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Rings, Modules, Fields

3 Localization and local rings

If \(R\) is a ring and \(X\) a set, \(R[X]\) is the set of polynomials with coefficients in \(R\) in the variables \(X\). This is the left adjoint to the forgetful functor from \(R\)-algebras to \(\Set \). Similarly we can construct the ring \(R[[X]]\) of formal power series. We’d like to say that \(R[X]\) is a UFR if \(R\) is, but to prove this we will need a simple case of a powerful technique called Localization.

Localized rings are in a sense dual to quotient rings, and both are examples of epis in the category of rings. Localization is essentially the universal process of formally adding inverses.

Given a multiplicative monoid \(M \subset R\), we can form the localization of \(R\) away from \(M\), denoted \(M^{-1}R\) as follows: The underlying set of \(M^{-1}R\) is \(R\times M\) (we can write \((r,m)\) as \(\frac r m\)) modded out by the equivalence relation \(\frac {r_1} {r_1} ~ \frac {r_2} {m_2}\) iff there is \(f \in M\) such that \(f(r_1m_2-r_2m_1) = 0\). Now it is easy to check this an equivalence relation, and we make \(M^{-1}R\) into a ring via \(\frac a b+\frac c d = \frac {ad+bc}{bd}\) and \(\frac a b \frac c d = \frac {ac}{bd}\). It is easy to check that these operations are well defined. Moreover there is a natural map \(R \to M^{-1}R\) sending \(r \mapsto \frac r 1\).

The localization is \(0\) iff \(1 =0\) in it iff \(0 \in M\). The map \(R \to M^{-1}R\) is injective iff \(M \subset R_z\) since \(\frac {a}{1} = \frac 0 1\) iff \(af = 0\) for some \(f \in M\), and \(f \in R_z \implies a = 0\).

The localized ring has the universal property any map sending \(M\) to units factors through \(M^{-1}R\). There are a few special instances of localization that deserve remark. First note that given a set \(S \subset R\), we can make sense of \(S^{-1}R\) as the localization away from the monoid generated by \(S\). If \(S\) has one element \(a\), we write the ring \(R_a\), the localization away from \(a\). If \(I\) is an ideal \(R-I\) is a monoid iff \(I\) is prime. Thus if \(\pp \) is a prime ideal, \(R_\pp = (R-\pp )^{-1}R\) is called the localization of \(R\) at \(\pp \). Finally if \(R\) is an integral domain, \(\Frac (R) = R_{(0)}\) is called the field of fractions. Indeed it is a field by the universal property, and since subrings of fields are integral domains, we see that integral domains are exactly those rings that are subrings of fields. For example, \(\Frac (\ZZ ) = \QQ \), \(\Frac (K[x]) = K(x)\), the field of rational functions (\(K\) is a field).

One can also localize modules. If \(N\) is an \(R\)-module, then \(M^{-1}N\) is the universal way of making \(N\) into a \(M^{-1}R\) module. Its underlying set is \(N \times M\) (once again written as a fraction) modded out by the equivalence relation \(\frac a b ~ \frac {c}{d}\) iff there is \(f \in M\) such that \(f(da-bc) = 0\). Addition is given by \(\frac a b + \frac c d = \frac {da+bc}{bd}\), and the action of \(M^{-1}R\) is given by \(\frac r m\frac a b = \frac {ra} {mb}\). There is a universal homomorphism \(N \to M^{-1}N\) given by \(n \mapsto \frac n 1\).

There is a correspondence of ideals for localized rings that is dual to quotients.

  • Definition 3.1. An ideal \(I\) in \(R\) is \(M\)-saturated if \(mr \in I\) implies \(r \in I\) for \(m \in M\).

  • Proposition 3.2. There is a natural isomorphism of lattices between ideals in \(M^{-1}R\) and \(M\)-saturated ideals in \(R\), sending an ideal in \(M^{-1}R\) to its preimage in \(R\). This isomorphism also gives a bijection between primes in \(M^{-1}R\) and primes in \(R\) not intersecting \(M\).

  • Proof. It is straightforward to verify that this isomorphism is as mentioned, and that primes correspond as mentioned which is easy to see since the preimage of primes are primes and primes not intersecting \(M\) are clearly \(M\)-saturated. □

This correspondence is the sense in which localization is dual to quotienting.

The name localization comes from algebraic geometry. We’d like to think of a ring as the ring of functions on some space. For example consider the ring of entire functions on \(\PP ^1\), \(\CC [x]\). We can now allow functions that are possibly meromorphic at the origin, giving \(\CC [x]_{x}\), so we have localized away from \(x\) (which corresponds to its zero set, the point \(0\)). Similarly, we can allow functions to be meromorphic everywhere except at the origin, in which case we get \(\CC [x]_{(x)}\), and we are localizing at the origin.

Now if we localize at a prime ideal \(\pp \), we get a local ring, one with a unique maximal ideal (in this case \(\pp \)).

  • Lemma 3.3. TFAE:

    • 1. \(R\) is a local ring with maximal ideal \(\mm \)

    • 2. Every element not in \(\mm \) is a unit

    • 3. \(R\) is nontrivial and the non-units form an additive subgroup

    • 4. \(R\) is nontrivial and either \(x\) or \(1-x\) is a unit for all \(x\).

    • 5. If a finite sum in \(R\) is a unit, one of the terms is a unit

  • Proof. It is clear that \((2)\) and \((1)\) are equivalent as well as \((3)\) and \((5)\) (for \((1) \implies (2)\) use existence of maximal ideals). Now \((2)\) implies \((4)\) as otherwise \(1 \in \mm \), and \((4)\) implies \((5)\) by taking a finite sum that is a unit, multiplying both sides by the inverse, and applying \((4)\) repeatedly. Finally \((3)\) implies \((2)\) by noting that the set of non units is an ideal. □

Another example of a local ring is the formal power series \(K[[x]]\), where \(K\) is a field, where the maximal ideal is \((x)\).

Add in things about “local properties" for example a module is 0 iff localization at all prime/maximals is 0. An integral domain is the intersection of its localizations. A