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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 2nd 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarted a minimum at _[[E-nilpotent completion]]_ (the thing that an $E$-Adams tower converges to).

   started a minimum at E-nilpotent completion (the thing that an $E$-Adams tower converges to).

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 15th 2016
   • (edited Jul 15th 2016)
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   Author: Urs
   Format: MarkdownItexI have added [here](https://ncatlab.org/nlab/show/nilpotent+completion+of+spectra#ENilpotentCompletion) the actual definition given by Bousfield. (I still don't really have the proof that this coincides with the Tot-tower construction of the cosimplicial spectrum.) Let $(E, \mu, e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) 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.](Adams+spectral+sequence#HomotopyFiberOfUnitOfCommutativeRingSpectrum) such that the $E$-Adams filtration of $Y$ ([def.](Adams+spectral+sequence#AdamsEAdamsSpectralSequence)) reads (according to [this lemma](Adams+spectral+sequence#Wp)) $$ \array{ \vdots \\ \downarrow \\ \overline{E}^3 \wedge Y \\ \downarrow \\ \overline{E}^2 \wedge Y \\ \downarrow \\ \overline{E} \wedge Y \\ \downarrow \\ Y } \,. $$ For $n \in \mathbb{N}$, write $$ \overline{E}_n \coloneqq hocof( \overline{E}^n \overset{i^n}{\longrightarrow} \mathbb{S}) $$ for the homotopy cofiber. Here $\overline{E}_0 \simeq 0$. By the [[tensor triangulated category|tensor triangulated]] structure of $Ho(Spectra)$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory)), this homotopy cofiber is preserved by forming [[smash product of spectra|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.](Introduction+to+Stable+homotopy+theory+--+1-1#CategoryWithCofiberSequences), [prop.](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsTriangulated)): $$ \array{ \overline{E}^{s+1} &\overset{i}{\longrightarrow}& \overline{E}^s &\longrightarrow& E \wedge \overline{E}^s &\longrightarrow& \Sigma \overline{E}^{s+1} \\ {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{i^s}} && \downarrow^{} && \downarrow \\ \overline{E}^{s+1} &\longrightarrow& \mathbb{S} &\longrightarrow& \overline{E}_s &\longrightarrow& \Sigma \overline{E}^{s+1} \\ && \downarrow && \downarrow^{\mathrlap{p_{s-1}}} \\ && \overline{E}_{s-1} &\overset{=}{\longrightarrow}& \overline{E}_{s-1} \\ && \downarrow && \downarrow \\ && \Sigma \overline{E}^s &\longrightarrow& \Sigma E \wedge \overline{E}^s } \,. $$ By the [[commuting square]] in the middle and using again the [[tensor triangulated category|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.

   I have added here the actual definition given by Bousfield. (I still don’t really have the proof that this coincides with the Tot-tower construction of the cosimplicial spectrum.)

   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

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

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 20th 2016
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   Author: Urs
   Format: MarkdownItexI have finally found a reference with a proof that Bousfield's original definition is equivalent to the one in terms of the Tot-tower of the cosimplicial spectrum: it is proposition 2.14 in the (neat) article * [[Akhil Mathew]], [[Niko Naumann]], [[Justin Noel]], _Nilpotence and descent in equivariant stable homotopy theory_ ([arXiv:1507.06869](http://arxiv.org/abs/1507.06869)) As the authors notice on p. 9 > This result is certainly not new, but we have included it for lack of a convenient reference. Indeed. I have briefly added a corresponding paragraph to the entry [here](https://ncatlab.org/nlab/show/nilpotent+completion+of+spectra#AsLimitOverTotTower).

   I have finally found a reference with a proof that Bousfield’s original definition is equivalent to the one in terms of the Tot-tower of the cosimplicial spectrum:

   it is proposition 2.14 in the (neat) article

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

   As the authors notice on p. 9

   This result is certainly not new, but we have included it for lack of a convenient reference.

   Indeed.

   I have briefly added a corresponding paragraph to the entry here.