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localisation of a commutative ring away from an element

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Contents

Idea

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

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

Definition

Let AA be a commutative ring, and let aa be an element of AA.

Definition

The localisation of AA away from aa, usually denoted A aA_{a} or A[1/a]A[1/a], is the commutative ring A[x]/(ax1)A[x] / (a x - 1).

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

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

Example

If AA is the polynomial ring [x]\mathbb{Z}[x], and aa is xx, then A aA_{a} is the ring of Laurent polynomial [x,x 1]\mathbb{Z}[x, x^{-1}].

Last revised on July 15, 2019 at 15:59:41. See the history of this page for a list of all contributions to it.