1}{M}^{\prime \prime }\to 0$ is exact.

Equivalently, $S^{-1}A$ is a flat $A$-module: the functor $S^{-1}A{\otimes }_A-$ is exact. This is a consequence of the fraction-level description of kernels and images in $S^{-1}M$.

Theorem 3 (Prime ideal correspondence). The maps $\mathfrak{q}\mapsto {\iota }^{-1}(\mathfrak{q})$ and $\mathfrak{p}\mapsto \mathfrak{p}(S^{-1}A)$ give a bijection between prime ideals of $S^{-1}A$ and prime ideals of $A$ that do not meet $S$.

Under this correspondence, maximal ideals of $S^{-1}A$ correspond to prime ideals of $A$ that are maximal among those disjoint from $S$. In particular, when $S=A\setminus \mathfrak{p}$ for a prime $\mathfrak{p}$, the only maximal ideal of $A_{\mathfrak{p}}$ is $\mathfrak{p}{A}_{\mathfrak{p}}$.

Theorem 4 (Nakayama's lemma). Let $(A,\mathfrak{m})$ be a local ring and $M$ a finitely generated $A$-module. If $\mathfrak{m}M=M$, then $M=0$. Equivalently, elements $m_1,\dots ,m_n$ generate $M$ if and only if their images generate $M/\mathfrak{m}M$ as a vector space over the residue field $A/\mathfrak{m}$.

The equivalence-of-generators formulation is the practical version: to check that elements generate $M$, it suffices to check that their images generate $M/\mathfrak{m}M$.

Theorem 5 (Support of a module). The support of an $A$-module $M$ is $\mathrm{Supp}(M)=\{\mathfrak{p}\in \mathrm{Spec}A:M_{\mathfrak{p}}\ne 0\}$. If $M$ is finitely generated, then $\mathrm{Supp}(M)=V(\mathrm{Ann}M)$, the set of primes containing the annihilator of $M$.

The support detects where a module is "visible" after localisation. For a finitely generated module over a Noetherian ring, the support is a closed subset of $\mathrm{Spec}A$ ^{[Eisenbud 1995]}.

Theorem 6 (Local-to-global principle). Let $M$ be a finitely generated module over a Noetherian ring $A$. The following are equivalent:

(i) $M=0$.

(ii) $M_{\mathfrak{p}}=0$ for every prime ideal $\mathfrak{p}$ of $A$.

(iii) $M_{\mathfrak{m}}=0$ for every maximal ideal $\mathfrak{m}$ of $A$.

Similarly, a homomorphism $f:M\to N$ of finitely generated modules over a Noetherian ring is injective (resp. surjective, bijective) if and only if $f_{\mathfrak{m}}$ is injective (resp. surjective, bijective) for every maximal ideal $\mathfrak{m}$.

Synthesis. The localisation functor is the foundational reason that local algebra reduces global questions about modules to computations at each prime ideal. The central insight is that exactness of $S^{-1}(-)$ [Theorem 2] makes localisation a flat base change, and this is exactly the mechanism that identifies local data at $\mathfrak{p}$ with the fibre of the structure sheaf on $\mathrm{Spec}A$. Putting these together with the support [Theorem 5] and the local-to-global principle [Theorem 6], the bridge is between the module-theoretic condition $M_{\mathfrak{p}}=0$ for all $\mathfrak{p}$ and the global vanishing $M=0$.

This is exactly the algebraic foundation of scheme theory: the assignment $\mathfrak{p}\mapsto A_{\mathfrak{p}}$ defines the structure sheaf on $\mathrm{Spec}A$, and the pattern generalises from rings to schemes, where quasi-coherent sheaves are locally given by modules and gluing is automatic by the exactness of localisation. The local-to-global principle appears again in 01.02.17, where Noetherianness is detected locally, and the pattern recurs in 01.02.11 where exactness of localisation turns short exact sequences of modules into short exact sequences of stalks.

Full proof set [Master]

Proposition 1 (Exactness of localisation). The functor $S^{-1}(-)$ is exact.

Proof. Let $M^{\prime }\stackrel{f}{\to }M\stackrel{g}{\to }{M}^{\prime \prime }$ be exact at $M$, meaning $\mathrm{im}f=\ker g$. We show $S^{-1}(\mathrm{im}f)=S^{-1}(\ker g)$.

For any submodule $N\subseteq M$, the localisation $S^{-1}N$ is the image of $N$ under the map $M\to S^{-1}M$ sending $m\mapsto m/1$. Concretely, $S^{-1}N=\{n/s:n\in N,s\in S\}$.

Since $f(M^{\prime })\subseteq \ker g$, every element $f(m^{\prime })/s$ maps to $g(f(m^{\prime }))/s=0/s=0$ under $S^{-1}g$. So $S^{-1}(\mathrm{im}f)\subseteq S^{-1}(\ker g)$.

Conversely, let $m/s\in S^{-1}(\ker g)$, so $m\in \ker g$ and $s\in S$. Then $m\in \mathrm{im}f$, say $m=f(m^{\prime })$. So $m/s=f(m^{\prime })/s=(S^{-1}f)(m^{\prime }/s)$. Hence $S^{-1}(\ker g)\subseteq S^{-1}(\mathrm{im}f)$.

Therefore $S^{-1}(\mathrm{im}f)=S^{-1}(\ker g)$, and $S^{-1}{M}^{\prime }\stackrel{S^{-1}f}{\to }{S}^{-1}M\stackrel{S^{-1}g}{\to }{S}^{-1}{M}^{\prime \prime }$ is exact at $S^{-1}M$. $\square$

Proposition 2 (Nakayama's lemma — equivalence of generators formulation). Let $(A,\mathfrak{m})$ be a local ring and $M$ a finitely generated $A$-module. Elements $m_1,\dots ,m_n$ generate $M$ if and only if their images generate $M/\mathfrak{m}M$ over $A/\mathfrak{m}$.

Proof. Let $N=A{m}_1+\cdots +A{m}_n$ be the submodule generated by $m_1,\dots ,m_n$. The quotient $M/N$ satisfies $\mathfrak{m}(M/N)=(\mathfrak{m}M+N)/N$. If the images of $m_1,\dots ,m_n$ generate $M/\mathfrak{m}M$, then $M=\mathfrak{m}M+N$, so $\mathfrak{m}(M/N)=M/N$.

The module $M/N$ is finitely generated (since $M$ is). By Nakayama's lemma [Theorem 4], $\mathfrak{m}(M/N)=M/N$ implies $M/N=0$, hence $M=N$.

Conversely, if $m_1,\dots ,m_n$ generate $M$, their images generate $M/\mathfrak{m}M$. $\square$

Proposition 3 (Prime ideal correspondence). The maps $\phi :\{\text{primes of }{S}^{-1}A\}\to \{\text{primes of }A\text{ disjoint from }S\}$ defined by $\phi (\mathfrak{q})={\iota }^{-1}(\mathfrak{q})$, and $\psi :\{\text{primes of }A\text{ disjoint from }S\}\to \{\text{primes of }{S}^{-1}A\}$ defined by $\psi (\mathfrak{p})=\{a/s:a\in \mathfrak{p},s\in S\}$, are mutually inverse bijections.

Proof. First, $\psi (\mathfrak{p})$ is a prime ideal of $S^{-1}A$. It is an ideal because $(a/s)(b/t)=(ab)/(st)$ and $\mathfrak{p}$ is an ideal. It is proper because $1/1\notin \psi (\mathfrak{p})$: if $1/1=a/s$ with $a\in \mathfrak{p}$, then $t(s-a)=0$ for some $t\in S$, giving $ts=ta\in \mathfrak{p}$; since $t\notin \mathfrak{p}$ (as $\mathfrak{p}\cap S=\varnothing$) and $\mathfrak{p}$ is prime, $s\in \mathfrak{p}$, contradicting $s\in S$. Primality: if $(a/s)(b/t)=(ab)/(st)\in \psi (\mathfrak{p})$, then $u\cdot ab\in \mathfrak{p}$ for some $u\in S$; since $u\notin \mathfrak{p}$ and $\mathfrak{p}$ is prime, $ab\in \mathfrak{p}$, hence $a\in \mathfrak{p}$ or $b\in \mathfrak{p}$.

Now check the composites. For $\phi \circ \psi$: given a prime $\mathfrak{p}$ of $A$ with $\mathfrak{p}\cap S=\varnothing$, $$ \varphi(\psi(\mathfrak{p})) = {a \in A : a/1 \in \psi(\mathfrak{p})} = {a \in A : \exists, s \in S,, a' \in \mathfrak{p} \text{ with } a/1 = a'/s}. $$ The equality $a/1=a^{\prime }/s$ means $t(sa-a^{\prime })=0$ for some $t\in S$, so $tsa=t{a}^{\prime }\in \mathfrak{p}$. Since $ts\notin \mathfrak{p}$, we get $a\in \mathfrak{p}$. Hence $\phi (\psi (\mathfrak{p}))\subseteq \mathfrak{p}$. The reverse inclusion $\mathfrak{p}\subseteq \phi (\psi (\mathfrak{p}))$ is immediate: if $a\in \mathfrak{p}$ then $a/1\in \psi (\mathfrak{p})$ so $a\in \phi (\psi (\mathfrak{p}))$.

For $\psi \circ \phi$: given a prime $\mathfrak{q}$ of $S^{-1}A$, $$ \psi(\varphi(\mathfrak{q})) = {a/s : a \in \iota^{-1}(\mathfrak{q}),, s \in S}. $$ If $a\in {\iota }^{-1}(\mathfrak{q})$ then $a/1\in \mathfrak{q}$, so $a/s=(a/1)(1/s)\in \mathfrak{q}$ (since $1/s$ is a unit). Hence $\psi (\phi (\mathfrak{q}))\subseteq \mathfrak{q}$. Conversely, if $a/s\in \mathfrak{q}$, then $a/1=(a/s)(s/1)\in \mathfrak{q}$, so $a\in {\iota }^{-1}(\mathfrak{q})$, giving $a/s\in \psi (\phi (\mathfrak{q}))$. $\square$

Connections [Master]

• Tensor product of modules 01.02.10. The localisation of a module satisfies $S^{-1}M\cong S^{-1}A{\otimes }_{A}M$, making localisation a special case of extension of scalars along the canonical map $A\to S^{-1}A$. The tensor-hom adjunction then identifies ${\mathrm{Hom}}_{S^{-1}A}(S^{-1}M,S^{-1}N)$ with $S^{-1}{\mathrm{Hom}}_A(M,N)$ when $M$ is finitely presented, linking localisation to the homological algebra developed in the tensor product unit.

• Hilbert basis theorem; Noetherian rings 01.02.17. Localisation preserves Noetherianness: if $A$ is Noetherian then so is $S^{-1}A$, because every ideal of $S^{-1}A$ has the form $I(S^{-1}A)$ for some ideal $I$ of $A$, and the ascending chain condition on ideals of $A$ transfers to $S^{-1}A$. The local-to-global principle for Noetherian rings states that a module is zero if and only if it is zero after localising at every maximal ideal, and this closes the loop between the local and global finiteness conditions.

• Exact sequence, snake lemma 01.02.11. The exactness of $S^{-1}(-)$ means that applying localisation to any short exact sequence of $A$-modules produces a short exact sequence of $S^{-1}A$-modules. This is the ring-theoretic analogue of the flatness property for tensor products: localisation is an exact functor. When applied to the snake lemma diagram, localisation commutes with the connecting homomorphism, so the snake lemma descends to every localisation.

• Field 01.01.01. The field of fractions of an integral domain is the localisation at $S=A\setminus \{0\}$, recovering the construction of $\mathbb{Q}$ from $\mathbb{Z}$ as the simplest example. Every field is already local (the unique maximal ideal is $(0)$), so localisation at a prime ideal generalises the passage from an integral domain to its fraction field by allowing more selective inversion.

Historical & philosophical context [Master]

Grell 1927 introduced the construction of localisation for integral domains in Beziehungen zwischen den Idealen verschiedener Ringe ^{[Grell 1927]}, establishing that every homomorphism from a domain $A$ to a field in which a prescribed set of elements is invertible factors through the ring of fractions. Chevalley 1943--44 extended the construction to arbitrary commutative rings with zero-divisors in On the theory of local rings ^{[Chevalley 1944]}, introducing the modern form of the equivalence relation with the witness element $t\in S$ and proving the prime-ideal correspondence.

Serre 1955 made localisation the cornerstone of algebraic geometry in Faisceaux algebriques coherents ^{[Serre 1955]}, where the assignment $\mathfrak{p}\mapsto A_{\mathfrak{p}}$ defines the structure sheaf on $\mathrm{Spec}A$. The local-to-global principle for coherent sheaves — a module is zero if and only if all its stalks are zero — is the geometric expression of the algebraic local-to-global principle proved above. Nakayama's lemma, proved independently by Nakayama and Azumaya around 1950--51, became the primary tool for studying finitely generated modules over local rings.

Bibliography [Master]

@article{Grell1927,
  author = {Grell, Heinrich},
  title = {Beziehungen zwischen den Idealen verschiedener Ringe},
  journal = {Mathematische Annalen},
  volume = {97},
  year = {1927},
  pages = {490--523},
}

@article{Chevalley1944,
  author = {Chevalley, Claude},
  title = {On the theory of local rings},
  journal = {Annals of Mathematics},
  volume = {44},
  year = {1943},
  pages = {690--708},
}

@article{Serre1955,
  author = {Serre, Jean-Pierre},
  title = {Faisceaux algebriques coherents},
  journal = {Annals of Mathematics},
  volume = {61},
  year = {1955},
  pages = {197--278},
}

@book{AtiyahMacdonald1969,
  author = {Atiyah, Michael F. and Macdonald, Ian G.},
  title = {Introduction to Commutative Algebra},
  publisher = {Addison-Wesley},
  year = {1969},
}

@book{Eisenbud1995,
  author = {Eisenbud, David},
  title = {Commutative Algebra with a View Toward Algebraic Geometry},
  publisher = {Springer},
  year = {1995},
  series = {Graduate Texts in Mathematics 150},
}