nLab nilpotent completion of spectra

under construction

Idea

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

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

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

Definition

Bousfield’s definition

Definition

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

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

For $n \in \mathbb{N}$, write

$$\overline{E}_n \coloneqq hocof( \overline{E}^{n+1} \overset{i^{n+1}}{\longrightarrow} \mathbb{S})$$

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

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

Now let

$$\overline{E}_s \overset{p_{s-1}}{\longrightarrow} \overline{E}_{s-1}$$

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

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

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

$$Y \simeq \mathbb{S} \wedge Y \longrightarrow \cdots \overset{p_3 \wedge id}{\longrightarrow} \overline{E}_3 \wedge Y \overset{p_2 \wedge id}{\longrightarrow} \overline{E}_2 \wedge Y \overset{p_1 \wedge id}{\longrightarrow} \overline{E}_1 \wedge Y$$

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

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

or rather the canonical morphism into it

$$Y \longrightarrow Y^\wedge_E \,.$$

Concretely, if

$$Y \simeq \mathbb{S} \wedge Y \longrightarrow \cdots \overset{p_3 \wedge id}{\longrightarrow} \overline{E}_3 \wedge Y \overset{p_2 \wedge id}{\longrightarrow} \overline{E}_2 \wedge Y \overset{p_1 \wedge id}{\longrightarrow} \overline{E}_1 \wedge Y$$

is presented by a tower of fibrations between fibrant spectra in the model structure on topological sequential spectra, then $Y^\wedge_E$ 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 $E$, its corresponding cosimplicial spectrum is the augmented cosimplicial spectrum

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

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

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

Proposition

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

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

over the tower of homotopy-totalizations of the skeleta of the cosimplicial spectrum $E^\bullet \otimes Y$ (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 $E$-Adams spectral sequence may equivalently be regarded as computing descent of quasicoherent infinity-stacks in E-infinity geometry along the canonical morphisms $Spec(E)\longrightarrow$ Spec(S). See at Adams spectral sequence – As derived descent.

Properties

Relation to $E$-localization

Remark

There is a canonical map

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

from the $E$-Bousfield localization of spectra of $X$ into the totalization.

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

Definition

For $R$ a ring, its core $c R$ is the equalizer in

$$c R \longrightarrow R \stackrel{\longrightarrow}{\longrightarrow} R \otimes R \,.$$

Proposition

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

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

(Bousfield 79).

Examples

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

Examples

For $X$ a connective spectrum, its $H \mathbb{F}_p$-nilpotent completion is the formal completion $X^{\wedge}_p$.

The MU-nilpotent completion of any connective spectrum $X$ is $X$.

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

(Ravenel 84, example 1.16)

Related concepts

• formal completion

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

• Jacob Lurie, Chromatic Homotopy Theory (2010) lecture 30: Localizations and the Adams-Novikov Spectral Sequence (pdf)

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

• Akhil Mathew, Niko Naumann, Justin Noel, Nilpotence and descent in equivariant stable homotopy theory (arXiv:1507.06869)

See also

• Andrew Baker, Andrey Lazarev, On the Adams Spectral Sequence for $R$-modules, Algebr. Geom. Topol. 1 (2001) 173-199 (arXiv:math/0105079)

• Gunnar Carlsson, Derived completions in stable homotopy theory (arXiv:0707.2585)

• Tyler Lawson, Completed representation ring spectra of nilpotent groups, Algebraic & Geometric Topology 6 (2006) 253–285 (arXiv:0902.4867)