symmetric monoidal (∞,1)-category of spectra
The localization of a commutative ring $R$ at a set $S$ of its elements is a new ring $R[S]^{-1}$ in which the elements of $S$ become invertible (units) and which is universal with this property.
When interpreting a ring under Isbell duality as the ring of functions on some space $X$ (its spectrum), then localizing it at $S$ corresponds to restricting to the complement of the subspace $Y \hookrightarrow X$ on which the elements in $S$ vanish.
See also commutative localization and localization of a ring (noncommutative).
Let $R$ be a commutative ring. Let $S \hookrightarrow U(R)$ be a multiplicative subset of the underlying set.
The following gives the universal property of the localization.
The localization $L_S \colon R \to R[S^{-1}]$ is a homomorphism to another commutative ring $R[S^{-1}]$ such that
for all elements $s \in S \hookrightarrow R$ the image $L_S(s) \in R[S^{-1}]$ is invertible (is a unit);
for every other homomorphism $R \to \tilde R$ with this property, there is a unique homomorphism $R[S^{-1}] \to \tilde R$ such that we have a commuting diagram
The special case of inverting an element $r$ of $R$, in which $S$ is the set $\{ r, r^{2}, r^{3}, \ldots \}$, is discussed at localisation of a commutative ring at an element. See also for example Sullivan 70, first pages.
The formal duals $Spec(R[S^{-1}]) \longrightarrow Spec(R)$ of the localization maps $R \longrightarrow R[S^{-1}]$ (under forming spectra) serve as the standard open immersions that define the Zariski topology on algebraic varieties.
(…) By the lifting property of etale morphisms for $E_\infty$-rings, see here. (…)
A classical set of lecture notes is
Other accounts of the basics include
Andreas Gathmann, Localization (pdf)
The Stacks Project, 10.9. Localization
Yoshifumi Tsuchimoto, Review of commutative (usual) affine schemes. Localization of a commutative ring
Last revised on December 9, 2020 at 02:02:24. See the history of this page for a list of all contributions to it.