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).
Localization “at” and “away from”
The common terminology in algebra is as follows.
For $S$ a set of primes, “localize at $S$” means “invert what is not divisible by $S$”; so for $p$ prime, localizing “at $p$” means considering only $p$-torsion.
Adjoining inverses $[S^{-1}]$ is pronounced “localized away from $S$”. Inverting a prime $p$ is localizing away from $p$, which means ignoring $p$-torsion.
See also lecture notes such as (Gathmann) and see at localization of a space for more discussion of this.
Evidently, this conflicts with more-categorial uses of “localized”; “inverting weak equivalences” is called localization, by obvious analogy, and is written as “localizing at weak equivalences”. This is confusing! It’s also weird: since a ring is a one-object Ab-enriched category with morphisms “multiply-by”, the localization-of-the-category $R$ “at $p$” (or its Ab-enriched version, if saying that is necessary) really means the localization-of-the-ring R “away from p”.
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.
Explicitly:
The localization of a commutative ring $R$ at a multiplicative subset $S$ is the commutative ring whose underlying set is the set of equivalence classes on $R \times S$ under the equivalence relation
Write $r s^{-1}$ for the equivalence class of $(r,s)$. On this set, addition and multiplication is defined by
(e.g. Stacks Project, def. 10.9.1)
The above definitions also work for non-commutative rings $R$ as well, so long as the multiplicative subset $S$ is a submonoid of the center $Z(R)$ of the multiplicative monoid of $R$.
(…) By the lifting property of etale morphisms for $E_\infty$-rings, see here. (…)
Localization away from a suitably tame ideal may be understood as the dR-shape modality in the cohesion of E-infinity arithmetic geometry:
cohesion in E-∞ arithmetic geometry:
cohesion modality | symbol | interpretation |
---|---|---|
flat modality | $\flat$ | formal completion at |
shape modality | $ʃ$ | torsion approximation |
dR-shape modality | $ʃ_{dR}$ | localization away |
dR-flat modality | $\flat_{dR}$ | adic residual |
the differential cohomology hexagon/arithmetic fracture squares:
A classical set of lecture notes:
Concretely on localization of commutative rings:
by Andrew Ranicki (pdf)
Discussion in constructive mathematics:
Further review:
Andreas Gathmann, Localization (pdf)
Joseph NeisendorferA Quick Trip through Localization (pdf)
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 July 29, 2023 at 19:11:03. See the history of this page for a list of all contributions to it.