nLab localisation of a commutative ring away from an element

1. Idea

The localisation of a commutative ring $A$ away from an element $a\in A$ is a universal means to ‘invert $a$’. The resulting ring captures information that is relevant ‘away from $a$’, i.e. ‘locally on the complement of $a$’.

Algebraically, it might make more sense to call this localization at $a$, but in algebraic geometry it really does correspond to “local behavior on the complement of (the zero-set of) $a$”, and the “away from” terminology is traditional.

2. Definition

Let $A$ be a commutative ring, and let $a$ be an element of $A$.

Definition 2.1. The localisation of $A$ away from $a$, usually denoted $A_{a}$ or $A[1/a]$, is the commutative ring $A[x] / (a x - 1)$.

Here (the equivalence class of) $x$ is to be thought of as $a^{-1}$.

Equivalently, we can define $A_{a}$ to be the localisation of $A$ with respect to the multiplicative system $S = { a, a^{2}, a^{3}, \ldots }$. This general notion of localisation is discussed at localisation of a commutative ring.

Example 2.2. If $A$ is the ring of integers $\mathbb{Z}$, and $a$ is $10$, then $A_{a}$ is the ring of decimal rational numbers $\mathbb{Z}[1/10]$.

Example 2.3. If $A$ is the polynomial ring $\mathbb{Z}[x]$, and $a$ is $x$, then $A_{a}$ is the ring of Laurent polynomial $\mathbb{Z}[x, x^{-1}]$.