Manifolds in Ink: The Craft of Localization

Prologue~
“A man sets himself the task of drawing the world.”
— Jorge Luis Borges

In this post, we discuss the concept of localization, a term which originated in algebraic geometry. For a discussion on the term `localization', we refer to the following passage which comes from Eisenbud's "Commutative Algebra with a View Towards Algebraic Geometry."

... localization as a general procedure was defined rather late: In the case of integral domains it was described by Grell, a student of Noether, in 1927, and it was not defined for arbitrary commutative rings until the work of Chevalley [1944] and Uzkov [1948], long after the basic ideas of commutative algebra were established. Perhaps this is because interest was focused on finitely generated algebras on the one hand, and power series rings on the other, and neither of these classes of rings are closed under localization. Instead of passing to a localized rings, as we would now, people often used ideal quotients as a substitute.

Verse I: Surveying the Terrain~
Localization as a philosophy refers to emphasizing on a specific part of a bigger object. So we essentially look closer on a smaller area of it while we tend to, for the time being, ignore the rest. We will discuss the algebraic view of localization, specifically in the context of rings.
Localization of a commutative ring by a multiplicative set is a new ring whose elements are fractions with numerators coming from the ring and the denominators coming from the multipicative set (well, there's a bit more to it).
We begin by defining a multiplicative set, also known as a multiplicatively closed set, say $S$, which, as the name suggests, consists of elements of the ring, say $R$, such that $S$ forms a subset of $R$ that is closed under multipication and contains the multiplicative identity. The set $S^{-1} R$ will be the resulting localization of $R$ at $S$.

Verse II: Drawing the Map~
More precisely, a subset $S \subseteq R$ is called a multiplicative set if the following conditions hold:

$1 \in S$,
$0 \notin S$,
for all $a,b \in S$, we have $ab \in S$ (multiplicatively closed).

Then, let $R$ be a commutative ring with unity and let $S \subseteq R$ be a multiplicative set. The localization of $R$ at $S$ is the ring $S^{-1}R$ consisting of equivalence classes of formal fractions $$ \frac{r}{s}, \quad r\in R,\ s\in S, \hspace{0.2in} \text{ where } \hspace{0.2in}\frac{r}{s} \sim \frac{r'}{s'} $$ if there exists $u\in S$ such that $u(s'r-r's)=0$.
Addition and multiplication are defined by $$\frac{r}{s}+\frac{r'}{s'} = \frac{s' r+r's}{ss'} \hspace{0.2in}\text{ and } \hspace{0.2in} \frac{r}{s}\cdot\frac{r'}{s'}=\frac{rr'}{ss'}$$ There is a natural ring homomorphism $$ \varphi:R\to S^{-1}R \hspace{0.2in} \text{given by} \hspace{0.2in} \varphi(r)=\frac{r}{1}$$

Verse III: A different view~

Another way to look at localization is by defining it via the universal property (i.e., in the categorical terms).

Let $R$ be a commutative ring and $S \subseteq R$ a multiplicative subset. The localization $S^{-1}R$ is characterized categorically by a universal property in the category of commutative rings.
The localization consists of:

A ring $S^{-1}R$
A canonical homomorphism $\iota_S \colon R \to S^{-1}R$

satisfying:

Inversion: For every $s \in S$, $\iota_S(s)$ is a unit in $S^{-1}R$.
Universality: For any ring homomorphism $f \colon R \to T$ such that $f(s)$ is a unit in $T$ for all $s \in S$, there exists a unique homomorphism $\tilde{f} \colon S^{-1}R \to T$ making the diagram commute:

This universal property says that $S^{-1}R$ is the initial object in the comma category $(R \downarrow \mathcal{U}_S)$, where $\mathcal{U}_S$ is the full subcategory of the category of commutative rings (say $\mathbf{CRings}$), modulo $R$, i.e., $\mathbf{CRings}/ R$ consisting of morphisms $R \to T$ that invert $S$.

Verse IV: Another (different) view~

Equivalently, localization is the left adjoint to the inclusion functor:

$$L_S \colon \mathbf{CRings} \longrightarrow \mathbf{CRings}_S \quad \text{and} \hspace{0.2in} R \mapsto S^{-1}R$$

where $\mathbf{CRings}_S$ is the category of commutative rings equipped with a homomorphism from $R$ that inverts $S$. The adjunction is: $$ \operatorname{Hom}_{\mathbf{CRings}}(S^{-1}R, T) \cong \operatorname{Hom}_{\mathbf{CRings},\,S^{-1}}(R, T)$$ where the RHS denotes homomorphisms inverting $S$.

Any two objects satisfying the universal property are uniquely isomorphic via a unique isomorphism commuting with the structure maps.
Localization is functorial in both $R$ and $S$. A ring homomorphism $\varphi \colon R \to R'$ with $\varphi(S) \subseteq S'$ induces $\varphi_S \colon S^{-1}R \to (S')^{-1}R'$.

If $S \subseteq T \subseteq R$, then $(T^{-1}R) \cong T^{-1}(S^{-1}R)$. This follows from the universal property by checking that both sides invert $T$.

As a set: $S^{-1}R = \left\{ \frac{r}{s} \mid r \in R, \, s \in S \right\} / \sim $ where $\frac{r}{s} \sim \frac{r'}{s'}$ iff $\exists \ t \in S$ such that $t(rs' - r's) = 0$. Operations are:

$$\frac{r}{s} + \frac{r'}{s'} = \frac{rs' + r's}{ss'}, \quad \frac{r}{s} \cdot \frac{r'}{s'} = \frac{rr'}{ss'}$$

The map $\iota_S(r) = \frac{r}{1}$ inverts $S$ since $\iota_S(s) \cdot \frac{1}{s} = 1$.

Verse V: Toward Ore Localization~

In $\mathbf{CRings}$, every multiplicative subset $S \subseteq R$ admits a localization $S^{-1}R$ because commutativity ensures the fraction calculus works smoothly.
For noncommutative rings, arbitrary multiplicative subsets $S$ may not admit a well-behaved localization. The Ore condition is the necessary and sufficient condition for constructing a ring of fractions $S^{-1}R$ (or $RS^{-1}$) when $R$ is noncommutative. We will be discussing this in depth in the upcomign post.

Epilogue~
“A man sets himself the task of drawing the world. As the years pass, he populates a space with images of provinces, kingdoms, mountains, bays, ships, islands, fishes, rooms, instruments, stars, horses, and people. Shortly before he dies, he discovers that that patient labyrinth of lines traces the image of his own face.”
— Jorge Luis Borges

Pages

The Craft of Localization