group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
Bousfield localization of spectra refers generally to localizations of the stable (∞,1)-category of spectra (hence Bousfield localization of model categories of spectra) at the collection of morphisms which become equivalences under smash product with a given spectrum $E$. Since any such $E$ represents a generalized homology theory, this may also be thought of $E$-homology localization.
More specifically, if the stable (∞,1)-category of spectra is presented by a (stable) model category, then the ∞-categorical localization can be presented by the operation of Bousfield localization of model categories. The original article (Bousfield 79) essentially considered localization at the level of homotopy categories.
Specifically, for $E \in Spec$ a spectrum, the Bousfield localization at $E$ of another spectrum $X$ is the universal map
to the $E$-local spectrum $L_E X$, with the property that for $Y$ any $E$-acyclic spectrum in that $Y \wedge E \simeq 0$, every morphism $Y \longrightarrow L_E X$ is null-homotopic (a zero morphism in the stable (∞,1)-category of spectra). (see for instance Lurie 10, Example 4)
For $E = M \mathbb{Z}_p$ the Moore spectrum of the cyclic group $\mathbb{Z}_p \coloneqq \mathbb{Z}/p\mathbb{Z}$ for some prime number $p$, this $E$-localization is p-completion. (see for instance Lurie 10, Examples 7 and 8)
We write $Ho(Spectra)$ for the stable homotopy category. For $X,Y \in Ho(Spectra)$ two spectra we write $[X,Y] \in$ Ab for its hom-abelian groups, and $[X,Y]_\bullet \coloneqq [\Sigma^\bullet X,Y]$ for the corresponding graded abelian group (def.).
Let $E \in Ho(Spectra)$ be a spectrum. Say that
a spectrum $X$ is $E$-acyclic if the smash product with $E$ is zero, $E \wedge X \simeq 0$;
a morphism $f \colon X \to Y$ of spectra is an $E$-equivalence if $E \wedge f \;\colon\; E \wedge X \to E \wedge Y$ is an isomorphism in $Ho(Spectra)$, hence if $E_\bullet(f)$ is an isomorphism in $E$-generalized homology;
a spectrum $X$ is $E$-local if the following equivalent conditions hold
(Bousfield 79, §1) see also for instance (Lurie, Lecture 20, example 4)
Notice that $A \in Ho(Spectra)$ being $E$-acyclic means equivalently that the unique morphism $0 \longrightarrow A$ is an $E$-equivalence.
Hence one direction of the claim is trivial. For the other direction we need to show that for $[-,X]_\bullet$ to give an isomorphism on all $E$-equivalences $f$, it is sufficient that it gives an isomorphism on all $E$-equivalences of the form $0 \to A$.
Given a morphism $f \colon A \to B$, write $B \longrightarrow B/A$ for its homotopy cofiber. Then since $Ho(Spectra)$ is a triangulated category (prop.) the defining axioms of triangulated categories (def., lemma) give that there is a commuting diagram of the form
where both the top as well as the bottom are homotopy cofiber sequences. Hence applying $[-,X]_\bullet$ to this diagram in $Ho(Spectra)$ yields a diagram of graded abelian groups of the form
where now both horizontal sequences are long exact sequences (prop.).
Hence if $[B/A,X]_\bullet \longrightarrow 0$ is an isomorphism, then all four outer vertical morphisms in this diagram are isomorphisms, and then the five-lemma implies that also $[f,X]_\bullet$ is an isomorphism.
Hence it is now sufficient to observe that with $f \colon A \to B$ an $E$-equivalence, then its homotopy cofiber $B/A$ is $E$-acyclic.
To see this, notice that by the tensor triangulated structure on $Ho(Spectra)$ (prop.) the smash product with $E$ preserves homotopy cofiber sequences, so that there is a homotopy cofiber sequence
But if the first morphism here is an isomorphism, then the axioms of a triangulated category (def.) imply that $E \wedge B / A \simeq 0$. In detail: by the axioms we may form the morphism of homotopy cofiber sequences
Then since two of the three vertical morphisms on the left are isomorphisms, so is the third (lemma).
Given $E,X \in Ho(Spectra)$, then an $E$-localization of $X$ is
an $E$-local spectrum $L_E X$
an $E$-equivalence $X \longrightarrow L_E X$.
We discuss now that $E$-Localizations always exist. The key to this is the following lemma , which asserts that a spectrum being $E$-local is equivalent to it being $A$-null, for some “small” spectrum $A$:
For every spectrum $E$ there exists a spectrum $A$ such that any spectrum $X$ is $E$-local (def. ) precisely if it is $A$-null, i.e.
and such that
there exists an infinite cardinal number $\kappa$ such that $A$ is a $\kappa$-CW spectrum (hence a CW spectrum (def.) with at most $\kappa$ many cells);
the class of $E$-acyclic spectra (def. ) is the class generated by $A$ under
the relation that if in a homotopy cofiber sequence $X_1 \to X_2 \to X_3$ two of the spectra are in the class, then so is the third.
(Bousfield 79, lemma 1.13 with lemma 1.14) review includes (Bauer 11, p.2,3, VanKoughnett 13, p. 8)
For $E \in Ho(Spectra)$ any spectrum, every spectrum $X$ sits in a homotopy cofiber sequence of the form
and natural in $X$, such that
$G_E(X)$ is $E$-acyclic,
$L_E(X)$ is $E$-local,
(Bousfield 79, theorem 1.1) see also for instance (Lurie, Lecture 20, example 4)
Consider the $\kappa$-CW-spectrum spectrum $A$ whose existence is asserted by lemma . Let
denote the set containing as its single element the canonical morphism (of sequential spectra) from $A$ into the standard cone of $A$, i.e. the cofiber
of the inclusion of $A$ into its standard cylinder spectrum (def.).
Since the standard cylinder spectrum on a CW-spectrum is a good cylinder object (prop.) this means (lemma) that for $X$ any fibrant sequential spectrum, and for $A \longrightarrow X$ any morphism, then an extension along the cone inclusion
equivalently exhibits a null-homotopy of the top morphism. Hence the $(A \to Cone(A))$-injective objects in $Ho(Spectra)$ are precisely those spectra $X$ for which $[A,X]_\bullet \simeq 0$.
Moreover, due to the degreewise nature of the smash tensoring $Cone(A) = A \wedge I$ (def), the inclusion morphism $A \to Cone(A)$ is degreewise the inclusion of a CW-complex into its standard cone, which is a relative cell complex inclusion (prop.).
By this lemma the $\kappa$-cell spectrum $A$ is $\kappa$-small object (def.) with respect to morphisms of spectra which are degreewise relative cell complex inclusion small object argument .
Hence the small object argument applies (prop.) and gives for every $X$ a factorization of the terminal morphism $X \to \ast$ as an $I_A$-relative cell complex (def.) followed by an $I_A$-injective morphism (def.)
By the above, this means that $[A, L_E X] = 0$, hence by lemma that $L_E X$ is $E$-local.
It remains to see that the homotopy fiber of $X \to L_E X$ is $E$-acyclic: By the tensor triangulated structure on $Ho(Spectra)$ (prop.) it is sufficient to show that the homotopy cofiber is $E$-acyclic (since it differs from the homotopy fiber only by suspension). By the pasting law, the homotopy cofiber of a transfinite composition is the transfinite composition of a sequence of homotopy pushouts. By lemma and applying the pasting law again, all these homotopy pushouts produce $E$-acyclic objects. Hence we conclude by observing that the the transfinite composition of the morphisms between these $E$-acyclic objects is $E$-acyclic. Since by construction all these morphisms are relative cell complex inclusions, this follows again with the compactness of the $n$-spheres (lemma).
It only remains to show that $X \to L_E X$ is an $E$-equivalence. By the tensor triangulated structure on $Ho(Spectra)$ (prop.) the smash product with $E$ preserves homotopy cofiber sequences, so that
is also a homotopy cofiber sequence. But now $E \wedge G_E X \simeq 0$ by prop. , and so the axioms (def.) of the triangulated structure on $Ho(Spectra)$ (prop.) imply that $E \wedge \eta$ is an isomorphism.
Hence where $L_E$ is traditionally called “$E$-localization”, $G_E$ might be called “$E$-acyclification”, though that terminology is not used commonly.
A basic special case of $E$-localization of spectra is given for $E = S A$ the Moore spectrum of an abelian group $A$ (Bousfield 79, section 2). For $A = \mathbb{Z}_{(p)}$ this is p-localization and for $A = \mathbb{F}_p$ this is p-completion, see examples and below for more.
For $A_1$ and $A_2$ two abelian groups then the following are equivalent
the Bousfield localizations at their Moore spectra are equivalent
$A_1$ and $A_2$ have the same type of acyclicity, meaning that
every prime number $p$ is invertible in $A_1$ precisely if it is in $A_2$;
$A_1$ is a torsion group precisely if $A_2$ is.
(Bousfield 79, prop. 2.3) recalled e.g. in (VanKoughnett 13, prop. 4.2).
This means that given an abelian group $A$ then
either $A$ is not torsion, then
where $I$ is the set of primes invertible in $A$ and $\mathbb{Z}[I^{-1}] \hookrightarrow \mathbb{Q}$ is the localization at these primes of the integers;
or $A$ is torsion, then
where the direct sum is over all cyclic groups of order $q$, for $q$ a prime not invertible in $A$.
Let $E$ be an E-∞ ring and let $X$ be any spectrum
There is a canonical map
from the $E$-Bousfield localization of spectra of $X$ into the totalization of the canonical cosimplicial spectrum (see at nilpotent completion).
We now consider conditions for this morphism to be an equivalence.
For $R$ a ring, its core $c R$ is the equalizer in
Let $E$ be a connective E-∞ ring such that the core of $\pi_0(E)$, def. , is either of
the localization of the integers at a set $J$ of primes, $c \pi_0(E) \simeq \mathbb{Z}[J^{-1}]$;
$\mathbb{Z}_n$ for $n \geq 2$.
Then the map in remark is an equivalence
(Bousfield 79) see also for instance (Bauer 11, p.2)
For more discussion of E-infinity (derived) formal completions via totalizations of Amitsur complexes, see (Carlsson 07).
The fracture theorem says how Bousfield localization at a coproduct/wedge sum of spectra is a homotopy pullback of Bousfield localization separately. See at fracture theorem for more on this.
(e.g. Lurie, Lecture 20, example 5)
For $E$ an E-∞ algebra over an E-∞ ring $S$ and for $X$ an $S$-∞-module, consider the dual Cech nerve cosimplicial object
By example each term is $E$-local, so that the map to the totalization
factors through the $E$-localization of $X$
Under suitable condition the second map here is indeed an equivalence, in which case the totalization of the dual Cech nerve exhibits the $E$-localization. This happens for instance in the discussion of the Adams spectral sequence, see the examples given there.
(see also e.g. Bauer 11, p. 2)
Bousfield localization at the Moore spectrum/Eilenberg-MacLane spectrum $S \mathbb{Q}\simeq H\mathbb{Q}$ of the rational numbers is rationalization to rational homotopy theory.
The corresponding $\mathbb{Q}$-acyclification (remark ) is torsion approximation.
e.g. (Bauer 11, example 1.7)
For $p$ a prime number write $\mathbb{Z}_{(p)}$ for the localization of the integers at $(p)$, for the ring of integers localized at $p$, hence with all primes except $p$ inverted; equivalently the subring of the rational numbers with denominator not divisible by $p$.
The Bousfield localization at the Moore spectrum $S \mathbb{Z}_{(p)}$ is p-localization.
(Bousfield 79), Bauer 11, example 1.7). See at localization of a space for details on this.
$p$-localization is a smashing localization:
(Bousfield 79, prop. 2.4) recalled e.g. as (van Koughnett 13, prop. 4.3).
For $p \in \mathbb{N}$ a prime number, write
for the cyclic group/finite field of order $p$.
Write
for the localization of the integers away from $p$ followed by the quotient by $\mathbb{Z}$.
e.g. (Bousfield 79, p. 6)
The short exact sequence of abelian groups
induces the homotopy fiber sequence (in spectra) of Moore spectra
As in Bousfield 79, p. 6 one also writes
The localization of spectra at the Moore spectrum $S\mathbb{F}_p$ is given by the mapping spectrum out of $S^{-1} \mathbb{Z}/p^\infty$:
Fact: $\mathbb{F}_p$-localizaton is p-completion, e.g. Lurie “Proper Morphisms…”, section 4.
Let
be the corresponding Eilenberg-MacLane spectrum. Then a spectrum which corresponds to a chain complex under the stable Dold-Kan corespondence is $E$-local, def. , if that chain complex has chain homology groups being $\mathbb{Z}[p^{-1}]$-modules.
The $E$-localization of a spectrum in this case is p-completion.
(e.g. Lurie, Lecture 20, example 8)
More generally
Bousfield localization at the Moore spectrum $S \mathbb{F}_p$ is p-completion to p-adic homotopy theory.
E.g. (Bauer 11, example 1.7). See at localization of a space for more on this.
Original articles are
Aldridge Bousfield, The localization of spectra with respect to homology, Topology Volume 18 Issue 4 (1979) (doi:10.1016/0040-9383(79)90018-1, pdf)
Douglas Ravenel, Localization with respect to certain periodic homology theories, American Journal of Mathematics, Vol. 106, No. 2, (Apr., 1984), pp. 351-414 (jstor:2374308, pdf)
Discussion in terms of Bousfield localization of model categories of spectra appears in
Anthony Elmendorf, Igor Kriz, Michael Mandell, Peter May, chapter VIII of Rings, modules and algebras in stable homotopy theory 1997 (pdf)
Stefan Schwede, Brooke Shipley, lemma 4.1 in A uniqueness theorem for stable homotopy theory, Mathematische Zeitschrift 239 (2002), 803-828 (arXiv:math/0012021)
see also
Vol. 357, No. 7 (Jul., 2005), pp. 2753-2770 (jstor)
Review:
Nerses Aramian, Bousfield Localization (pdf)
Tilman Bauer, Bousfield localization and the Hasse square (pdf, pdf), Chapter 6 in: Christopher Douglas, John Francis, André Henriques, Michael Hill (eds.), Topological Modular Forms, Mathematical Surveys and Monographs Volume 201, AMS 2014 (ISBN:978-1-4704-1884-7)
Paul VanKoughnett, Spectra and localization, 2013 (pdf)
Tyler Lawson, An introduction to Bousfield localization (arXiv:2002.03888) in: Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill (eds.), Stable categories and structured ring spectra MSRI Book Series, Cambridge University Press
Discussion in the general context of higher algebra/stable homotopy theory:
Jacob Lurie, Chromatic Homotopy Theory, Lecture notes (2010) (web), Lecture 20 Bousfield localization (pdf)
Jacob Lurie, section 4 of Proper Morphisms, Completions, and the Grothendieck Existence Theorem
Discussion specifically of K(n)-local spectra includes
See also
section 2.4 of
Discussion specifically of KU-local stable homotopy theory:
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