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nilpotent completion of spectra

under construction

Contents

Idea

Given a spectrum XX, and a ring spectrum EE, then the EE-nilpotent completion of XX at EE is, for any choice X XX_\bullet \to X of EE-Adams tower, the homotopy limit limX \underset{\longleftarrow}{\lim} X_\bullet over that tower (Ravenel 84, def. 1.13).

Under certain finiteness conditions (see below), but not generally, this is equivalent to the EE-Bousfield localization L EXL_E X (which, in turn, is in special cases given by formal completion, see at fracture theorem).

The EE-Adams spectral sequence induced by the given Adams tower conditionally converges to the EE-nilpotent completion.

Definition

Bousfield’s definition

Definition

Let (E,μ,e)(E, \mu, e) be a homotopy commutative ring spectrum (def.) and YHo(Spectra)Y \in Ho(Spectra) any spectrum. Write E¯\overline{E} for the homotopy fiber of the unit 𝕊eE\mathbb{S}\overset{e}{\to} E as in this def. such that the EE-Adams filtration of YY (def.) reads (according to this lemma)

E¯ 3Y E¯ 2Y E¯Y Y. \array{ \vdots \\ \downarrow \\ \overline{E}^3 \wedge Y \\ \downarrow \\ \overline{E}^2 \wedge Y \\ \downarrow \\ \overline{E} \wedge Y \\ \downarrow \\ Y } \,.

For nn \in \mathbb{N}, write

E¯ nhocof(E¯ ni n𝕊) \overline{E}_n \coloneqq hocof( \overline{E}^n \overset{i^n}{\longrightarrow} \mathbb{S})

for the homotopy cofiber. Here E¯ 00\overline{E}_0 \simeq 0. By the tensor triangulated structure of Ho(Spectra)Ho(Spectra) (prop.), this homotopy cofiber is preserved by forming smash product with YY, and so also

E¯ nYhocof(E¯ nYY). \overline{E}_n \wedge Y \simeq hocof( \overline{E}^n \wedge Y \overset{}{\longrightarrow} Y) \,.

Now let

E¯ sp s1E¯ s1 \overline{E}_s \overset{p_{s-1}}{\longrightarrow} \overline{E}_{s-1}

be the morphism implied by the octahedral axiom of the triangulated category Ho(Spectra)Ho(Spectra) (def., prop.):

E¯ s+1 i E¯ s EE¯ s ΣE¯ s+1 = i s E¯ s+1 𝕊 E¯ s ΣE¯ s+1 p s1 E¯ s1 = E¯ s1 ΣE¯ s ΣEE¯ s. \array{ \overline{E}^{s+1} &\overset{i}{\longrightarrow}& \overline{E}^s &\longrightarrow& E \wedge \overline{E}^s &\longrightarrow& \Sigma \overline{E}^{s+1} \\ {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{i^s}} && \downarrow^{} && \downarrow \\ \overline{E}^{s+1} &\longrightarrow& \mathbb{S} &\longrightarrow& \overline{E}_s &\longrightarrow& \Sigma \overline{E}^{s+1} \\ && \downarrow && \downarrow^{\mathrlap{p_{s-1}}} \\ && \overline{E}_{s-1} &\overset{=}{\longrightarrow}& \overline{E}_{s-1} \\ && \downarrow && \downarrow \\ && \Sigma \overline{E}^s &\longrightarrow& \Sigma E \wedge \overline{E}^s } \,.

By the commuting square in the middle and using again the tensor triangulated structure, this yields an inverse sequence under YY:

Y𝕊Yp 3idE¯ 3Yp 2idE¯ 2Yp 1idE¯ 1Y Y \simeq \mathbb{S} \wedge Y \longrightarrow \cdots \overset{p_3 \wedge id}{\longrightarrow} \overline{E}_3 \wedge Y \overset{p_2 \wedge id}{\longrightarrow} \overline{E}_2 \wedge Y \overset{p_1 \wedge id}{\longrightarrow} \overline{E}_1 \wedge Y

The E-nilpotent completion Y E Y^\wedge_E of YY is the homotopy limit over the resulting inverse sequence

Y E lim nE¯ nY Y^\wedge_E \coloneqq \mathbb{R}\underset{\longleftarrow}{\lim}_n \overline{E}_n \wedge Y

or rather the canonical morphism into it

YY E . Y \longrightarrow Y^\wedge_E \,.

Concretely, if

Y𝕊Yp 3idE¯ 3Yp 2idE¯ 2Yp 1idE¯ 1Y Y \simeq \mathbb{S} \wedge Y \longrightarrow \cdots \overset{p_3 \wedge id}{\longrightarrow} \overline{E}_3 \wedge Y \overset{p_2 \wedge id}{\longrightarrow} \overline{E}_2 \wedge Y \overset{p_1 \wedge id}{\longrightarrow} \overline{E}_1 \wedge Y

is presented by a tower of fibrations between fibrant spectra in the model structure on topological sequential spectra, then Y E Y^\wedge_E is represented by the ordinary sequential limit over this tower.

(Bousfield 79, top, middle and bottom of page 272)

As the totalization of the cosimplicial spectrum

Definition

Given a E-infinity ring spectrum EE, its corresponding cosimplicial spectrum is the augmented cosimplicial spectrum

E (𝕊eEμideeidEEideididμμideedgeidideEEE). E^\bullet \;\coloneqq\; \left( \mathbb{S} \overset{e}{\longrightarrow} E \underoverset {\underset{id \wedge e}{\longrightarrow}} {\overset{e \wedge id}{\longrightarrow}} {\overset{\mu}{\longleftarrow}} E \wedge E \underoverset {\underset{e \edge id}{\longrightarrow}} {\overset{id \wedge e}{\longrightarrow}} { \underoverset {\underset{id \wedge \mu}{\longleftarrow}} {\overset{\mu \wedge id}{\longleftarrow}} {\overset{id \wedge e \wedge id}{\longrightarrow}} } E \wedge E \wedge E \cdots \right) \,.

(This is the formal dual of the Cech nerve of Spec(E)Spec(𝕊)Spec(E) \to Spec(\mathbb{S}) in the opposite category, where we write Spec(E)Spec(E) for the object EE regarded in the opposite category.)

Moreover, for XHo(Spectra)X \in Ho(Spectra) any spectrum, then there is the corresponding augmented cosimiplicial spectrum E XE^\bullet \wedge X.

Proposition

Given an E-infinity ring spectrum EE and any spectrum XX, then the EE-nilpotent completion X E X^\wedge_E (according to def. ) is equivalently the homotopy limit

X E Tot(E X) =holim nΔ(E nX) holim nTot n(E X) \begin{aligned} X^\wedge_E &\simeq Tot(E^\bullet \wedge X) \\ & = \underset{\leftarrow}{holim}_{n \in \Delta} (E^n\wedge X) \\ & \simeq \underset{\leftarrow}{holim}_{n \in \mathbb{N}}Tot^n(E^\bullet \wedge X) \end{aligned}

over the tower of homotopy-totalizations of the skeleta of the cosimplicial spectrum E YE^\bullet \otimes Y (def. ).

This claim originates in (Hopkins 99, remark 5.5 (ii)). It is taken for granted in (Lurie 10, lecture 8, lecture 30). The first published proof is (Mathew-Naumann-Noel 15, prop. 2.14). See also (Carlsson 07, e.g. remark 3.1).

Remark

Prop. implies that the EE-Adams spectral sequence may equivalently be regarded as computing descent of quasicoherent infinity-stacks in E-infinity geometry along the canonical morphisms Spec(E)Spec(E)\longrightarrow Spec(S). See at Adams spectral sequence – As derived descent.

Properties

Relation to EE-localization

Remark

There is a canonical map

L EXlim n(E S n+1 SX) L_E X \overset{}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)

from the EE-Bousfield localization of spectra of XX into the totalization.

We consider now conditions for this morphism to be an equivalence.

Definition

For RR a ring, its core cRc R is the equalizer in

cRRRR. c R \longrightarrow R \stackrel{\longrightarrow}{\longrightarrow} R \otimes R \,.
Proposition

Let EE be a connective E-∞ ring such that the core of π 0(E)\pi_0(E), def. , is either of

  • the localization of the integers at a set JJ of primes, cπ 0(E)[J 1]c \pi_0(E) \simeq \mathbb{Z}[J^{-1}];

  • n\mathbb{Z}_n for n2n \geq 2.

Then the map in remark is an equivalence

L EXlim n(E S n+1 SX). L_E X \stackrel{\simeq}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X) \,.

(Bousfield 79).

Examples

The nilpotent completion of a connective spectrum at the Eilenberg-MacLane spectrum HH \mathbb{Z}, happens to be the spectrum itself (by a Postnikov tower argument).

Examples

For XX a connective spectrum, its H𝔽 pH \mathbb{F}_p-nilpotent completion is the formal completion X^{\hat}_p.

The MU-nilpotent completion of any spectrum XX^{\hat}X is XX.

The BP-nilpotent completion at prime pp of any spectrum XX is X (p)X_{(p)}.

(Ravenel 84, example 1.16)

References

The concept originates with

  • Aldridge Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281. (pdf)

  • Douglas Ravenel, Localization with respect to certain periodic homology theories, American Journal of Mathematics, Vol. 106, No. 2, (Apr., 1984), pp. 351-414 (pdf)

The re-interpretation in terms of totalization of the cosimplicial spectrum is briefly mentioned in

  • Mike Hopkins, section 4 of Complex oriented cohomology theories and the language of stacks, course notes 1999 (pdf)

and tacitly assumed in

A proof of the equivalence of this re-interpretation appears in

See also

Last revised on July 21, 2016 at 05:20:44. See the history of this page for a list of all contributions to it.