Localization and Completion of Modules

Introduction

Two constructions enlarge a ring so that a module over it acquires better behaviour. Localisation inverts a chosen set of elements, producing a ring in which those elements are units; it is exact, it is a flat extension of scalars, and it is the tool that reduces a global statement to the local statements at the prime ideals. Completion passes to an inverse limit of quotients by powers of an ideal; it is the algebraic side of taking the limit of an ever finer filtration, it produces the $p$-adic integers from $\mathbb{Z}$, and over a Noetherian ring it is again flat. This article develops both as operations on modules, records their exactness and flatness properties, and explains how they behave when the base ring is closed further.

Throughout, $R$ is a commutative ring with $1 \neq 0$ and $M$ is an $R$-module. The extension-of-scalars formalism of the companion article of this category is used throughout: localisation is the base change along $R \to S^{-1}R$, and completion is the base change along $R \to \hat R$ for the classes of rings where it is flat.

Localization of Modules

Definition by Fractions

Definition. Let $S \subseteq R$ be a multiplicatively closed subset containing $1$. The localisation of $M$ at $S$ is the $R$-module $S^{-1}M$ whose elements are fractions $m/s$ with $m \in M$, $s \in S$, subject to

$$ \frac{m}{s}=\frac{m'}{s'} \quad \Longleftrightarrow \quad t(s'm-sm')=0 \text{ for some } t \in S , $$

with addition $m/s+m'/s'=(s'm+sm')/(ss')$ and scalar action $r(m/s)=(rm)/s$. The localisation of the ring is $S^{-1}R$, and $S^{-1}M$ is an $S^{-1}R$-module by $(r/s)(m/s')=(rm)/(ss')$.

Proposition. The operations are well defined, $S^{-1}M$ is an $S^{-1}R$-module, and the map $\lambda:M \to S^{-1}M$, $m \mapsto m/1$, is $R$-linear. If $M$ is generated by a set $E$, then $S^{-1}M$ is generated over $S^{-1}R$ by the image of $E$.

Proof. The relation is an equivalence relation because $S$ is multiplicatively closed and contains $1$; the standard verification of well-definedness of addition and of the module axioms is finite in nature. Generation follows from $m/s=(1/s)(m/1)$ and the linearity of $\lambda$ from the definitions.

The Universal Property

Theorem. Let $N$ be an $R$-module such that multiplication by every $s \in S$ is invertible as an endomorphism of $N$. Then every $R$-linear map $f:M \to N$ factors uniquely as $f=\bar f \circ \lambda$ with $\bar f:S^{-1}M \to N$ an $S^{-1}R$-linear map.

Proof. Define $\bar f(m/s)=s^{-1}f(m)$, where $s^{-1}$ denotes the inverse of the endomorphism $s$ on $N$. The definition is independent of the representation: if $m/s=m'/s'$ then $t(s'm-sm')=0$, hence $t(s'f(m)-sf(m'))=0$ and, since $t$ and $s,s'$ are invertible on $N$, $s'f(m)=sf(m')$. Additivity and $S^{-1}R$-linearity are immediate; uniqueness holds because every element of $S^{-1}M$ is of the form $s^{-1}(m/1)$ and $\bar f$ is forced on such elements.

Corollary. $S^{-1}M \cong S^{-1}R \otimes_R M$, and localisation is the extension of scalars along $R \to S^{-1}R$; in particular it is right exact, and it is exact because $S^{-1}R$ is a flat $R$-module.

Proof. The balanced map $S^{-1}R \times M \to S^{-1}M$, $(r/s,m) \mapsto rm/s$, gives a map $S^{-1}R \otimes_R M \to S^{-1}M$, and both the module $S^{-1}M$ and the tensor product have the universal property above. Flatness is the proposition of the companion article on flatness and exactness; exactness then follows.

Exactness

Theorem. Localisation is an exact functor: if $M' \xrightarrow{\ \alpha\ } M \xrightarrow{\ \beta\ } M''$ is exact, then $S^{-1}M' \xrightarrow{\ S^{-1}\alpha\ } S^{-1}M \xrightarrow{\ S^{-1}\beta\ } S^{-1}M''$ is exact. Moreover $S^{-1}$ commutes with finite direct sums, with tensor products and with the formation of kernels and cokernels, and for a finitely presented module $N$ there is a natural isomorphism $S^{-1}\operatorname{Hom}_R(N,M) \cong \operatorname{Hom}_{S^{-1}R}(S^{-1}N,S^{-1}M)$.

Proof. Since $S^{-1}M \cong S^{-1}R \otimes_R M$ and $S^{-1}R$ is flat, the functor is exact; this gives exactness and compatibility with kernels and cokernels, while compatibility with finite direct sums and tensor products holds because these are colimits or because the identification is immediate on fractions. For the Hom statement, present a finitely presented $N$ by $R^p \to R^q \to N \to 0$; applying the exact functors $\operatorname{Hom}(-,M)$ and $S^{-1}$ and using the compatibility with finite limits for finite presentation gives the isomorphism.

Local–Global Principles

Definition. For a prime ideal $\mathrm{P} \subseteq R$, write $M_{\mathrm{P}}=S^{-1}M$ with $S=R \setminus \mathrm{P}$; this is the localisation at $\mathrm{P}$, an $R_{\mathrm{P}}$-module. The support of $M$ is $\operatorname{Supp}M=\{\mathrm{P} : M_{\mathrm{P}} \neq 0\}$.

Example. The localisations of a finite module are computed from its primary decomposition. For $M=\mathbb{Z}/12\mathbb{Z}$ and the prime $\mathrm{P}=(2)$, the multiplicatively closed set is the odd integers; the Chinese remainder theorem writes $M \cong \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$, inverting odd integers leaves $\mathbb{Z}/4\mathbb{Z}$ unchanged because every odd integer is a unit modulo $4$, and kills $\mathbb{Z}/3\mathbb{Z}$ because $3$ is odd and therefore becomes invertible, so $M_{(2)} \cong \mathbb{Z}/4\mathbb{Z}$. The same computation gives $M_{(3)} \cong \mathbb{Z}/3\mathbb{Z}$ and $M_{(5)}=0$, since $12$ annihilates every element of $M$ and $12$ is not a multiple of $5$, so it belongs to the multiplicatively closed set $\mathbb{Z}\setminus(5)$ and kills every fraction. The support is therefore $\operatorname{Supp}M=\{(2),(3)\}$, the primes dividing $12$, and the rank of $M_{\mathrm{P}}$ over $R_{\mathrm{P}}$ is $0$ at every prime, even though $M$ is nonzero: localising detects the support, not the size.

Theorem. (i) $M=0$ if and only if $M_{\mathrm{M}}=0$ for every maximal ideal $\mathrm{M}$. (ii) $M$ is flat over $R$ if and only if $M_{\mathrm{M}}$ is flat over $R_{\mathrm{M}}$ for every maximal ideal $\mathrm{M}$. (iii) If $M$ is finitely generated and $M_{\mathrm{P}}$ is free of rank $n(\mathrm{P})$ over $R_{\mathrm{P}}$ for every prime $\mathrm{P}$, then $n(\mathrm{P})$ is locally constant on $\operatorname{Spec}R$, and $M$ is a projective module if in addition $R$ is Noetherian or $M$ is of finite presentation.

Proof. (i) If $M \neq 0$ choose $x \neq 0$; the annihilator of $x$ is contained in a maximal ideal $\mathrm{M}$, and then $x/1 \neq 0$ in $M_{\mathrm{M}}$ because $t x=0$ would put $t$ in the annihilator, hence in $\mathrm{M}$, contradicting $t \in R \setminus \mathrm{M}$. (ii) is the local character of flatness, proved by testing the ideal criterion at each maximal ideal. (iii) Local freeness of finite rank means that for every prime $\mathrm{P}$ there is an element $f \notin \mathrm{P}$ such that $M_f$ is free of rank $n(\mathrm{P})$ over $R_f$, by Nakayama's lemma; the local trivialisations glue because $M$ is finitely generated and projective is equivalent to being a direct summand of a free module of finite rank.

I-Adic Completion

Definition

Definition. Let $I \subseteq R$ be an ideal. The $I$-adic completion of $M$ is the inverse limit

$$ \widehat M = \varprojlim_n M/I^nM , $$

with the projections $\widehat M \to M/I^nM$; the $I$-adic completion of $R$ is $\widehat R=\varprojlim_n R/I^n$. There is a natural map $\iota:M \to \widehat M$, $m \mapsto (m+I^nM)_n$.

Definition. The chain $M \supseteq IM \supseteq I^2M \supseteq \cdots$ is the $I$-adic filtration of $M$. The module $M$ is $I$-adically separated if $\bigcap_n I^nM=0$, and $I$-adically complete if $\iota$ is an isomorphism, equivalently if every coherent sequence $(m_n)$ with $m_n \in M/I^nM$ and $m_{n+1} \equiv m_n \bmod I^nM$ is the sequence of truncations of a unique element of $M$. No distance and no topology is used: separatedness and completeness are statements about the filtration.

Example. Take $R=\mathbb{Z}$ and $I=(p)$. Then $\varprojlim_n \mathbb{Z}/p^n\mathbb{Z}=\mathbb{Z}_p$, the ring of $p$-adic integers, and the $I$-adic completion of $\mathbb{Z}$ is separated and complete, and the natural map $\mathbb{Z} \to \mathbb{Z}_p$ is injective. Similarly, for $R=k[x]$ and $I=(x)$, the completion is the formal power series ring $k[[x]]=\varprojlim_n k[x]/(x^n)$.

Exactness

Theorem. (i) Let $0 \to M' \xrightarrow{\ \alpha\ } M \xrightarrow{\ \beta\ } M'' \to 0$ be exact and put $K_n=\ker(M'/I^nM' \to M/I^nM)$. Then the completed sequence

$$ 0 \to \widehat{M'} \xrightarrow{\ \widehat\alpha\ } \widehat M \xrightarrow{\ \widehat\beta\ } \widehat{M''} $$

is exact at $\widehat M$ always, and is exact at $\widehat{M'}$ exactly when $\varprojlim_nK_n=0$; the cokernel of $\widehat\beta$ is $\varprojlim^1_nK_n$. In particular completion is left exact on the sequences for which $M'/I^nM' \to M/I^nM$ is injective for every $n$, and it is not exact in general. (ii) If $R$ is Noetherian and $M$ is finitely generated, completion is exact, and $\widehat M \cong \widehat R \otimes_R M$; moreover $\widehat R$ is a flat $R$-module, so completion is a flat base change along $R \to \widehat R$.

Proof. (i) Since $M \mapsto M/I^nM$ is right exact, the sequences

$$ 0 \to K_n \to M'/I^nM' \to M/I^nM \to M''/I^nM'' \to 0 $$

are exact, and the inverse limit is left exact, so $0 \to \varprojlim_nK_n \to \widehat{M'} \to \widehat M \to \widehat{M''}$ is exact; this gives the exactness at $\widehat M$, the criterion at $\widehat{M'}$, and, with the derived long exact sequence of $\varprojlim$ at the following term, the cokernel statement, the systems $M'/I^nM'$ being Mittag-Leffler with vanishing $\varprojlim^1$. (ii) is the Artin–Rees lemma: it gives a stabilisation of the filtrations which makes the passage to the limit exact, and the identification $\widehat M \cong \widehat R \otimes_R M$ follows for finitely generated $M$ once $\widehat R$ is flat; the flatness of $\widehat R$ over a Noetherian $R$ is the standard consequence of Artin–Rees.

Example. Completion is not left exact in general. Take $R=\mathbb{Z}$, $I=(p)$ and the inclusion of $\mathbb{Z}$-modules $\mathbb{Z} \subseteq \mathbb{Z}[\tfrac1p]$. Since $p$ is invertible in $\mathbb{Z}[\tfrac1p]$ one has $I^n\mathbb{Z}[\tfrac1p]=\mathbb{Z}[\tfrac1p]$ for every $n \ge 1$, so the completion of $\mathbb{Z}[\tfrac1p]$ is $0$, while the completion of $\mathbb{Z}$ is $\mathbb{Z}_p$; the completed map is the zero map $\mathbb{Z}_p \to 0$, which is not injective. Here $K_n=\ker(\mathbb{Z}/p^n\mathbb{Z} \to 0)=\mathbb{Z}/p^n\mathbb{Z}$, with $\varprojlim_nK_n=\mathbb{Z}_p \neq 0$ and $\varprojlim^1_nK_n=0$, in accordance with the theorem.

Remark. That the inverse limit is left exact but not right exact is the mechanism of every exactness failure of completion. For a system of exact sequences $0 \to K_n \to A_n \to B_n \to 0$ the limit $0 \to \varprojlim K_n \to \varprojlim A_n \to \varprojlim B_n$ can fail to be exact on the right: for $A_n=\mathbb{Z}$, $K_n=p^n\mathbb{Z}$ and $B_n=\mathbb{Z}/p^n\mathbb{Z}$ the limit is $0 \to 0 \to \mathbb{Z} \to \mathbb{Z}_p$, and $\mathbb{Z} \to \mathbb{Z}_p$ is not surjective. The derived functor $\varprojlim^1$ measures the defect, and the Artin–Rees lemma is exactly the statement that for finitely generated modules over a Noetherian ring the defect vanishes; for arbitrary modules over a ring in which Artin–Rees fails, the same defect can destroy the surjectivity of a completed surjection, so completion of arbitrary modules is not exact in either direction. The details are in the standard accounts of completion and the Artin–Rees lemma cited below.

Flatness and the Local–Global Picture

Theorem (Krull). Let $R$ be Noetherian, $I \subseteq R$ an ideal and $M$ a finitely generated $R$-module. Then $\bigcap_n I^nM$ consists of the elements annihilated by $1+i$ for some $i \in I$. In particular, if $R$ is a Noetherian local ring with maximal ideal $\mathrm{M}$ and $M$ is finitely generated, then $M$ is $\mathrm{M}$-adically separated, so $M \hookrightarrow \widehat M$.

Proof. The statement is Krull's intersection theorem, quoted as standard; its proof uses the Artin–Rees lemma and the Nakayama lemma.

Corollary. Let $R$ be Noetherian and $M$ finitely generated. Then (i) $\widehat M \cong \widehat R \otimes_R M$ and $\widehat R$ is flat over $R$; (ii) completion is exact on finitely generated modules; (iii) $\widehat M$ is a finitely generated $\widehat R$-module, and $\widehat R$ is a Noetherian local ring when $R$ is local; (iv) $\widehat M$ is complete for the $I$-adic filtration.

Proof. All four are standard consequences of Artin–Rees and Krull's theorem, quoted from the commutative algebra literature.

Behaviour under Closure of the Base

Proposition (localisation of a localisation). For multiplicatively closed sets $S \subseteq T \subseteq R$ there is a natural isomorphism $T^{-1}M \cong (T^{-1}R) \otimes_{S^{-1}R} S^{-1}M$. In particular $S^{-1}$ and $T^{-1}$ may be composed, and the composite is a flat base change.

Proof. Both sides satisfy the same universal property: an $R$-linear map from $M$ to a module on which every element of $T$ acts invertibly.

Proposition (localisation of a completion and completion of a localisation). Let $R$ be Noetherian, $I \subseteq R$ an ideal and $S \subseteq R$ multiplicatively closed. Then $S^{-1}\widehat M \cong \widehat{S^{-1}R} \otimes_{S^{-1}R} S^{-1}M$ for finitely generated $M$, where the completion on the right is $S^{-1}I$-adic; completion commutes with localisation in this sense.

Proof. Combine $\widehat M \cong \widehat R \otimes_R M$ with the compatibility of localisation with tensor products and with quotients by powers of $I$.

Proposition (closure properties). The classes of flat and of exact base changes are closed under composition and under base change: if $R \to S$ and $S \to T$ are flat, then $R \to T$ is flat, and if $M$ is flat over $R$ then $S \otimes_R M$ is flat over $S$ for every $R$-algebra $S$.

Proof. The first statement is the identity $T \otimes_R M \cong T \otimes_S (S \otimes_R M)$ and the fact that tensoring flat modules is flat; the second is the associativity of the tensor product.

Primes under Localisation

Definition. For an ideal $I \subseteq R$ write $I^e=S^{-1}R\cdot\lambda(I)$ for the extension of $I$ to $S^{-1}R$, and for an ideal $J \subseteq S^{-1}R$ write $J^c=\lambda^{-1}(J)$ for its contraction to $R$.

Theorem. (i) $\lambda:R \to S^{-1}R$ is a ring homomorphism, every element of $S$ is a unit of $S^{-1}R$, and $S^{-1}R$ is the initial ring with this property.

(ii) Contraction $\mathrm{Q} \mapsto \mathrm{Q}^c$ is a bijection from the prime ideals of $S^{-1}R$ onto the prime ideals $\mathrm{P}$ of $R$ with $\mathrm{P} \cap S=\varnothing$, with inverse $\mathrm{P} \mapsto \mathrm{P}^e=S^{-1}\mathrm{P}$.

(iii) If $\mathrm{P}$ is a prime and $S=R\setminus\mathrm{P}$, then $R_{\mathrm{P}}$ is a local ring with unique maximal ideal $\mathrm{P} R_{\mathrm{P}}$ and residue field $\operatorname{Frac}(R/\mathrm{P})$.

Proof. (i) The operations on fractions make $S^{-1}R$ a ring with unit $1/1$, and $s^{-1}=1/s$ for $s \in S$; any ring homomorphism $f:R \to A$ inverting the elements of $S$ sends $r/s$ to $f(r)f(s)^{-1}$, which is the unique possible value, so $S^{-1}R$ is initial. (ii) The preimage of a prime ideal under a ring homomorphism is prime, and a prime $\mathrm{Q}$ of $S^{-1}R$ meets $S$ in no element, since its elements are units; conversely, if $\mathrm{P}$ is prime with $\mathrm{P} \cap S=\varnothing$ then $S^{-1}(R/\mathrm{P})$ is the fraction field of the domain $R/\mathrm{P}$, so $S^{-1}\mathrm{P}$ is prime. (iii) The primes of $R_{\mathrm{P}}$ correspond by (ii) to the primes of $R$ contained in $\mathrm{P}$, the largest of which is $\mathrm{P}$; hence $\mathrm{P}R_{\mathrm{P}}$ is the unique maximal ideal, and $R_{\mathrm{P}}/\mathrm{P}R_{\mathrm{P}} \cong (R/\mathrm{P})_{(0)}=\operatorname{Frac}(R/\mathrm{P})$.

Example. For $R=\mathbb{Z}$ and $\mathrm{P}=(p)$, the ring $\mathbb{Z}_{(p)}=\{a/b:p \nmid b\}$ is local with maximal ideal $p\mathbb{Z}_{(p)}$ and residue field $\mathbb{F}_p$; localising instead at $S=\mathbb{Z}\setminus\{0\}$ gives $\mathbb{Q}$. If $R$ is a domain with fraction field $K$, then $R_{(0)}=K$.

Summary

Localisation at a multiplicatively closed set $S$ is the extension of scalars along $R \to S^{-1}R$, and $S^{-1}M \cong S^{-1}R \otimes_R M$. It is characterised by the universal property that $R$-linear maps from $M$ to a module on which every $s \in S$ acts invertibly factor uniquely through $M \to S^{-1}M$. It is an exact functor and so is flat base change; it commutes with kernels, cokernels, finite direct sums and tensor products, and with $\operatorname{Hom}$ for finitely presented first arguments. Localising at the maximal ideals detects vanishing and flatness, so $M=0$ if and only if $M_{\mathrm{M}}=0$ for all maximal $\mathrm{M}$, and $M$ is flat if and only if all of its localisations are; finitely generated locally free modules are projective.

The $I$-adic completion is $\widehat M=\varprojlim M/I^nM$, with $\mathbb{Z}_p$ and $k[[x]]$ as the standard examples. Completion is exact for finitely generated modules over a Noetherian ring by the Artin–Rees lemma, so that $\widehat M \cong \widehat R \otimes_R M$ and $\widehat R$ is flat over $R$, making completion a flat base change; on arbitrary modules it is not exact, since the completed version of an inclusion need not be injective and a completed surjection need not be surjective, the obstruction in each case being measured by the derived functors of $\varprojlim$ applied to the kernels $\ker(M'/I^nM' \to M/I^nM)$. Krull's intersection theorem gives the separatedness of finitely generated modules over a Noetherian local ring, so that $M \hookrightarrow \widehat M$, and it makes the completion a Noetherian local ring when $R$ is one. Localisation and completion compose and commute in the sense stated, and the flat base changes are closed under composition and under further base change.

Summary of Notation

Symbol	Meaning
$R$	a commutative ring with $1 \neq 0$
$S \subseteq R$	a multiplicatively closed subset
$S^{-1}M$, $S^{-1}R$	localisation of a module and of the ring
$\lambda:M \to S^{-1}M$, $\lambda:R \to S^{-1}R$	the localisation maps
$M_{\mathrm{P}}$, $R_{\mathrm{P}}$	localisation at a prime ideal $\mathrm{P}$
$\operatorname{Supp}M$	support of a module
$I \subseteq R$	an ideal for the $I$-adic filtration
$\widehat M = \varprojlim M/I^nM$	$I$-adic completion
$\widehat R$	$I$-adic completion of the ring
$\mathbb{Z}_p$	the $p$-adic integers
$k[[x]]$	formal power series ring
$\iota:M \to \widehat M$	the natural completion map
$I^nM$	the $I$-adic filtration of $M$

Localization and Completion of Modules

Introduction

Localization of Modules

Definition by Fractions

The Universal Property

Exactness

Local–Global Principles

I-Adic Completion

Definition

Exactness

Flatness and the Local–Global Picture

Behaviour under Closure of the Base

Primes under Localisation

Summary

Summary of Notation

Further Reading