Localization

Section 10.2 Localization

“The best way to be global is to be local.”
―Alex Atala

Example 10.11. Powers of Element is Multiplicatively Closed.

Let \(R\) be a ring. For any \(f\in R\text{,}\) the set \(W=\{1, f, f^2, f^3, \dots\}\) is a multiplicative set.

Example 10.12. Complement of Prime Ideal is Multiplicative.

Let \(R\) be a ring. If \(P \subseteq R\) is a prime ideal, the set \(W=R \setminus P\) is multiplicative.

Definition 10.13. Regular Element.

Let \(R\) be a ring. An element that is not a zerodivisor is called a nonzerodivisor or a regular element.

Example 10.14. Regular Elements are Multiplicative.

Let \(R\) be a ring. The set of regular elements in \(R\) forms a multiplicatively closed subset.

Example 10.15. Intersection of Multiplicative Sets is Multiplicative.

An arbitrary intersection of multiplicatively closed subsets is multiplicatively closed. In particular, for any family of primes \(\{\fp _{\lambda} \}\text{,}\) the complement of \(\bigcup_{\lambda} \fp _{\lambda}\) is multiplicatively closed.

Definition 10.16. Localization of a Ring.

Let \(R\) be a ring, and \(W\) be a multiplicative set with \(0\notin W\text{.}\) The localization of \(R\) at \(W\) is the ring

\begin{equation*} W^{-1} R := \left\{ \frac{r}{w} \ \Big| \ r\in R, w\in W \right\} / \sim \end{equation*}

where \(\sim\) is the equivalence relation

\begin{equation*} \displaystyle \frac{r}{w}\sim \frac{r'}{w'} \textrm{ if there exists } u\in W : u(rw'-r'w)=0. \end{equation*}

The operations are given by

\begin{equation*} \frac{r}{v}+\frac{s}{w}=\frac{rw+sv}{vw} \quad \text{and} \quad \frac{r}{v}\frac{s}{w}=\frac{rs}{vw}. \end{equation*}

The zero in \(W^{-1}R\) is \(\frac{0}{1}\) and the identity is \(\frac{1}{1}\text{.}\)

EMPTY

Remark 10.17.

Note that we write elements in \(W^{-1}R\) in the form \(\frac{r}{w}\) even though they are equivalence classes of such expressions.

Example 10.18. Field of Fractions is a Localization.

Note that if \(R\) is a domain, the equivalence relation simplifies to \(rw'=r'w\text{,}\) so \(R \subseteq W^{-1}R \subseteq \mathrm{Frac}(R)\text{,}\) and in particular \(W^{-1}R\) is a domain too. In particular, \(\mathrm{Frac}(R)\) is a localization of \(R\text{.}\)

Remark 10.19.

In the localization of \(R\) at \(W\text{,}\) every element of \(W\) becomes a unit. The following universal property says roughly that \(W^{-1}R\) is the smallest \(R\)-algebra in which every element of \(W\) is a unit.

Proposition 10.20. UMP of Localizations.

Let \(R\) be a ring, and \(W\) a multiplicative set with \(0\notin W\text{.}\) Let \(S\) be an \(R\)-algebra in which every element of \(W\) is a unit. Then there is a unique homomorphism \(\alpha\) such that the following diagram commutes:

\begin{equation*} \begin{CD} R@>>>W\inv R\\ @VVV \\ S \end{CD} \end{equation*}

#fix where the vertical map is the structure homomorphism and the horizontal map is the canonical homomorphism.

Example 10.21. Localization of Powers of Element.

For \(f\in R\) and \(W=\{1, f, f^2, f^3, \dots\} = \{ f^n \mid n \geqslant 0 \}\text{,}\) we usually write \(R_f\) for \(W^{-1}R\text{.}\)

Example 10.22. Total Ring of Fractions.

When \(W\) is the set of nonzerodivisors on \(R\text{,}\) we call \(W^{-1}R\) the total ring of fractions of \(R\text{.}\) When \(R\) is a domain, this is just the fraction field of \(R\text{,}\) and in this case this coincides with the localization at the prime \((0)\text{.}\)

Example 10.23. Localization at a Prime.

For a prime ideal \(P\) in \(R\text{,}\) we generally write \(R_{P}\) for \((R\setminus P)^{-1} R\text{,}\) and call it the localization of \(R\) at \(P\). Given an ideal \(I\) in \(R\text{,}\) we sometimes write \(I_P\) to refer to \(I R_P\text{,}\) the image of \(I\) via the canonical map \(R \to R_P\text{.}\) Notice that when we localize at a prime \(P\text{,}\) the resulting ring is a local ring \((R_P,P_{P})\text{.}\) We can think of the process of localization at \(P\) as zooming in at the prime \(P\text{.}\) Many properties of an ideal \(I\) can be checked locally, by checking them for \(I R_P\) for each prime \(P \in V(I)\text{.}\)

Example 10.24. \(k[x_1,\dots,x_d]_{(x_1,\dots,x_d)}\) is Local.

A local ring one often encounters is \(k[x_1,\dots,x_d]_{(x_1,\dots,x_d)}\text{.}\) We can consider this as the ring of rational functions that in lowest terms have a denominator with nonzero constant term. Note that we can talk about lowest terms since the polynomial ring is a UFD.

Example 10.25. Local Ring of the Point.

If \(k\) is algebraically closed and \(I\) is a radical ideal, then \(k[x_1,\dots,x_d]/I= k[X]\) is the coordinate ring of some affine variety, and \((x_1,\dots,x_d)=\fm_{\underline{0}}\) is the ideal defining the origin (as a point in \(X \subseteq \mathbb{A}^d\)). Then we call

\begin{equation*} k[X]_{\fm_{\underline{0}}} := (k[x_1,\dots,x_d]/I)_{(x_1,\dots,x_d)} \end{equation*}

the local ring of a point \(\underline{0} \in X\text{;}\) some people write this as \(\mathcal{O}_{X,\underline{0}}\text{.}\) The radical ideals of this ring consist of radical ideals of \(k[X]\) that are contained in \(\fm_{\underline{0}}\text{,}\) which by the Nullstellensatz correspond to subvarieties of \(X\) that contain \(\underline{0}\text{.}\) Similarly, we can define the local ring at any point \(\underline{a} \in X\text{.}\)

Lemma 10.26. Localization Ideals.

Let \(W\) be multiplicatively closed in \(R\text{.}\)

If \(I\) is an ideal in \(R\text{,}\) then \(W^{-1} I = I W^{-1} R\text{.}\)
If \(I\) is an ideal in \(R\text{,}\) then \(W^{-1} I \cap R = \{ r \in R \ | \ wr\in I \textrm{ for some } w\in W\}\text{.}\)
If \(J\) is an ideal in \(W^{-1}R\text{,}\) then \(W^{-1}(J \cap R) = J\text{.}\)
If \(\fp\) is prime and \(W \cap \fp=\es\text{,}\) then \(W^{-1}\fp =\fp (W^{-1}R)\) is prime.
The map \(\Spec(W^{-1} R) \to \Spec(R)\) is injective, with image

\begin{equation*} \{\fp \in \Spec(R) \ | \ \fp \cap W = \varnothing\}. \end{equation*}

Proof.

Note that

\begin{equation*} W^{-1} I = \left\{ \frac{a}{w} \mid a \in I, w \in W \right\}, \end{equation*}

while \(I W^{-1}\) is the ideal generated by all the elements of the form

\begin{equation*} a \cdot \frac{s}{w} \quad \textrm{where } s \in R, w \in W, a \in I. \end{equation*}

Since we can rewrite

\begin{equation*} a \cdot \frac{s}{w} = \frac{sa}{w} \end{equation*}

and \(sa \in I\text{,}\) we conclude that \(W^{-1} I = I W^{-1} R\text{.}\)
If \(wr \in I\) for some \(w \in W\text{,}\) then

\begin{equation*} \frac{r}{1} = \frac{wr}{w} \in W^{-1} I. \end{equation*}

Conversely, if \(r \in W^{-1} I \cap R\text{,}\) then that means that

\begin{equation*} \frac{r}{1} = \frac{a}{w} \textrm{ for some } a \in I, w \in W. \end{equation*}

By definition of the equivalence relation defining \(W^{-1}R\text{,}\) this means that there exists \(u \in W\) such that

\begin{equation*} u(rw - a \cdot 1) = 0 \Leftrightarrow r(uw) = wa \in I. \end{equation*}

Since \(W\) is multiplicatively closed, \(uw \in W\text{,}\) and thus the element \(t := uw \in W\) satisfies \(tr \in I\text{.}\)
The containment \(W^{-1}(J \cap R) \subseteq J\) holds for general reasons: given any map \(f\text{,}\) and a subset \(J\) of the target of \(f\text{,}\) \(f(f^{-1}(J)) \subseteq J\text{.}\) On the other hand, if \(\frac{a}{w} \in J\text{,}\) then \(\frac{a}{1} \in J\text{,}\) since its a unit multiple of an element of \(J\text{,}\) and thus \(a \in J \cap R\text{,}\) so \(\frac{a}{w} \in W^{-1}(J \cap R)\text{.}\)
First, since \(W\cap P =\varnothing\text{,}\) and \(P\) is prime, no element of \(W\) kills \(\bar{1}=1 + P\) in \(R/P\text{,}\) so \(\bar{1}/1\) is nonzero in \(W^{-1}(R/P)\text{.}\) Thus, \(W^{-1}R / W^{-1}P \cong W^{-1}(R/P)\) is nonzero, and a localization of a domain, hence is a domain. Thus, \(W^{-1}P\) is prime.
First, by part 2), the map \(P \mapsto W^{-1}P\text{,}\) for \(S=\{P \in \Spec(R) \ | \ P \cap W = \varnothing\}\) sends primes to primes. We claim that

\(\Spec(W^{-1} R)\) \(Q\) \(W^{-1}P\)

#empty

\(S\) \(Q \cap R\) \(P\) are inverse maps.

We have already seen that \(J=(J\cap R) W^{-1}R\) for any ideal \(J\) in \(W^{-1}R\text{.}\)

If \(W\cap P=\varnothing\text{,}\) then using part a) and the definition of prime, we have that

\begin{equation*} W^{-1}P \cap R = \lbrace r \in R \, \mid \, rw \in P \textrm{ for some } w \in W \rbrace = \lbrace r \in R \, \mid \, r \in P \rbrace = P. \end{equation*}

Corollary 10.27. Map on Spec Localization Prime.

Let \(R\) be a ring and \(P\) be a prime ideal in \(R\text{.}\) The map on spectra induced by the canonical map \(R \to R_P\) corresponds to the inclusion

\begin{equation*} \{ \fq \in \Spec(R) \, \mid \, \fq \subseteq P \} \subseteq \Spec(R). \end{equation*}

Definition 10.28. Module Localization.

Let \(R\) be a ring, \(W\) be a multiplicative set, and \(M\) an \(R\)-module. The localization of \(M\) at \(W\) is the \(W^{-1}R\)-module

\begin{equation*} W^{-1} M := \left\{ \frac{m} {w} \ \Big| \ m\in M, w\in W \right\} / \sim \end{equation*}

where \(\sim\) is the equivalence relation \(\displaystyle \frac{m}{w}\sim \frac{m'}{w'}\) if \(u(mw'-m'w)=0\) for some \(u\in W\text{.}\) The operations are given by

\begin{equation*} \frac{m}{v}+\frac{n}{w}=\frac{mw+nv}{vw} \quad \text{and} \quad \frac{r}{v} \frac{m}{w}=\frac{rm}{vw}. \end{equation*}

We will use the notations \(M_f\) and \(M_P\) analagously to \(R_f\) and \(R_P\text{.}\)

If \(R\) is not a domain, the canonical map \(R\to W\inv R\) is not necessarily injective.

Example 10.29. \(k[x,y]/(xy)\) has Non Injective Canonical Maps.

Consider \(R = k[x,y]/(xy)\text{.}\) The canonical maps \(R \longrightarrow R_{(x)}\) and \(R \longrightarrow R_y\) are not injective, since in both cases \(y\) is invertible in the localization, and thus

\begin{equation*} x \mapsto \frac{x}{1} = \frac{xy}{y} = \frac{0}{y} = \frac{0}{1}. \end{equation*}

Definition 10.30. Annihilator.

The annihilator of a module \(M\) is the ideal

\begin{equation*} \ann(M) := \lbrace r \in R \mid rm = 0 \textrm{ for all } m \in M \rbrace. \end{equation*}

Definition 10.31. Colon Ideal.

Let \(I\) and \(J\) be ideals in a ring \(R\text{.}\) The colon of \(I\) and \(J\) is the ideal

\begin{equation*} (J : I) := \lbrace r \in R \mid rI \subseteq J \rbrace. \end{equation*}

More generally, if \(M\) and \(N\) are submodules of some \(R\)-module \(A\text{,}\) the colon of \(N\) and \(M\) is

\begin{equation*} (N :_R M) := \lbrace r \in R \mid rM \subseteq N \rbrace. \end{equation*}

Example 10.32. Annihilators and Colons are Ideals.

The annihilator of \(M\) is an ideal in \(R\text{,}\) and

\begin{equation*} \ann (M) = (0 :_R M). \end{equation*}

Moreover, any colon \((N :_R M)\) is an ideal in \(R\text{.}\)

Example 10.33. \(M\cong R/\ann(M)\).

If \(M = R m\) is a cyclic \(R\)-module, then \(M \cong R/I\) for some ideal \(I\text{.}\)

Solution.

Notice that \(I \cdot (R/I) = 0\text{,}\) and that given an element \(g \in R\text{,}\) we have \(g (R/I) = 0\) if and only if \(g \in I\text{.}\) Therefore, \(M \cong R/\ann(M)\text{.}\)

Example 10.34. Ideals Contained in Annihilator.

Let \(M\) be an \(R\)-module. If \(I\) is an ideal in \(R\) such that \(I \subseteq \ann(M)\text{,}\) then \(I M = 0\text{,}\) and thus \(M\) is naturally an \(R/I\)-module with the same structure it has as an \(R\)-module, meaning

\begin{equation*} (r+I) \cdot m = rm \end{equation*}

for each \(r \in R\text{.}\)

Example 10.35. Annihilator of Quotient is Colon.

If \(N \subseteq M\) are \(R\)-modules, then \(\ann(M/N) = (N :_R M)\text{.}\)

Lemma 10.36. Zero in Localization.

Let \(M\) be an \(R\)-module, and \(W\) a multiplicative set. Then the following are equivalent:

\(\displaystyle \frac{m}{w}\in W^{-1}M \ \text{ is zero}\)
\(\displaystyle vm=0 \textrm{ for some } v\in W\)
\(\displaystyle \ann_R(m) \cap W \neq \es.\)

Note in particular that this holds for \(w=1\text{.}\)

Proof.

For the first equivalence, we use the equivalence relation defining \(W^{-1}R\) to note that \(\frac{m}{w}=\frac{0}{1}\) in \(W^{-1}M\) if and only if there exists some \(v\in W\) such that \(0=v(1m-0w)=vm\text{.}\) The second equivalence just comes from the definition of the annihilator.

Example 10.37. Canonical Localization Map and Injectivity.

As a consequence of Lemma 10.36, it follows that if \(R\) is a domain, then the canonical map \(R \to W^{-1}R\) is always injective for any multiplicatively closed set \(W\text{,}\) since every nonzero \(r \in R\) has \(\ann(r) = 0\text{.}\)

Notice, however, that even when \(R\) is a domain, the elements in a module \(M\) may still have nontrivial annihilators, and thus \(M \to W^{-1} M\) may fail to be injective.

Example 10.38. Induced Localization Module Homomorphism.

If \(M\xrightarrow{\alpha} N\) is an \(R\)-module homomorphism, then there is a \(W^{-1}R\)-module homomorphism \(W^{-1}M \xrightarrow{W^{-1}\alpha} W^{-1}N\)given by the rule \(W^{-1}\alpha (\frac{m}{w}) = \frac{\alpha(m)}{w}\text{.}\)

Solution.

Proving this lemma actually requires some homological algebra that we do not have, so for now we will just believe it.

Lemma 10.39. Hom and Localization.

Let \(R\) be a noetherian ring, \(W\) be a multiplicative set, \(M\) be a finitely generated \(R\)-module, and \(N\) an arbitrary \(R\)-module. Then,

\begin{equation*} \Hom_{W^{-1}R}(W^{-1}M , W^{-1}N) \cong W^{-1} \Hom_R(M,N). \end{equation*}

In particular, if \(P\) is prime,

\begin{equation*} \Hom_{R_{P}}(M_{P} , N_{P}) \cong \Hom_R(M,N)_{P}. \end{equation*}

Theorem 10.40. Localization is Exact.

Given a short exact sequence of \(R\)-modules

\begin{equation*} \begin{CD} 0@>>>A@>>>B@>>>C@>>>0 \end{CD} \end{equation*}

and a multiplicative set \(W\text{,}\) the sequence

\begin{equation*} \begin{CD} 0@>>>W\inv A@>f>>W\inv B@>g>>W\inv C@>>>0 \end{CD} \end{equation*}

is also exact.

Example 10.41. Localization Preserves Inclusions.

It follows from this [provisional cross-reference: cite] [[Mathematics/Commutative Algebra/Results/Lemma - Zero in Localization|Lemma]] that if \(\begin{CD}N@>\a>>M\end{CD}\) is injective, then \(W^{-1}\alpha\) is also injective, since

\begin{equation*} 0=W^{-1}\alpha \left( \frac{n}{w} \right) = \frac{\alpha(n)}{w} \ \implies \ 0=u\alpha(n)=\alpha(un) \textrm{ for some } u \in W \implies un=0 \implies \frac{n}{w}=0. \end{equation*}

So this explains some of [provisional cross-reference: cite] [[Mathematics/Commutative Algebra/Results/Theorem - Localization is Exact|Theorem]], since it shows that localization preserves inclusions.

Example 10.42. Localizations and Quotients Commute.

Given a submodule \(N\) of \(M\text{,}\) we can apply the [provisional cross-reference: cite] [[Mathematics/Commutative Algebra/Results/Theorem - Localization is Exact|Theorem]] to the short exact sequence

\begin{equation*} \begin{CD} 0@>>>N@>>>M@>>>M/N@>>>0 \end{CD} \end{equation*}

and conclude that that \(W^{-1}(M/N) \cong W^{-1}M / W^{-1}N\text{.}\)

Lemma 10.43. Localization Commutes with Finite Intersections.

Let \(M\) be a module, and \(N_1,\dots,N_t\) be a finite collection of submodules. Let \(W\) be a multiplicative set. Then,

\begin{equation*} W^{-1} (N_1 \cap \cdots \cap N_t) = W^{-1} N_1 \cap \cdots \cap W^{-1} N_t \subseteq W^{-1}M. \end{equation*}

Proof.

The containment \(W^{-1} (N_1 \cap \cdots \cap N_t) \subseteq W^{-1} N_1 \cap \cdots \cap W^{-1} N_t\) is clear. Elements of \(W^{-1} N_1 \cap \cdots \cap W^{-1} N_t\) are of the form \(\frac{n_1}{w_1}= \cdots = \frac{n_t}{w_t}\text{;}\) we can find a common denominator to realize this in \(W^{-1} (N_1 \cap \cdots \cap N_t)\text{.}\)

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