text:
\section{Introduction}
\vspace{22pt}

The telescope conjecture, first stated by Ravenel~\cite[Conj.~10.5]{[Rav84]}, is a claim about two classes of localization functors in the $p$-local stable homotopy category of spectra.  First, one can localize away from a finite type $n+1$ spectrum $F(n+1)$; the acyclics are the smallest localizing subcategory containing $F(n+1)$, and we denote this functor $L_{n}^f$.  Second, one can localize at the wedge of the first $n+1$ Morava $K$-theories $K(0)\vee\cdots \vee K(n)$; the acyclics are all spectra that smash with $K(0)\vee\cdots\vee K(n)$ to zero and this is denoted $L_{n}$.  Both these localizations are smashing, i.e.~they commute with coproducts.  The telescope conjecture (${\sf TC}_n$), basically, claims that $L_n^f$ and $L_n$ are isomorphic.  In fact, here we consider three slightly different versions, ${\sf TC1_n}$, ${\sf TC2_n}$, and ${\sf TC3_n}$, of the telescope conjecture.  In Section~\ref{sect-TCs} we articulate them carefully and show implications between them.

The conjecture is known to hold for $n=0$~\cite[p.~79]{[Rav92]}, and for $n=1$ when $p=2$~\cite{[Mah82]} and $p>2$~\cite{[Mil81]}.  A valiant but unsuccessful effort at a counterexample, for $n\geq 2$, was undertaken by Mahowald, Ravenel, and Shick, as outlined in~\cite{[MRS]}.  Since then little progress has been made, and the original conjecture remains open.

A generalization of the telescope conjecture can be stated for spectra, as well as other triangulated categories.  Localization away from a finite spectrum, i.e.~a compact object of the category, always yields a smashing localization functor (see e.g.~\cite[Prop.~2.9]{[Bou79a]} or~\cite{[Miller]} or~\cite[Thm.~3.3.3]{[HPS]}). The Generalized Smashing Conjecture (${\sf GSC}$) is that every smashing localization arises in this way.  If true, then every smashing localization is determined by its compact acyclics; if the ${\sf GSC}$ holds in spectra, then so must the ${\sf TC}_n$ for all $n$.

The ${\sf GSC}$, essentially stated for spectra decades ago by Bousfield \cite[Conj.~3.4]{[Bou79b]}, has been formulated in many other triangulated categories, in many cases labeled as the telescope conjecture, and in many cases proven to hold.  Neeman~\cite{[Neeman92]} made the conjecture for the derived category $D(R)$ of a commutative ring $R$, and showed it holds when the ring is Noetherian.  See also~\cite[Thm.~6.3.7]{[HPS]} or~\cite{[KrSt10]} for a generalization.  On the other hand, Keller~\cite{[keller]} gave an example of a non-Noetherian ring for which the ${\sf GSC}$ fails.  Benson, Iyengar, and Krause have shown that the ${\sf GSC}$ holds in a stratified category~\cite{[BIKstcat]}, such as the stable module category of a finite group~\cite{[BIKstcat2]}.  Balmer and Favi~\cite{[BF]} show that in a tensor triangulated category with a good notion of support, the ${\sf GSC}$ is a ``local" question.

It is worth noting that there are further variations of the {\sf GSC}, which we won't consider here.  Krause~\cite{[krause-sstc]} formulated a variation of the ${\sf GSC}$, in terms of subcategories generated by sets of maps, that makes sense (and holds) for any compactly generated triangulated category.  Krause and Solberg give a variation for stable module categories, stated in terms of cotorsion pairs~\cite{[KS03]}.  See also~\cites{[krause-cqsl],[HST08],[Bruning07],[Stov10]}.

To date, Keller's ring yields the only category where the ${\sf GSC}$ is known to fail.  In this paper we give several more examples.  Incidentally, each is a well generated triangulated category that is not compactly generated.

One of our main results is the following.  We weaken the assumptions for ``finite localization", and show that in many categories, localization away from any set of strongly dualizable objects yields a smashing localization.  (Recall that an object $X$ is strongly dualizable if $F(X,Y) \cong F(X,\boldone)\wedge Y$ for all $Y$, where $\boldone$ is the tensor unit and $F(-,-)$ the function object bifunctor.)  Let ${\sf loc}(X)$ denote the smallest localizing subcategory containing $X$.  We prove the following as Theorem~\ref{sdsl}.\\

\noindent \textbf{Theorem A:} Let ${\sf T}$ be a well generated tensor triangulated category such that  ${\sf loc}(\boldone) = {\sf T}$.  Let $A=\{B_\alpha\}$ be a (possibly infinite) set of strongly dualizable objects.  Then there exists a smashing localization functor $L:{\sf T}\rightarrow {\sf T}$ with Ker~$L = {\sf loc}(A)$. \\

Thus we are led to conjecture the following.\\

\noindent \textbf{Strongly Dualizable Generalized Smashing Conjecture (${\sf SDGSC}$):} Every smashing localization is localization away from a set of strongly dualizable objects.\\

We give several examples of categories where the ${\sf GSC}$ fails, but the ${\sf SDGSC}$ holds.  In fact, we consider a topological setting, where one can also formulate a version (or versions, rather) of the original telescope conjecture.

Specifically, we consider localized categories of spectra.  Let $\mathcal{S}$ be the $p$-local stable homotopy category, and let $\wedge$ denote the smash, i.e. tensor, product.  Take $Z$ an object of $\mathcal{S}$, and let $L=L_Z:\mathcal{S}\rightarrow \mathcal{S}$ be the localization functor that annihilates $Z_*$-acyclic objects.  The full subcategory of $L$-local objects, that is, objects $X$ for which $X {\rightarrow} LX$ is an equivalence, has a tensor triangulated structure induced by that of $\mathcal{S}$.  Let $\mathcal{L}$ denote this category; the triangles are the same as in $\mathcal{S}$, the coproduct is $X\coprod_\mathcal{L} Y = L(X\coprod Y)$ and the tensor is $X\wedge_\mathcal{L} Y = L(X\wedge Y)$.

In Definition~\ref{defn-l}, we define localization functors ${\it l}_n^f$ and ${\it l}_n$ on $\mathcal{L}$ that are localized versions of $L_n^f$ and $L_n$.  The localized telescope conjecture (${\sf LTC}$), basically, is that ${\it l}_n^f$ and ${\it l}_n$ are isomorphic.  In fact, we give three versions of the ${\sf LTC}$, and in Theorems~\ref{LTCs} and~\ref{LTCstwo} establish implications between them.  Then, examining specific examples of localized categories of spectra, we conclude the following in Theorems~\ref{LTCs-harm}, \ref{LTCs-kn}, \ref{LTCs-hfp}, \ref{i-loc-tc} and \ref{LTCs-bp} and Corollary~\ref{LTCs-en}.  \\

\noindent \textbf{Theorem B:} All versions of the localized telescope conjecture, ${\sf LTC1_i}$, ${\sf LTC2_i}$, and ${\sf LTC3_i}$ hold for all $i\geq 0$, in the $\bigvee_{n\geq 0}K(n)$-local (i.e. harmonic), $K(n)$-local, ${H\bf_p}$-local, $BP$-local, and $I$-local categories, where $I$ is the Brown-Comenetz dual of the sphere spectrum.\\

In order to consider the ${\sf GSC}$ and ${\sf SDGSC}$ in $\mathcal{L}$, we must classify the smashing localizations on $\mathcal{L}$.  We are able to do this in several examples.\\

\noindent \textbf{Theorem C:} In the harmonic category, the ${\sf GSC}$ fails but the ${\sf SDGSC}$ holds.  Likewise in the ${H\bf_p}$-local and $I$-local categories.  In the $BP$-local category the ${\sf GSC}$ fails but the ${\sf SDGSC}$ is open.  In the $E(n)$-local and $K(n)$-local categories the ${\sf GSC}$ and ${\sf SDGSC}$ both hold.\\

\begin{pf} This is Theorems~\ref{harm-sdgsc} and \ref{LTCs-kn}, Propositions~\ref{gsc-hfp} and \ref{gsc-bp}, and Corollaries~\ref{LTCs-en} and \ref{gsc-i}. \end{pf}

One novelty in our approach is our use of Bousfield lattice arguments.  Given an object $X$ in a tensor triangulated category ${\sf T}$, the Bousfield class of $X$ is $\<X\> = \{W\;|\; W\wedge X =0\}$.  It is now known~\cite{[IK]} that every well generated tensor triangulated category has a set of Bousfield classes.  This set has the structure of a lattice, and is called the Bousfield lattice of ${\sf T}$.  One can now attempt to calculate the Bousfield lattices of categories of localized spectra.  Furthermore, every smashing localization yields a pair of so-called complemented Bousfield classes.  Information about the Bousfield lattice of a category gives information about its complemented classes, which gives information about the smashing localization functors on the category.

Moreover, the first version of the telescope conjecture ${\sf TC1_n}$ is that two spectra $T(n)$ and $K(n)$ have the same Bousfield class.  In the localized version this becomes (${\sf LTC1_n}$) the claim that $\<LT(n)\>=\<LK(n)\>$ in the Bousfield lattice of $\mathcal{L}$.  One is thus led to investigating Bousfield lattices of localized spectra.

Corollary~\ref{bl-size} gives an upper bound, $2^{2^{\aleph_0}}$, on the cardinality of such lattices.  Jon Beardsley has calculated the Bousfield lattice of the harmonic category to be isomorphic to the power set of $\mathbb{N}$; we give this calculation in Proposition~\ref{jb}.  In Corollary~\ref{ln-bl} and Proposition~\ref{realize} we show that one can realize this lattice as an inverse limit of the Bousfield lattices of $E(n)$-local categories, as $n$ ranges over $\mathbb{N}$.  Then in Corollary~\ref{bl-kn} and Propositions~\ref{bl-hfp} and ~\ref{bl-i}, we show that the $K(n)$-local, ${H\bf_p}$-local, and $I$-local categories all have two-element Bousfield lattices.  In Proposition~\ref{bl-bp} we give a lower bound, $2^{\aleph_0}$, on the cardinality of the Bousfield lattice of the $BP$-local category.

One immediate object-level application of these Bousfield lattice calculations is the following.  Call an object $X$ square-zero if it is nonzero, but $X\wedge X=0$.  Then Proposition~\ref{no-sz} shows that there are no square-zero objects in the harmonic, $E(n)$-, $K(n)$-, ${H\bf_p}$-, or $I$-local categories.

We are also able to answer the analogue of a conjecture by Hovey and Palmieri, originally stated for the stable homotopy category.  Conjecture 9.1 in~\cite{[HP]} is that every localizing subcategory is a Bousfield lattice.  Proposition~\ref{loc-not-bc} demonstrates that this fails in the ${H\bf_p}$-local category, by giving two localizing subcategories that are not Bousfield classes.

Section~\ref{sect-prelims} establishes the categorical setting, and provides background on localization, Bousfield lattices, and stable homotopy theory.  Section~\ref{sect-TCs} defines the various versions of the telescope conjecture, for spectra and for localized spectra, and establishes implications among them.  The remainder of the paper is devoted to examining specific examples: the harmonic category (Section~\ref{sect-harm}), the $E(n)$-local and $K(n)$-local categories (Section~\ref{sect-enkn}), and the ${H\bf_p}$-local, $I$-local, $BP$-local, and $F(n)$-local categories (Section~\ref{sect-other}).  All results are new unless cited.  Most of the results on the $E(n)$-local and $K(n)$-local categories in Section~\ref{sect-enkn} follow in a straightforward way from Hovey and Strickland's work in~\cite{[HovS]}, and are included for completeness.  \\

We would like to thank Dan Christensen for extensive discussions and suggestions, and Jon Beardsley for Proposition~\ref{jb}.

\vspace{22pt}
\section{Preliminaries} \label{sect-prelims}
\vspace{22pt}

\subsection{Categorical setting} We start with the notion of a tensor triangulated category ${\sf C}$; i.e.~ a triangulated category with set-indexed coproducts and a closed symmetric monoidal structure compatible with the triangulation~\cite[App.A]{[HPS]}.  Let $\Sigma: {\sf C}\rightarrow{\sf C}$ denote the shift, $[X,Y]$ the morphisms from $X$ to $Y$, and $[X, Y]_n = [\Sigma^n X, Y]$ for any $n\in\mathbb{Z}$.

Let $-\wedge-$ denote the smash (tensor) product, $\boldone$ denote the unit, and $F(-,-)$ denote the function object bifunctor; $F(X,-)$ is the right adjoint to $X\wedge-$.  Recall that an object $X$ in ${\sf C}$ is said to be {\it strongly dualizable} if the natural map $DX\wedge Y\rightarrow F(X,Y)$ is an isomorphism for all $Y$, where $DX=F(X,\boldone)$ is the Spanier-Whitehead dual.  Since $F(\boldone, X)\cong X$ for all $X$, the map $F(\boldone,\boldone)\wedge Y\rightarrow F(\boldone, Y)$ is an equivalence and $\boldone$ is always strongly dualizable.

For a regular cardinal $\alpha$, we say an object $X$ is {\it $\alpha$-small} if every morphism $X\rightarrow \coprod_{i\in I} Y_i$ factors through $\coprod_{i\in J}Y_i$ for some $J\subseteq I$ with $|J|<\alpha$.  If $X$ is $\aleph_0$-small we say $X$ is {\it compact} (\cite{[HPS]} calls this {\it small}); this is equivalent to the condition that the natural map $\oplus_{i\in K}[X,Z_i]\rightarrow [X,\coprod_{i\in K} Z_i]$ is an isomorphism for any set-indexed coproduct $\coprod_{i\in K} Z_i$.  We say ${\sf C}$ is {\it $\alpha$-well generated} if it has a set of perfect generators~\cite[Sect.~5.1]{[krause-loc-survey]} which are $\alpha$-small, and ${\sf C}$ is {\it well generated} if it is $\alpha$-well generated for some $\alpha$.  See~\cite{[krause-loc-survey]} for more details.

A {\it localizing} subcategory is a triangulated subcategory of ${\sf C}$ that is closed under retracts and coproducts; a {\it thick} subcategory is a triangulated subcategory that is closed under retracts.  Given an object or set of objects $X$, let ${\sf loc}(X)$ (resp. ${\sf th}(X)$) denote the smallest localizing (resp. thick) subcategory containing $X$.  We say that ${\sf loc}(X)$ is {\it generated by }$X$.

\begin{definition-notation}  Throughout this paper let ${\sf T}$ be a well generated tensor triangulated category such that  ${\sf loc}(\boldone) = {\sf T}$. \label{cat}
\end{definition-notation}

In the language of~\cite{[HPS]}, such a ${\sf T}$ is almost a ``monogenic stable homotopy category", except that we do not insist that the unit $\boldone$ is compact.

In practice, in this paper ${\sf T}$ will always be either the $p$-local stable homotopy category of spectra $\mathcal{S}$ or the category $\mathcal{L}_Z$ of $L_Z$-local objects derived from a localization functor $L_Z:\mathcal{S}\rightarrow \mathcal{S}$.  The former satisfies Definition
~\ref{cat} by~\cite[Ex.~1.2.3(a)]{[HPS]}, and the latter by Theorem~\ref{local-cat} and Lemma~\ref{local-cat-blah} below.\\

\subsection{Background on localization}  Recall that a {\it localization functor} (or simply {\it localization}) on a tensor triangulated category ${\sf C}$ is an exact functor $L:{\sf C}\rightarrow {\sf C}$, along with a natural transformation $\eta: 1\rightarrow L$ such that $L\eta$ is an equivalence and $L\eta=\eta L$.  We call Ker~$L$ the {\it $L$-acyclics}.  It follows that there is an exact functor $C:{\sf C}\rightarrow {\sf C}$, called {\it colocalization}, such that every $X$ in ${\sf C}$ fits into an exact triangle $CX\rightarrow X \rightarrow LX$, with $CX$ $L$-acyclic.  An object $Y$ is {\it $L$-local} if it is in the essential image of $L$, and this is equivalent to satisfying $[Z,Y] = 0$ for all $L$-acyclic $Z$.  See~\cite[Ch.~3]{[HPS]} or~\cite{[krause-loc-survey]} for further background.

We also recall two special types of localizations.  A localization $L:{\sf C}\rightarrow {\sf C}$ is said to be {\it smashing} if $L$ preserves coproducts, equivalently if $LX\cong L\boldone\wedge X$ for all $X$.

Given a set $A$ of objects of ${\sf C}$, we say that a localization functor $L:{\sf C}\rightarrow{\sf C}$ is {\it localization away from $A$} if the $L$-acyclics are precisely ${\sf loc}(A)$.  If such a localization exists, we also say it is {\it generated by $A$}.  When ${\sf C}=\mathcal{S}$, it is well known (e.g.~\cites{[Miller], [MS]}) that localization away from a set of compact objects exists, and yields a smashing localization functor.  As mentioned in the introduction, this result has been generalized to other categories as well (e.g.~\cite[Thm.~3.3.3]{[HPS]}, \cite[Thm.~4.1]{[BF]}).  We present a further generalization in Theorem~\ref{sdsl}.

In this paper we will restrict our attention to homological localizations, which we now describe.  Given an object $Z$ in a tensor triangulated category ${\sf C}$, the {\it Bousfield class} of $Z$ is defined to be
$$\<Z\> = \{W\in {\sf C}\;|\; W\wedge Z=0\}.$$

Extending a classical result of Bousfield's for $\mathcal{S}$, Iyengar and Krause recently showed~\cite[Prop.~2.1]{[IK]} that for every object $Z$ in a well generated tensor triangulated category ${\sf C}$, there is a localization functor $L_Z:{\sf C}\rightarrow {\sf C}$ with $L_Z$-acyclics precisely $\<Z\>$.  We call such an $L_Z$ {\it homological localization at $Z$}.

\begin{notation} \label{loc-notation} Let ${\sf T}$ be as in Definition~\ref{cat}, with tensor unit $\boldone$.  For an object $Z$ in ${\sf T}$, let $L_Z:{\sf T}\rightarrow {\sf T}$ be homological localization at $Z$.  Let $\mathcal{L}_Z$ denote the category of $L_Z$-local objects, the essential image of $L_Z$.\\
\end{notation}

\begin{theorem}~\cite[3.5.1,3.5.2]{[HPS]} \label{local-cat} Let $L=L_Z:{\sf T}\rightarrow {\sf T}$ be a localization, and $\mathcal{L}_Z$ the category of $L_Z$-local objects.  Then $\mathcal{L}_Z$ has a natural structure as a tensor triangulated category, generated by $L_Z\boldone$, which is the unit.  Considered as a functor from ${\sf T}$ to $\mathcal{L}_Z$, $L$ preserves triangles, the tensor product and its unit, coproducts, and strong dualizability.  Furthermore, $L$ preserves compactness if and only if $L$ is a smashing localization.
\end{theorem}

Explicitly, for $L$-local objects $X$, $X_i$ and $Y$, in $\mathcal{L}$ we have $\coprod_{\mathcal{L}} X_i = L(\coprod_{{\sf T}} X_i)$ and $X \wedge_\mathcal{L} Y = L(X\wedge_{\sf T} Y)$ and $F_\mathcal{L}(X,Y) = F(X,Y)$.  Note that $L_Z\boldone$ is strongly dualizable but may not be compact in $\mathcal{L}_Z$.

\begin{lemma} \label{local-cat-blah} The category $\mathcal{L}_Z$ is well generated.
\end{lemma}

\begin{pf} By Proposition 2.1 of~\cite{[IK]}, the $L_Z$-acyclics $\<Z\>$ form a well generated localizing subcategory of ${\sf T}$.  Then by~\cite[Thm.~7.2.1]{[krause-loc-survey]}, the Verdier quotient ${\sf T}/\<Z\>$, which is equivalent to the local category $\mathcal{L}_Z$, is well generated.  \end{pf}

We conclude this subsection with a lemma with four useful well-known facts.  Recall that a {\it ring object} in a tensor triangulated category is an object $R$ with an associative multiplication map $\mu: R\wedge R\rightarrow R$ and a unit $\iota: \boldone\rightarrow R$, making the evident diagrams commute.  If $R$ is a ring object, then an {\it $R$-module object} is an object $M$ with a map $m: R\wedge M\rightarrow M$ along with evident commutative diagrams.  Note that $R\wedge X$ is an $R$-module object for every $X$.  A {\it skew field object} is a ring object such that every $R$-module object is {\it free}, i.e.~isomorphic to a coproduct of suspensions of $R$~\cite[Def.~3.7.1]{[HPS]}.

\begin{lemma} \label{useful} Let ${\sf C}$ be a tensor triangulated category with ${\sf loc}(\boldone)={\sf C}$, and $L:{\sf C}\rightarrow {\sf C}$ a localization.
\begin{enumerate}
     \item Every localizing subcategory ${\sf S}$ of ${\sf C}$ is tensor-closed; that is, given $X\in {\sf S}$ and $Y\in {\sf C}$, $X\wedge Y\in {\sf S}$.
     \item	For all $X$ and $Y$ in ${\sf C}$, $L(X\wedge Y) = L(LX\wedge LY)$.
     \item Considered as a functor from ${\sf C}$ to $\mathcal{L}$, $L$ also preserves ring objects and  module objects.
		 \item If $R$ is a ring object and $M$ is an $R$-module object (in particular, if $M=R$), then $M$ is $R$-local.
\end{enumerate}
\end{lemma}

\begin{pf} For $(1)$, note that $Y\in {\sf loc}(\boldone) = {\sf C}$, so $X\wedge Y\in {\sf loc}( X\wedge \boldone)={\sf loc}(X)\subseteq {\sf S}$.

For $(2)$, consider the exact triangle $X\wedge CY\rightarrow X\wedge Y\rightarrow X\wedge LY$.  Since $CY$ is $L$-acyclic and these form a localizing subcategory, $L(X\wedge CY)=0$, so $L(X\wedge Y) = L(X\wedge LY)$.  Using the same reasoning with the triangle $CX\wedge LY\rightarrow X\wedge LY\rightarrow LX\wedge LY$, the result follows.

 If $R \in {\sf C}$, is a ring object, then $L(\mu): L(R\wedge R) = L(LR\wedge LR) = LR\wedge_{\mathcal{L}} LR\rightarrow LR$, and all the localized diagrams commute.  Similarly for module objects.

Part (4) is~\cite[Prop.~1.17(a)]{[Rav84]}. \end{pf}

\subsection{Background on Bousfield lattices} Every well generated tensor triangulated category, and hence every localized category of spectra, has a set (rather than a proper class) of Bousfield classes~\cite[Thm.~3.1]{[IK]}.  This was also recently shown for every tensor triangulated category with a combinatorial model~\cite{[CGR]}.  This set is called the {\it Bousfield lattice} $\mathsf{BL}({\sf T})$ and has a lattice structure which we now recall.  Refer to~\cites{[HP],[preprint]} for more details.

The partial ordering is given by reverse inclusion: we say $\<X\>\leq \<Y\>$ when $W\wedge Y=0 \implies W\wedge X=0$.  It is also helpful to remember that, unwinding definitions, $\<X\>\leq \<Y\>$ precisely when every $L_X$-local object is also $L_Y$-local.  Clearly $\<0\>$ is the minimum and $\<\boldone\>$ is the maximum class.  The join of any set of classes is $\bigvee_{i\in I} \<X_i\> = \<\coprod_{i\in I} X_i\>$, and the meet is defined to be the join of all lower bounds.

The smash product induces an operation on Bousfield classes, where $\<X\>\wedge \<Y\> = \<X\wedge Y\>$.  This is a lower bound, but in general not the meet.  However, if we restrict to the subposet $\mathsf{DL} = \{\<W\>\;|\; \<W\wedge W\> = \<W\>\}$, then the meet and smash agree.  Since coproducts distribute across the smash product, $\mathsf{DL}$ is a distributive lattice.

We say a class $\<X\>$ is {\it complemented} if there exists a class $\<X^c\>$ such that $\<X\>\wedge \<X^c\>=\<0\>$ and $\<X\>\vee\<X^c\> = \<\boldone\>$.  The collection of complemented classes is denoted $\mathsf{BA}$.  For example, every smashing localization $L:{\sf T}\rightarrow {\sf T}$ gives a pair of complemented classes, namely $\<C\boldone\>$ and $\<L\boldone\>$.  Because every complemented class is also in $\mathsf{DL}$, $\mathsf{BA}$ is a Boolean algebra.

\begin{proposition} \label{lattice-map} Let ${\sf T}$ be as in Definition~\ref{cat}, and $L_Z:{\sf T}\rightarrow {\sf T}$ a localization functor as in Notation~\ref{loc-notation}.  Then $L_Z$ induces a well-defined order-preserving map of lattices $\mathsf{BL}({\sf T})\rightarrow \mathsf{BL}(\mathcal{L}_Z)$, where $\<X\>\mapsto \<L_ZX\>$.  This map is surjective, and sends $\mathsf{DL}({\sf T})$ onto $\mathsf{DL}(\mathcal{L}_Z)$ and $\mathsf{BA}({\sf T})$ onto $\mathsf{BA}(\mathcal{L}_Z)$.
\end{proposition}

\begin{pf} Most of this was proved in Lemma $3.1$ of~\cite{[preprint]}.  For $\<X\>\in \mathsf{DL}({\sf T})$, using Lemma~\ref{ugly} we get
$$\<LX\> = \<L(X\wedge X)\> = \<L(LX\wedge LX)\> = \<LX \wedge_\mathcal{L} LX\>.$$  Likewise, one can check that for $\<X\>\in \mathsf{BA}({\sf T})$, the class $\<LX\>\in \mathsf{BL}(\mathcal{L})$ is complemented by $\<L(X^c)\>$, keeping in mind that $\<L\boldone\>$ is the top class in $\mathsf{BL}(\mathcal{L})$. \end{pf}

\begin{corollary} \label{bl-size} For any $Z\in \mathcal{S}$, we have $|\mathsf{BL}(\mathcal{L}_Z)|\leq 2^{2^{\aleph_0}}$.
\end{corollary}

\begin{pf} $|\mathsf{BL}(\mathcal{L}_Z)|\leq |\mathsf{BL}(\mathcal{S})|\leq 2^{2^{\aleph_0}}$, where the second inequality was proved in~\cite{[ohkawa]}.
\end{pf}

\begin{lemma}\label{compose}  Let ${\sf T}$ be as in Definition~\ref{cat}, and $X$ and $Y$ objects of ${\sf T}$.  Then $\<X\>\leq \<Y\>$ if and only if $L_X=L_XL_Y=L_YL_X$ and in this case the following diagram commutes (also with $\mathsf{BL}$ replaced by $\mathsf{DL}$ or $\mathsf{BA}$).

$$
\UseComputerModernTips
\xymatrix{
 & \mathsf{BL}({\sf T}) \ar@{->>}[dl]_{L_Y} \ar@{->>}[dr]^{L_X}  \\
\mathsf{BL}(\mathcal{L}_Y) \ar@{->>}[rr]^{L_X} && \mathsf{BL}(\mathcal{L}_X)
}
$$
\end{lemma}

\begin{pf} The first equivalence is straightforward; it follows from~\cite[Prop.~1.22]{[Rav84]} and the observation that $\<X\>\leq \<Y\>$ precisely when all $L_X$-locals are $L_Y$-locals.  The last remark follows from Proposition~\ref{lattice-map}. \end{pf}

Here we mention one object-level application of the Bousfield lattice calculations of Sections~\ref{sect-enkn} and \ref{sect-other}.  Call an object $X$ {\it square-zero} if it is nonzero, but $X\wedge X=0$.  For example, $I\wedge I=0$ in $\mathcal{S}$.

\begin{proposition} There are no square-zero objects in the harmonic, $E(n)$-, $K(n)$-, ${H\bf_p}$-, or $I$-local categories. \label{no-sz}
\end{proposition}

\begin{pf} In Corollary 2.8 of~\cite{[preprint]}, we show that in a well generated tensor triangulated category, there are no square-zero objects if and only if $\mathsf{BA}=\mathsf{DL}=\mathsf{BL}$.  The claim then follows from Corollaries~\ref{ln-bl} and~\ref{bl-kn} and Propositions~\ref{bl-hfp} and~\ref{bl-i}.
\end{pf}

\subsection{Background on spectra}  We quickly review some relevant background on the stable homotopy category.  See~\cites{[Rav92],[Hov95a],[MS], [RavLaTC]} for more details.  Fix a prime $p$ and let $\mathcal{S}$ denote the $p$-local stable homotopy category of spectra.  Let $S^0$ denote the sphere spectrum.  The {\it finite} spectra $\mathcal{F}$ are the compact objects of $\mathcal{S}$, and $\mathcal{F} = {\sf th}(S^0)$.  The structure of $\mathcal{F}$ is determined by the Morava $K$-theories $K(i)$.  For each $i\geq 0$, $K(i)$ is a skew field object in $\mathcal{S}$, such that $K(i)\wedge K(j)=0$ when $i\neq j$.  If $X$ is a finite spectrum and $K(j)\wedge X=0$, then $K(j-1)\wedge X=0$.  We say a finite spectrum $X$ is {\it type $n$} if $n$ is the smallest integer such that $K(n)\wedge X\neq 0$.  Define $\mathcal{C}_n = \<K(n-1)\>\cap \mathcal{F}$.  Then every thick subcategory of $\mathcal{F}$ is $\mathcal{C}_n$ for some $n$.  It follows that any two spectra of type $n$ generate the same thick subcategory, and hence Bousfield class; let $F(n)$ denote a generic type $n$ spectrum.

Given a type $n$ spectrum $X$, there is a $v_n$ self-map $f:\Sigma^d X\rightarrow X$ with $[S^0, K(n)\wedge f]_i$ an isomorphism for all $i$ and $[S^0, K(m)\wedge f]_j=0$ for all $j$ and $m\neq n$.  We define $f^{-1}X$ to be the telescope, i.e.\ sequential colimit, i.e.\ homotopy colimit, of the diagram  $X\rightarrow \Sigma^{-d} X\rightarrow \cdots$.  By the periodicity theorem, any choice of $v_n$ self-map $f$ yields an isomorphic telescope.  The telescopes of different type $n$ spectra are Bousfield equivalent; denote this class by $\<T(n)\>$.

As mentioned above, localization away from a finite spectrum $F(n+1)$ exists, and is smashing.  This localization functor is denoted $L_n^f$, and is the same as homological localization at $T(0)\vee\cdots\vee T(n)$.

Let $E(n)$ denote the Johnson-Wilson spectrum; this is a ring spectrum with $\<E(n)\> = \<K(0)\vee\cdots\vee K(n)\>$.  Define $L_n:\mathcal{S}\rightarrow \mathcal{S}$ to be homological localization at $E(n)$.  A deep theorem of Ravenel~\cite[Thm.~7.5.6]{[Rav92]} shows that $L_n$ is smashing for all $n$.  The $L_n^f$ and $L_n$ are the only known smashing localization functors on $\mathcal{S}$.

The $L_n^f$-acyclics are ${\sf loc}(F(n+1)) = {\sf loc}(\mathcal{C}_{n+1}) = {\sf loc}(\<K(n)\>\cap \mathcal{F}) = {\sf loc}(\<E(n)\>\cap \mathcal{F})$.  The $L_n$-acyclics are $\<E(n)\> = {\sf loc}(\<E(n)\>)$.  Thus every $L_n^f$-acyclic is $L_n$-acyclic, and we have $\<K(0)\vee\cdots\vee K(n)\>\leq \<T(0)\vee\cdots \vee T(n)\>$ for all $n$.  It follows that there is a natural map $L_n^f \rightarrow L_n$.

For convenience later, we collect some calculations in $\mathcal{S}$.

\begin{lemma} \label{calcs} In $\mathsf{BL}(\mathcal{S})$ we have the following.
\begin{enumerate}
    \item $\<F(m)\>\leq \<F(n)\>$ if and only if $m\geq n$.  For all $n$ and $m$, $\<F(m)\wedge F(n)\>\neq \<0\>$.  Furthermore, $\<F(n)\wedge F(n)\> = \<F(n)\>$ for all $n$.
		\item $\<F(m) \wedge T(n)\> = \<0\>$ when $m> n$, and $\<F(m) \wedge T(n)\> = \<T(n)\>$ when $m\leq n$.
		\item $\<T(m) \wedge T(n)\> = \<0\>$ when $m\neq n$, and $\<T(n) \wedge T(n)\> = \<T(n)\>$.
		\item $\<F(m) \wedge K(n)\> = \<0\>$ when $m > n$, and $\<K(n)\> = \<F(m) \wedge K(n)\> \leq \<F(m)\>$ when $m\leq n$.
		\item $\<T(m) \wedge K(n)\> = \<0\>$ when $m \neq n$, and $\<K(n)\> = \<T(n) \wedge K(n)\> \leq \<T(n)\>$.
		\item $\<K(m) \wedge K(n)\> = \<0\>$ when $m \neq n$, and $\<K(n) \wedge K(n)\> = \<K(n)\>$.
	\end{enumerate}
\end{lemma}

\begin{pf} Part $(1)$ is Theorem $14$ of~\cite{[HS]}, along with the observation that $\<F(n)\>$ is complemented by $\<L_{n-1}^fS^0\>$ and hence is in $\mathsf{DL}$.

Part (2) is in ~\cite[2.8(i)]{[RavLaTC]}, \cite[6.2]{[MS]}, and \cite[Sect.~5]{[HP]}.  Part $(3)$ is also in~\cite[Sect.~5]{[HP]}.

Part (4) follows from the definition of type $m$ spectra.  Since each $K(i)$ is a skew field object, $F(m)\wedge K(i)\neq \<0\>$ implies this $K(i)$-module object $F(m)\wedge K(i)$ is a wedge of suspensions of $K(i)$.

From the periodicity theorem, $T(m)$ has nonzero $K(m)$ homology, so $T(m)\wedge K(m)\neq 0$.  The rest of Part $(5)$ is in~\cite[Prop.~A.2.13]{[Rav92]}.  Finally, Part $(6)$ is well known.
\end{pf}

\vspace{22pt}
\section{Local versions of the telescope conjecture} \label{sect-TCs}
\vspace{22pt}

In this section, let $L=L_Z:\mathcal{S}\rightarrow \mathcal{S}$ be a localization functor for some $Z\in\mathcal{S}$, and let $\mathcal{L}=\mathcal{L}_Z$ denote the category of $L$-locals.  First we state the various versions of the original telescope conjecture on $\mathcal{S}$.

\begin{definition} Fix an integer $n\geq 0$. On $\mathcal{S}$, we have the following versions of the telescope conjecture.\\

$\begin{array}{ll}
{\sf TC1_n}: & \<T(n)\> = \<K(n)\>. \\
{\sf TC2_n}: & L_n^f X\stackrel{\sim}{\rightarrow} L_n X \mbox{ for all } X. \\
{\sf TC3_n}: & \mbox{If }X\mbox{  is type }n\mbox{  and }f\mbox{  is a }v_n\mbox{  self-map, then }L_nX\cong f^{-1}X.\\
{\sf GSC}: & \mbox{Every smashing localization is generated by a set of compact objects.}\\
{\sf SDGSC}: & \mbox{Every smashing localization is generated by a set of }\\
 & \mbox{     strongly dualizable objects.}\\
\end{array}$
\end{definition}

\begin{theorem} \label{TCs} On the category $\mathcal{S}$ we have ${\sf TC1_n} \Leftrightarrow {\sf TC3_n}.$

Also, ${\sf TC2_n}$ holds if and only if ${\sf TC1_i}$ holds for all $i\leq n$.

Given ${\sf TC2_{n-1}}$ and ${\sf TC1_n}$, then ${\sf TC2_n}$ holds.

Furthermore, ${\sf GSC} \Leftrightarrow {\sf SDGSC}$, and this implies ${\sf TC2_n}$ for all $n$.
\end{theorem}

\begin{remark} Note that if we quantify over all $n$, the first three versions of the telescope conjecture are equivalent.  That is,
$$ {\sf TC1_n} \mbox{ for all }n \Leftrightarrow{\sf TC2_n} \mbox{ for all }n \Leftrightarrow {\sf TC3_n} \mbox{ for all }n .$$
\end{remark}

\begin{remark} \label{just-s} ${\sf TC2_n}$ holds if and only if $L_n^f S^0\stackrel{\sim}{\rightarrow} L_n S^0$.  Indeed, since both $L_n^f$ and $L_n$ are smashing, the subcategory of objects $W$ such that $L_n^f W \stackrel{\sim}{\rightarrow} L_n W$ is localizing.  Thus if it contains $S^0$, it contains ${\sf loc}(S^0) = \mathcal{S}$.
\end{remark}

\begin{pf} First we show the equivalence of ${\sf TC1_n}$ and ${\sf TC3_n}$.  This is also sketched in~\cite[1.13]{[MRS]}.  For any type $n$ spectrum $Y$, ${\sf th}(Y) = {\sf th}(F(n))$ and so we have ${\sf th}(L_n Y) = {\sf th} (L_n F(n))$, and $\<L_n Y\> = \<L_n F(n)\>$.  A construction in~\cite[8.3]{[Rav92]} gives a type $n$ spectrum $Y$ with $L_n Y\in {\sf th}(K(n))$.  Thus $\<L_n Y\>\leq \<K(n)\>$, and $0\neq \<L_n F(n)\> = \<K(n)\>$.  Suppose ${\sf TC3_n}$ holds.  Then $\<L_n Y\> = \<f^{-1} Y\> = \<T(n)\>$, and so $\<T(n)\> = \<K(n)\>$.

If $X$ is type $n$ and $f$ is a $v_n$ self-map, then~\cite[Prop.~3.2]{[MS]} implies that $L_n^f X\cong L_{T(n)}X\cong f^{-1}X$.  Thus assuming ${\sf TC1_n}$, we have $L_{K(n)}X\cong f^{-1}X$, and to prove ${\sf TC3_n}$ it suffices to show that $L_n X\cong L_{K(n)}X$.  This is known (see e.g.~\cite{[Hov95a]}), but we will give a proof that extends well to the localized setting.  Since $\<K(n)\>\leq \<E(n)\>$, localization at $K(n)$ gives a map $L_n X\rightarrow L_{K(n)}X$.  It suffices to show that this is an $L_n$-equivalence.  The fiber is $K(n)$ acyclic, so  $L_n X\wedge K(n) \rightarrow L_{K(n)}X\wedge K(n)$ is an isomorphism.  Consider $i<n$.  The triangle $C_n X \wedge K(i)\rightarrow X\wedge K(i)\rightarrow L_n X\wedge K(i)$ shows that $L_n X\wedge K(i)$ is zero, because $X$ is type $n$ and $C_n X$ is $K(i)$ acyclic.  Lemma $3.3.1$ in~\cite{[HPS]} states that $LW = LS^0\wedge W$ for any localization $L$ and strongly dualizable $W$.  Since every finite spectrum is strongly dualizable, $L_{K(n)}X\wedge K(i) = L_{K(n)}S^0\wedge X\wedge K(i) = 0$.  Thus $L_n X\wedge K(i) \rightarrow L_{K(n)}X\wedge K(i)$ is an isomorphism for all $i\leq n$, and hence $L_n X\rightarrow L_{K(n)}X$ is an $L_n$-equivalence.

For the second statement, note that ${\sf TC2_n}$ is equivalent to the statement $\<T(0)\vee\cdots\vee T(n)\> = \<K(0)\vee\cdots\vee K(n)\>$.  Smashing this with $\<T(i)\>$, for $0\leq i\leq n$, and using Lemma~\ref{calcs}, yields ${\sf TC1_i}$ for each $i$.  The third statement is also clear from this observation.

Finally, ${\sf GSC}\Leftrightarrow {\sf SDGSC}$ because objects in $\mathcal{S}$ are compact if and only if they are strongly dualizable.  Given ${\sf GSC}$, consider $L_n$.  The ${\sf GSC}$ would imply that the $L_n$-acyclics are ${\sf loc}(\<E(n)\>\cap\mathcal{F})$.  As observed earlier, this is the same as ${\sf loc}(F(n+1))$, which are the $L_n^f$-acyclics.  Therefore we would have $L_n^f\cong L_n$. \end{pf}

The ${\sf SDGSC}$ is new, and we will discuss it first.  As mentioned in the above proof, in $\mathcal{S}$ compactness is equivalent to strong dualizability.  It is well known that localization away from a set of compact objects is smashing.  The ${\sf GSC}$ is precisely the statement that the converse holds.  However, as we will show next, one only needs strong dualizability to generate a smashing localization functor.  We will state our result in slightly more general terms.

\begin{theorem} \label{sdsl} Let ${\sf T}$ be a well generated tensor triangulated category such that  ${\sf loc}(\boldone) = {\sf T}$, as in Definition~\ref{cat}.  Let $A=\{B_\alpha\}$ be a (possibly infinite) set of strongly dualizable objects.  Then there exists a smashing localization functor $L:{\sf T}\rightarrow {\sf T}$ with Ker~$L = {\sf loc}(A)$.
\end{theorem}

\begin{pf}  Let $E=\vee_\alpha B_\alpha$ and note that ${\sf loc}(E) = {\sf loc}(A)$.  The category ${\sf T}$ is well generated by hypothesis.  The localizing subcategory ${\sf S}={\sf loc}(E)$ is also well generated, by~\cite[Rmk.~2.2]{[IK]}, and is tensor-closed by Lemma~\ref{useful}.

By~\cite[Prop.~2.1]{[IK]} there exists a localization functor $L:{\sf T}\rightarrow {\sf T}$ with Ker~$L= {\sf S}$.  We will show that $L$ is a smashing localization.

First we claim that the $L$-locals are tensor-closed.  For any $Y\in {\sf T}$, we have the following.
$$\begin{array}{ll}
Y \mbox{ is }L\mbox{-local} & \Leftrightarrow [W,Y]_n=0 \mbox{ for all }W\in{\sf S} \mbox{ and all }n\in\mathbb{Z}\\
 & \Leftrightarrow [E, Y]_n=\prod[B_\alpha, Y]_n=0 \mbox{ for all }n\in\mathbb{Z}\\
 & \Leftrightarrow [B_\alpha, Y]_n=0 \mbox{ for all }\alpha \mbox{ and }n\in\mathbb{Z}\\
 & \Leftrightarrow DB_\alpha\wedge Y=0 \mbox{ for all }\alpha\\
\end{array}$$

The second equivalence follows from the fact that $\{X\;|\; [X,Y]_n=0 \mbox{ for all }n\in\mathbb{Z}\}$ is a localizing subcategory containing $E$, hence all of ${\sf S}$.  The final equivalence uses the fact that the $B_\alpha$ are strongly dualizable.

Now suppose $Y$ is $L$-local and $X$ is arbitrary.  Then $DB_\alpha\wedge Y=0$ for all $\alpha$, so $DB_\alpha\wedge Y\wedge X=0$ for all $\alpha$, and thus $Y\wedge X$ is $L$-local.  This shows that the $L$-locals are tensor-closed.

Consider the localization triangle $C\boldone\rightarrow \boldone \rightarrow L\boldone$, where $L\boldone$ is $L$-local and $C\boldone\in{\sf S}$.  For arbitrary $X\in {\sf T}$, tensoring gives an exact triangle
$$ C\boldone \wedge X \rightarrow X\rightarrow L\boldone\wedge X.$$

The object $L\boldone\wedge X$ is $L$-local, since the locals are tensor-closed.  Likewise, $C\boldone\wedge X\in {\sf S}$, since ${\sf S}$ is tensor-closed and so $L(C\boldone\wedge X)=0$.  Therefore $X\rightarrow L\boldone \wedge X$ is an $L$-equivalence from $X$ to an $L$-local object, and it follows that $LX\cong L\boldone\wedge X$.  This shows that $L$ is a smashing localization. \end{pf}

In the  stable homotopy category, and more generally whenever $\boldone\in {\sf T}$ is compact, this gives nothing new; by~\cite[Thm.~2.1.3(d)]{[HPS]} compact and strongly dualizable are equivalent.  Consider, however, the harmonic category, which has no nonzero compact objects.  In Section~\ref{sect-harm} we classify all smashing localizations on the harmonic category; they are indexed by $\mathbb{N}$.  Thus the ${\sf GSC}$ fails in the harmonic category, but as we show in Theorem~\ref{harm-sdgsc}, the ${\sf SDGSC}$ holds.  In fact, in the following sections we will give several examples of categories where the ${\sf GSC}$ fails but the ${\sf SDGSC}$ holds.

On the other hand, we don't expect the ${\sf SDGSC}$ to hold in complete generality, since Keller's counterexample~\cite{[keller]} to the ${\sf GSC}$ is also a counterexample to the ${\sf SDGSC}$; in the derived category of a ring $R$, the unit $R$ is compact and strongly dualizable, so the ${\sf GSC}$ and ${\sf SDGSC}$ are equivalent.

The ${\sf GSC}$ and ${\sf SDGSC}$ make sense in any localized category, but ${\sf TC1_n}$, ${\sf TC2_n}$, and ${\sf TC3_n}$ may not, since $T(n)$ and $K(n)$ may not be objects in $\mathcal{L}$.  Instead we make the following definitions.

\begin{definition} Let $L:\mathcal{S}\rightarrow \mathcal{S}$ be a localization, and $\mathcal{L}$ the category of $L$-locals. \label{defn-l}
 \begin{enumerate}
	\item Let ${\it l}_n^f:\mathcal{L}\rightarrow \mathcal{L}$ denote localization at $\<LT(0)\vee LT(1)\vee\cdots \vee LT(n)\>$.
	 \item Let ${\it l}_n: \mathcal{L}\rightarrow \mathcal{L}$ denote localization at $\<LK(0)\vee LK(1)\vee\cdots \vee LK(n)\>$.
	\end{enumerate}
\end{definition}

Before stating and proving a local version of Theorem~\ref{TCs}, we establish some results about ${\it l}_n^f$ and ${\it l}_n$.  First we make an observation about calculations in $\mathsf{BL}(\mathcal{L})$

\begin{lemma} \label{L-calcs} All the calculations in Lemma~\ref{calcs} are valid in $\mathsf{BL}(\mathcal{L})$ if we replace $F(n)$, $T(n)$, and $K(n)$ with $LF(n)$, $LT(n)$, and $LK(n)$.
\end{lemma}

\begin{pf} This follows from Theorem~\ref{local-cat} and the statements in Lemma~\ref{calcs}.
\end{pf}

\begin{proposition} \label{lnf-smash} ${\it l}_n^f$ is localization away from $LF(n+1)$, and hence is smashing.
\end{proposition}

\begin{pf} By Theorem~\ref{sdsl} we know that there is some smashing localization functor ${\it l}: \mathcal{L}\rightarrow \mathcal{L}$ that is localization away from $LF(n+1)$; we wish to show ${\it l}={\it l}_n^f$.  Let $\boldone=LS^0$ for simplicity of notation, and let ${\it c}$ denote the colocalization corresponding to ${\it l}$.  We claim that the ${\it l}$-acyclics are precisely ${\sf loc}({\it c}\boldone)$.  Clearly ${\it c}\boldone$ is ${\it l}$-acyclic, and these are a localizing subcategory, so ${\sf loc}({\it c}\boldone)\subseteq \{{\it l}\mbox{-acyclics}\}$.  On the other hand, suppose $W$ is ${\it l}$-acyclic.  Because ${\it l}$ is smashing, $LW = W\wedge {\it l}\boldone=0$, so $W=W\wedge c\boldone$.  Then since $W\in {\sf loc}(\boldone) = \mathcal{L}$, we have $W=W\wedge {\it c}\boldone\in{\sf loc}(\boldone\wedge{\it c}\boldone)={\sf loc}({\it c}\boldone)$, proving the claim.

By definition, the ${\it l}$-acyclics are also given by ${\sf loc}(LF(n+1))$.  Therefore $\<LF(n+1)\> = \<{\it c}\boldone\>$.

The class $\<F(n+1)\>$ is complemented by $\<T(0)\vee\cdots\vee T(n)\>$ in $\mathsf{BL}(\mathcal{S})$~\cite[Sect.~5]{[HP]}, so $\<LF(n+1)\>$ is complemented by $\<LT(0)\vee\cdots\vee LT(n)\>$ in $\mathsf{BL}(\mathcal{L})$.  At the same time, $\<{\it c}\boldone\>$ is complemented by $\<{\it l}\boldone\>$, and complements are unique.  We conclude that $\{{\it l}\mbox{-acylics}\} = \<{\it l}\boldone\> = \<LT(0)\vee\cdots\vee LT(n)\>$.  Since ${\it l}$ and ${\it l}_n^f$ are two localizations on $\mathcal{L}$ with the same acyclics, they are equal. \end{pf}

\begin{lemma} \label{sm-compose} If $L$ is smashing, then ${\it l}_n^f = LL_n^f=L_n^f L$ and ${\it l}_n = LL_n=L_n L$, and both are smashing.
\end{lemma}

\begin{pf} Smashing localization functors always commute, and compose to give a smashing localization.  The functor $LL_n:\mathcal{S}\rightarrow\mathcal{S}$, sending $X\mapsto L(L_n S^0\wedge X) =  LS^0\wedge L_n S^0 \wedge X$ is a smashing localization.  Since $LL_n$-locals are $L$-local, it also gives a smashing localization on $\mathcal{L}$.  The acyclics of this functor are $\<LL_n S^0\>$ in $\mathsf{BL}(\mathcal{L})$, which is $\<LK(0)\vee\cdots\vee LK(n)\>$.  Thus $LL_n$ and ${\it l}_n$ are localizations on $\mathcal{L}$ with the same acyclics, hence isomorphic.  The same proof works for ${\it l}_n^f = LL_n^f$. \end{pf}

In $\mathcal{S}$, for a type $n$ finite spectrum $X$ with a $v_n$ map $f:\Sigma^dX\rightarrow X$ and telescope $f^{-1}X$, it is known~\cite[Prop.~3.2]{[MS]} that $L_n^f X\cong L_{T(n)}X\cong f^{-1}X$.  The following proposition shows that the local version of this result holds as well.

\begin{lemma} \label{ugly} Let $L:\mathcal{S}\rightarrow \mathcal{S}$ be a localization, and ${\it l}_n^f$, $X$ and $f^{-1}X$ as above.  Then $${\it l}_n^f (LX)\cong L_{LT(n)}(LX)\cong L(f^{-1}X).$$
\end{lemma}

\begin{pf}  The proof parallels the~\cite{[MS]} result, one must only check that everything works when localized.  If $L_{LT(n)}(LX)\cong L(f^{-1}X)$ holds for a single type $n$ spectrum, then it holds for all type $n$ spectra.  So without loss of generality, we can choose $X$ to be a type $n$ spectrum that is a ring object in $\mathcal{S}$.  Then for any $v_n$ self-map $f$, the telescope $f^{-1}X$ is also a ring object~\cite[Lemma~2.2]{[MS]}.  By Lemma~\ref{useful}, $L(f^{-1}X)$ is a ring object in $\mathcal{L}$, and hence is local with respect to itself.

Lemma~2.2 in~\cite{[MS]} shows that $X\wedge f^{-1}X \cong f^{-1}X\wedge f^{-1}X$ in $\mathcal{S}$, so $LX \wedge_\mathcal{L} L(f^{-1}X) \cong L(f^{-1}X)\wedge_\mathcal{L} L(f^{-1}X)$ in $\mathcal{L}$ and the canonical map $LX\rightarrow L(f^{-1}X)$ is an $L(f^{-1}X)$-equivalence.  It follows that $L_{LT(n)}(LX)\cong L(f^{-1}X)$.

Since $\<LT(n)\>\leq \<LT(0)\vee\cdots\vee LT(n)\>$, $L(f^{-1}X)$ is ${\it l}_n^f$-local.  One then uses Lemma~\ref{L-calcs} to see that $LX\rightarrow L(f^{-1}X)$ is a ${\it l}_n^f$-equivalence, and so ${\it l}_n^f(LX) = {\it l}_n^f(L(f^{-1}X))=L(f^{-1}X)$. \end{pf}

\begin{definition} Let $L:\mathcal{S}\rightarrow \mathcal{S}$ be a localization, and consider the category $\mathcal{L}$ of locals.  Fix an $n\geq 0$.  We have the following versions of the telescope conjecture on $\mathcal{L}$.  \\

$\begin{array}{ll}
{\sf LTC1_n}: & \<LT(n)\> = \<LK(n)\>. \\
{\sf LTC2_n}: & {\it l}_n^f X\stackrel{\sim}{\rightarrow} {\it l}_n X \mbox{ for all } X. \\
{\sf LTC3_n}: & \mbox{If }X\in \mathcal{S} \mbox{  is type }n\mbox{  and }f\mbox{  is a }v_n\mbox{  self-map, then }{\it l}_n (LX)\cong L(f^{-1}X).\\
{\sf GSC}: & \mbox{Every smashing localization is generated by a set of compact objects.}\\
{\sf SDGSC}: & \mbox{Every smashing localization is generated by a set of }\\
 & \mbox{     strongly dualizable objects.}\\
\end{array}$
\end{definition}

\begin{theorem} \label{LTCs} On the category $\mathcal{L}$ we have ${\sf LTC1_n}\Rightarrow {\sf LTC3_n}.$

Also, ${\sf LTC2_n}$ holds if and only if ${\sf LTC1_i}$ holds for all $i\leq n$.

Given ${\sf LTC2_{n-1}}$ and ${\sf LTC1_n}$, then ${\sf LTC2_n}$ holds.
\end{theorem}

\begin{pf} Note that ${\sf LTC2_n}$ is equivalent to the statement $\<LT(0)\vee\cdots\vee LT(n)\> = \<LK(0)\vee\cdots\vee LK(n)\>$, so the last two statements are clear.  We will show that ${\sf LTC1_n}\Rightarrow {\sf LTC3_n}$, by mimicking the proof in Theorem~\ref{TCs}.

If $X$ is type $n$ and $f$ is a $v_n$ self-map, Lemma~\ref{ugly} shows that ${\it l}_n^f (LX)\cong L_{LT(n)}(LX)\cong L(f^{-1}X)$.  Then ${\sf LTC1_n}$ implies $L_{LK(n)}LX\cong L(f^{-1}X)$.  So it suffices to show that ${\it l}_n (LX) = L_{LK(n)} LX$.  We must show that the map $L_{LK(n)}: {\it l}_n (LX) \rightarrow L_{LK(n)} LX$ is an ${\it l}_n$-equivalence.  The same reasoning as in Theorem~\ref{TCs}, along with the computations of Lemma~\ref{L-calcs} and some definition unwinding, gives us that ${\it l}_n (LX)\wedge LK(i) \rightarrow L_{LK(n)} LX\wedge LK(i)$ is an equivalence for all $i\leq n$; we only need to notice that Lemma $3.3.1$ in~\cite{[HPS]} applies to strongly dualizable objects, and $LX$ is strongly dualizable.  \end{pf}

\begin{theorem} \label{LTCstwo} Furthermore, if $L:\mathcal{S}\rightarrow \mathcal{S}$ is a smashing localization, then on the category $\mathcal{L}$ of locals we have ${\sf LTC1_n}\Leftarrow {\sf LTC3_n},$ and
$${\sf GSC} \Leftrightarrow {\sf SDGSC} \Rightarrow {\sf LTC2_n} \mbox{ for all }n.$$
\end{theorem}

\begin{remark} In this case, ${\sf LTC2_n}$ is equivalent to ${\it l}_n^f (LS^0)\stackrel{\sim}{\rightarrow} {\it l}_n (LS^0)$, since by Lemma~\ref{sm-compose} both ${\it l}_n^f$ and ${\it l}_n$ are smashing, so the argument in Remark~\ref{just-s} applies.
\end{remark}

\begin{pf}  By~\cite[Thm.~2.1.3(d)]{[HPS]}, the compact objects and strongly dualizable objects in $\mathcal{L}$ coincide.  Thus ${\sf GSC}\Leftrightarrow{\sf SDGSC}$, and this implies ${\sf LTC2_n}$ just as in Theorem~\ref{TCs}.

Suppose $X$ has type $n$ and ${\sf LTC3_n}$ holds.  As in the proof of Theorem~\ref{TCs}, $\<L_n F(n)\>=\<K(n)\>$ in $\mathsf{BL}(\mathcal{S})$, so $\<LL_n LF(n)\> = \<LK(n)\>$ in $\mathsf{BL}(\mathcal{L})$.  By Lemma~\ref{sm-compose}, we have $\<{\it l}_n LF(n)\> = \<LK(n)\>$.  Then ${\sf LTC3_n}$ implies that $\<LT(n)\> = \<L(f^{-1}X\> = \<{\it l}_n(LX)\> = \<{\it l}_n LF(n)\>$, so ${\sf LTC1_n}$ holds.
\end{pf}

\begin{question} Is ${\it l}_n:\mathcal{L}\rightarrow\mathcal{L}$ always a smashing localization?
\end{question}

This is the case in all the local categories investigated in this paper, whether or not $L:\mathcal{S}\rightarrow \mathcal{S}$ is a smashing localization.  If one could show ${\it l}_n$ is always smashing, then most likely on $\mathcal{L}$ one would have ${\sf SDGSC}\implies {\sf LTC2_n} \forall n$.

We would of course like to know if and when information on localized telescope conjectures can help with those in the original category $\mathcal{S}$, where all versions remain open.

\begin{proposition} \label{implications} Let $L:\mathcal{S}\rightarrow \mathcal{S}$ be a localization, with localized category $\mathcal{L}$.
\begin{enumerate}
  \item If ${\sf TC1_n}$ holds on $\mathcal{S}$, then ${\sf LTC1_n}$ holds on $\mathcal{L}$.
  \item If ${\sf TC2_n}$ holds on $\mathcal{S}$, then ${\sf LTC2_n}$ holds on $\mathcal{L}$.
	\item If ${\sf TC3_n}$ holds on $\mathcal{S}$, then ${\sf LTC3_n}$ holds on $\mathcal{L}$.
\end{enumerate}

\noindent Furthermore, if $L$ is a smashing localization then we have the following.

\begin{enumerate}
	\item[(4)] If ${\sf GSC}$ holds on $\mathcal{S}$, then ${\sf GSC}$ holds on $\mathcal{L}$.
	\item[(5)] If ${\sf SDGSC}$ holds on $\mathcal{S}$, then ${\sf SDGSC}$ holds on $\mathcal{L}$.
\end{enumerate}
\end{proposition}

\begin{pf} Part $(1)$ follows immediately from Proposition~\ref{lattice-map}.  So does Part $(2)$, since ${\sf TC2_n}$ is equivalent to the statement $\<T(0)\vee\cdots\vee T(n)\>=\<K(0)\vee\cdots\vee K(n)\>$ and similarly for ${\sf LTC2_n}$.  From this and Theorems~\ref{TCs} and~\ref{LTCs} we have ${\sf TC3_n}\Leftrightarrow {\sf TC1_n}\Rightarrow {\sf LTC1_n}\Rightarrow  {\sf LTC3_n}.$

Now suppose $L$ is smashing, and the ${\sf GSC}$ holds on $\mathcal{S}$.  Let ${\it l}:\mathcal{L}\rightarrow \mathcal{L}$ be a smashing localization.  Thus ${\it l}$ is defined by ${\it l}(LY) = {\it l}(LS^0)\wedge_\mathcal{L} LY = {\it l} S^0\wedge LS^0\wedge Y$.  We can therefore extend ${\it l}$ to be a smashing localization on all of $\mathcal{S}$, with $X\mapsto {\it l} S^0\wedge LS^0\wedge X = {\it l} LS^0\wedge X$.  Since the ${\sf GSC}$ holds on $\mathcal{S}$ by assumption, the acyclics of this functor are $\<{\it l} LS^0\> = {\sf loc}(A)$, for some set of compact objects $A$ in $\mathcal{S}$.  Here $\<{\it l} LS^0\>$ refers to the Bousfield class in $\mathsf{BL}(\mathcal{S})$.

We must show that $\<{\it l} LS^0\>$ in $\mathsf{BL}(\mathcal{L})$ is generated by a set of objects that are compact in $\mathcal{L}$.  Note that $\<{\it l} LS^0\>$ in $\mathsf{BL}(\mathcal{L})$ is $\{LW\;|\; LW\wedge_\mathcal{L} {\it l} LS^0=0\} = \{LW\;|\; LW\wedge_\mathcal{S} {\it l} LS^0=0\} = \<{\it l} LS^0\> \cap \mathcal{L}$, where the latter $\<{\it l} LS^0\>$ is in $\mathsf{BL}(\mathcal{S})$.  Therefore $\<{\it l} LS^0\>$ in $\mathsf{BL}(\mathcal{L})$ is ${\sf loc}(A)\cap \mathcal{L}$.  We claim that this is ${\sf loc}(L(A))$.  Since $L$ sends compacts to compacts, this will show that ${\it l}$ is generated by a set of compacts.

If $X\in {\sf loc}(A)$, then $LX\in {\sf loc}(L(A))$.  If $X\in \mathcal{L}$ in addition, then $X\cong LX\in {\sf loc}(L(A))$.  For the other inclusion, note that the intersection of two localizing subcategories is a localizing subcategory, and $\mathcal{L}$ is a localizing subcategory of $\mathcal{S}$ because $L$ is smashing.  For $Y\in A$, $LY$ is in $\mathcal{L}$, and $LY = LS^0\wedge Y\in {\sf loc}(Y)\subseteq {\sf loc}(A)$.  Therefore $L(A)\subseteq {\sf loc}(A)\cap \mathcal{L}$, and ${\sf loc}(L(A))={\sf loc}(A)\cap \mathcal{L}$.

Part $(5)$ follows immediately, since if $L$ is smashing then ${\sf GSC}\Leftrightarrow {\sf SDGSC}$ in both $\mathcal{S}$ and $\mathcal{L}$. \end{pf}

Balmer and Favi~\cite[Prop.~4.4]{[BF]} have also recently proved Part $(4)$ in the slightly more general setting of a smashing localization on a unital algebraic stable homotopy category; the above proof would apply there as well.  One would like to prove Part $(5)$ without the assumption that $L$ is smashing, but it's not clear if this is possible.

Letting $L=L_Z$, for $Z = \bigvee_{i\geq 0}K(i)$, $E(n)$, $K(n)$, $BP$, $H\mathbb{F}_p$, or $I$ provides interesting examples of categories $\mathcal{L}$ on which to investigate these telescope conjectures.  Furthermore, ${\sf LTC1_n}$ suggests the relevance of Bousfield lattices to understanding these questions.  In the remaining sections, we focus on specific localized categories.

\vspace{22pt}
\section{The Harmonic category} \label{sect-harm}
\vspace{22pt}

Let $Q=\bigvee_{i\geq 0}K(i)$, and $L=L_Q:\mathcal{S}\rightarrow \mathcal{S}$, and consider the harmonic category $\mathcal{H}$ of $L$-locals.  Harmonic localization is not smashing.  An object is called {\it harmonic} if it is $L$-local, and {\it dissonant} if it is $L$-acyclic.  For example, finite spectra, suspension spectra, finite torsion spectra, and $BP$ are known to be harmonic~\cites{[Hov95a],[Rav84]}.  On the other hand, $I$ and ${H\bf_p}$ are dissonant.

In order to answer the telescope conjectures in $\mathcal{H}$, we will first calculate the Bousfield lattice of $\mathcal{H}$.  In this section all smash products are in $\mathcal{H}$, unless otherwise noted.  Given any set $P$, let $2^P$ denote the power set of $P$.

\begin{definition} Given $X\in \mathcal{H}$, define the {\it support} of $X$ to be $${\sf supp}(X) = \{i\;|\; X\wedge K(i)\neq 0\}\subseteq \mathbb{N}.$$
\end{definition}

The following result and proof was pointed out to us by Jon Beardsley.

\begin{proposition} \label{jb} The Bousfield lattice of $\mathcal{H}$ is $2^\mathbb{N}$.
\end{proposition}

\begin{pf} Each $K(n)$ is a ring object, hence $K(n)$-local by Lemma~\ref{useful}.  Because $\<K(n)\>\leq \<Q\>$, $K(n)$-locals are harmonic, thus each $K(n)$ is harmonic.  The argument hinges on the fact that $K(n)$ is a skew field object in $\mathcal{H}$: for $X=LX$ in $\mathcal{H}$, if $X\wedge K(n)\neq 0$ then $X\wedge K(n) = L(X\wedge_\mathcal{S} K(n))$ so $X\wedge_\mathcal{S} K(n)\neq 0$, and $X\wedge_\mathcal{S} K(n)$ must be a nonempty wedge of suspensions of $K(n)$'s.  Thus
$$X\wedge K(n) = L(X\wedge_\mathcal{S} K(n)) = L(\vee \Sigma^i K(n)) = L(\vee \Sigma^i LK(n)) = \coprod_\mathcal{L} \Sigma^i LK(n) = \coprod_\mathcal{L} \Sigma^i K(n).$$

It follows that $LX\wedge K(n)=0$ if and only if $LX\wedge_\mathcal{S} K(n)=0$.  Furthermore, if $LX\wedge K(n)\neq 0$, then $\<LX\wedge K(n)\> = \<K(n)\>$, where these are Bousfield classes in $\mathsf{BL}(\mathcal{H})$.

By the definition of $L$, for any $W\in \mathcal{S}$, if $W\wedge_\mathcal{S} K(n) =0$ for all $n$, then $LW=0$.  Combining this with the above observation, we get that a local object $W=LW$ has $W\wedge K(n)=0$ in $\mathcal{H}$ for all $n$ if and only if $W=0$.

Therefore, for any $X,Y\in\mathcal{H}$, we have
$$Y\wedge X=0 \Leftrightarrow Y\wedge X\wedge K(n)=0 \;\forall n \Leftrightarrow Y\wedge K(n)=0 \mbox{ for all }n\in {\sf supp}(X).$$

We conclude that there is a lattice isomorphism $F:\mathsf{BL}(\mathcal{H})\rightarrow 2^\mathbb{N}$, given by the following.
$$ \<X\> = \bigvee_{{\sf supp}(X)} \<K(i)\> \mapsto {\sf supp}(X),$$
$$ N\subseteq\mathbb{N} \mapsto  \bigvee_{i\in N} \<K(i)\>.$$ \end{pf}

\begin{theorem} On $\mathcal{H}$, for all $n\geq 0$ we have that ${\sf LTC1_n}, {\sf LTC2_n}$, and ${\sf LTC3_n}$ hold. \label{LTCs-harm}
\end{theorem}

\begin{pf} By Lemmas~\ref{calcs} and~\ref{L-calcs}, $LT(n)$ and $LK(n) = K(n)$ have the same support.  The above theorem then implies that $\<LT(n)\> = \<LK(n)\>$.  Thus ${\sf LTC1_n}$ holds for all $n$, and the claim follows from Theorem~\ref{LTCs}.
\end{pf}

Next, we classify all smashing localizations on $\mathcal{H}$, and show that the ${\sf GSC}$ fails but the ${\sf SDGSC}$ holds.  The proof is based on that of~\cite[Thm.~6.14]{[HovS]}, which classifies smashing localizations in the $E(n)$-local category.

\begin{theorem} \label{harm-sdgsc} If $L':\mathcal{H}\rightarrow \mathcal{H}$ is a smashing localization functor, then $L'={\it l}_n^f$ for some $n\geq 0$, or $L'=0$ or $L'=id$.  Therefore the ${\sf GSC}$ fails but the ${\sf SDGSC}$ holds on $\mathcal{H}$.
\end{theorem}

\begin{pf} Let $L':\mathcal{H}\rightarrow \mathcal{H}$ be a smashing localization functor, and let $\boldone=LS^0$ be the unit in $\mathcal{H}$.  The acyclics of $L'$ are given by $\<L'\boldone\>$.  From Proposition~\ref{jb}, $\<L'\boldone\>$ is equal to the wedge of $\<K(i)\>$ for all $i\in {\sf supp}(L'\boldone)$.  If ${\sf supp}(L'\boldone)=\emptyset$ then $\<L'\boldone\>=\<0\>$ and $L'=0$.

Assume now that ${\sf supp}(L'\boldone)$ is not empty, and take $j\in{\sf supp}(L'\boldone)$.  We will show that $\<L'\boldone\>\geq \<K(0)\vee\cdots\vee K(j)\>$.  It follows that either $\<L'\boldone\> = \bigvee_{i\geq 0}\<K(i)\> = \<\boldone\>$ and $L'=id$, or $L'={\it l}_n={\it l}_n^f$ for $n=max({\sf supp}(L'\boldone))$.

Since $\<K(j)\>\leq \<L'\boldone\>$, from Lemma~\ref{compose} we have $L_{K(j)}L' = L'L_{K(j)}=L_{K(j)}$.  Therefore $\<L_{K(j)}\boldone\> = \<L'\boldone\wedge L_{K(j)}\boldone\>\leq \<L'\boldone\>$.  Proposition 5.3 of~\cite{[HovS]} shows that in $\mathcal{S}$, $L_{K(j)}S^0\wedge_\mathcal{S} K(i)$ is nonzero for $0\leq i\leq j$ and zero for $i>j$.  Note that $L_{K(j)}S^0 = L_{K(j)}LS^0=L_{K(j)}\boldone$, and as remarked in the proof of Proposition~\ref{jb}, $LX\wedge K(i)=0$ if and only if $LX\wedge_\mathcal{S} K(i)=0$.  Therefore in $\mathsf{BL}(\mathcal{H})$ we have $\<L_{K(j)}\boldone\> = \<K(0)\vee\cdots\vee K(j)\>$, and so $\<L'\boldone\>\geq \<K(0)\vee\cdots\vee K(j)\>$ as desired.

Each ${\it l}_n^f$ is localization away from $LF(n+1)$ by Proposition~\ref{lnf-smash}, which is strongly dualizable by Theorem~\ref{local-cat}.  The identity is localization away from zero, and the zero functor is localization away from $LS^0$; these are both strongly dualizable.  Therefore the ${\sf SDGSC}$ holds.  On the other hand, Corollary B.13 in~\cite{[HovS]} shows that there are no nonzero compact objects in $\mathcal{H}$, so the ${\sf GSC}$ fails. \end{pf}

\begin{question} Classify localizing subcategories of $\mathcal{H}$.
\end{question}

It seems likely that every localizing subcategory of $\mathcal{H}$ is a Bousfield class, and so these are in bijection with $2^\mathbb{N}$, but we have been unable to prove this.

\vspace{22pt}
\section{The $E(n)$- and $K(n)$- local categories} \label{sect-enkn}
\vspace{22pt}

\subsection{The $E(n)$-local category} \label{sect-en}

Recall that $E(n)$ has $\<E(n)\> = \<K(0)\vee K(1)\vee\cdots \vee K(n)\>$.  In this section, fix $L=L_n = L_{E(n)}: \mathcal{S}\rightarrow \mathcal{S}$ and let $\mathcal{L}_n$ denote the local category.  The functor $L_n$ is smashing, and so by Theorem~\ref{cat} each $LF(i)$ is compact in $\mathcal{L}_n$. Hovey and Strickland~\cite{[HovS]} have studied $\mathcal{L}_n$ in detail, and determine the localizing subcategories, smashing localizations, and Bousfield lattice of $\mathcal{L}_n$.  We begin by recalling these results.

\begin{lemma} For $0\leq i\leq n$, we have $LK(i) = K(i)$, and for $i >n$ we have $LK(i)=0$.
\end{lemma}

\begin{pf}  This follows from $\<E(n)\> = \<K(0)\vee K(1)\vee\cdots \vee K(n)\>$. \end{pf}

\begin{theorem}~\cite[Thm.~6.14]{[HovS]} The lattice of localizing subcategories of $\mathcal{L}_n$, ordered by inclusion, is in bijection with the lattice of subsets of the set $\{0,1,...,n\}$, where a localizing subcategory ${\sf S}$ corresponds to
$$\{ i \;|\; K(i) \in {\sf S}\}.$$
\end{theorem}

\begin{corollary} Every localizing subcategory of $\mathcal{L}_n$ is a Bousfield class, in particular a localizing subcategory ${\sf S}$ is the Bousfield class $$\bigvee \< K(j)\;|\; K(j)\notin {\sf S}, 0\leq j\leq n\>.$$
\end{corollary}

\begin{corollary} \label{ln-bl} For every $n\geq 0$, there is a lattice isomorphism $$f_n: \mathsf{BL}(\mathcal{L}_n)\stackrel{\sim}{\longrightarrow} 2^{\{0,1,...,n\}}.$$
\end{corollary}

\begin{pf} The isomorphism is given by
$$ \<X\> = \bigvee_{X\wedge K(i)\neq 0} \<K(i)\> \mapsto \{ i\;|\; X\wedge K(i)\neq 0\},$$
$$ N\subseteq \{0,1,...,n\} \mapsto  \bigvee_{i\in N} \<K(i)\>.$$
\end{pf}

\begin{theorem}~\cite[Cor.~6.10]{[HovS]} If $L':\mathcal{L}_n\rightarrow \mathcal{L}_n$ is a smashing localization, then $L' = L_i=L_i^f$ for some $0\leq i\leq n$ or $L'=0$.  Thus the ${\sf GSC}$ holds on $\mathcal{L}_n$.
\end{theorem}

\begin{corollary} On $\mathcal{L}_n$, all of ${\sf LTC1_i}$, ${\sf LTC2_i}$, ${\sf LTC3_i}$ hold for all $i$, and ${\sf GSC}$ and  ${\sf SDGSC}$ also hold. \label{LTCs-en}
\end{corollary}

\begin{pf} This follows from the previous Theorem, and Theorem~\ref{LTCstwo}.  Note that for $i>n$, we have $LT(i)=0=LK(i)$ by Lemma~\ref{calcs}, and so ${\it l}_i={\it l}_n = {\it l}_n^f = {\it l}_i^f$.
\end{pf}

Recall that there is a natural map $L_nX\rightarrow L_{n-1}X$ for all $X$ in $\mathcal{S}$ and $n$, and by Proposition~\ref{lattice-map} this induces a surjective lattice map $\mathsf{BL}(\mathcal{L}_n)\rightarrow \mathsf{BL}(\mathcal{L}_{n-1})$, and an inverse system of lattice maps.

$$\cdots \rightarrow \mathsf{BL}(\mathcal{L}_n)\rightarrow \mathsf{BL}(\mathcal{L}_{n-1}) \rightarrow \cdots \rightarrow \mathsf{BL}(\mathcal{L}_1)\rightarrow \mathsf{BL}(\mathcal{L}_{0})$$

\begin{proposition} The lattice isomorphisms $F$ and $f_n$ from Proposition~\ref{jb} and Corollary~\ref{ln-bl} realize $\mathsf{BL}(\mathcal{H})$ as the inverse limit of the maps $\mathsf{BL}(\mathcal{L}_n)\rightarrow \mathsf{BL}(\mathcal{L}_{n-1})$. \label{realize}
\end{proposition}

\begin{pf} From Lemma~\ref{compose}, and the facts that $L_QK(i) = K(i)$ for all $i$, and $L_nK(i) = K(i)$ for $i\leq n$ and $L_nK(i)=0$ for $i>n$, we get the following diagram for all $n$.  The map $2^{\{0,1,...,n\}}\rightarrow 2^{\{0,1,...,n-1\}}$ is induced by sending $m\mapsto m$ for $m<n$ but $n\mapsto 0$, and the maps $2^\mathbb{N} \rightarrow 2^{\{0,1,...,i\}}$ are defined similarly.

$$
\UseComputerModernTips
\xymatrix{
 & \mathsf{BL}(\mathcal{H}) \ar@{->>}[dl] \ar@{->>}[dd] \ar@{<->}[rrr]_{F} &&& 2^\mathbb{N} \ar@{->>}[dl] \ar@{->>}[dd] \\
\mathsf{BL}(\mathcal{L}_n) \ar@{->>}[dr] \ar@{<->}[rrr]_{f_n} &&& 2^{\{0,1,...,n\}} \ar@{->>}[dr]\\
& \mathsf{BL}(\mathcal{L}_{n-1}) \ar@{<->}[rrr]_{f_{n-1}} &&& 2^{\{0,1,...,n-1\}}\\
}
$$
\end{pf}

\subsection{The $K(n)$-local category}  Although an incredibly complicated category in its own right, the $K(n)$-local category is quite basic from the perspective of localizing subcategories, Bousfield lattices, and telescope conjectures.  In this subsection, let $L=L_{K(n)}: \mathcal{S}\rightarrow \mathcal{S}$ be localization at $K(n)$, and let $\mathcal{K}_n$ denote the category of locals.  Hovey and Strickland classify the localizing subcategories of $\mathcal{K}_n$, and there are not many of them.\\

\begin{proposition}~\cite[Thm.~7.5]{[HovS]} There are no nonzero proper localizing subcategories of $\mathcal{K}_n$.
\end{proposition}

This Proposition implies that the Bousfield lattice of $\mathcal{K}_n$ is the two-element lattice $\{\<0\>, \<K(n)\>\}$.  We will prove a slightly more general result, that will be used again in Subsection~\ref{hfp}.

\begin{proposition} \label{sfobl} Consider a category ${\sf T}$ as in Definition~\ref{cat}, an object $Z$ in ${\sf T}$, and localization $L_Z:{\sf T}\rightarrow {\sf T}$ with localized category $\mathcal{L}_Z$.
\begin{enumerate}
   \item If $Z$ is a ring object, then $\<L_ZZ\>=\<Z\>$ is the maximum class in $\mathsf{BL}(\mathcal{L}_Z)$.
	 \item If $Z$ is a skew field object, then $\mathsf{BL}(\mathcal{L}_Z)$ is the two-element lattice $\{\<0\>, \<Z\>\}$.
\end{enumerate}
\end{proposition}

\begin{pf} For $(1)$, first note that Lemma~\ref{useful} implies $L_ZZ=Z$.  Consider $\<Z\>$ in $\mathsf{BL}(\mathcal{L}_Z)$.  By definition, this is the collection of all $W\in \mathcal{L}_Z$ with $L(Z\wedge_{\sf T} W)=0$.  But $Z\wedge_{\sf T} W$ is a $Z$-module object in ${\sf T}$, hence is $L_Z$-local.  The only object that is both local and acyclic with respect to any localization is zero, so $Z\wedge_{\sf T} W=0$.  But this says that $W$ is $L_Z$-acyclic, hence zero in $\mathcal{L}_Z$.  Therefore in $\mathsf{BL}(\mathcal{L}_Z)$ we have $\<Z\> = \{0\}$.

Now suppose $Z$ is a skew field object in ${\sf T}$.  In particular, it is a ring object, so $\<Z\>$ is the maximum class in $\mathsf{BL}(\mathcal{L}_Z)$.  Consider $\<LX\>$ in $\mathsf{BL}(\mathcal{L}_Z)$, for arbitrary $X\in {\sf T}$.  If $X\wedge_{\sf T} Z=0$, then $LX=0$.  Otherwise, $X\wedge_{\sf T} Z$ is a wedge of suspensions of $Z$, so $\<Z\> = \<X\wedge_{\sf T} Z\>\leq \<X\>$ in $\mathsf{BL}({\sf T})$.  Then $\<Z\> = \<L_ZZ\>\leq \<L_ZX\>$ in $\mathsf{BL}(\mathcal{L}_Z)$, so $\<L_ZX\>=\<Z\>$. \end{pf}

\begin{corollary} The Bousfield lattice of $\mathcal{K}_n$ is $\{\<0\>, \<K(n)\>\}$. \label{bl-kn}
\end{corollary}

\begin{theorem} In $\mathcal{K}_n$, all of ${\sf LTC1_i}$, ${\sf LTC2_i}$, ${\sf LTC3_i}$ hold for all $i$, and ${\sf GSC}$ and  ${\sf SDGSC}$ also hold. \label{LTCs-kn}
\end{theorem}

\begin{pf} In light of Theorem~\ref{LTCs}, we will show that ${\sf LTC1_i}$ holds for all $i$.  This follows from Lemma~\ref{calcs}: for $i\neq n$ we have $LT(i)=0=LK(i)$, but $LT(n)\neq 0$ so by the last corollary $\<LT(n)\> = \<K(n)\> = \<LK(n)\>$.

There are exactly two smashing localizations on $\mathcal{K}_n$.  The identity functor is smashing, and is localization away from $0$, which is compact and strongly dualizable.  The zero functor is smashing, and is localization away from $LS^0$, which is strongly dualizable.  It is not compact, but by Theorem 7.3 in~\cite{[HovS]}, $LF(n)$ is compact in $\mathcal{K}_n$ and ${\sf loc}(LF(n)) = {\sf loc}(LS^0)=\mathcal{K}_n$.  Therefore the zero functor is also generated by a compact object.  This shows that both the ${\sf GSC}$ and ${\sf SDGSC}$ hold. \end{pf}

\vspace{22pt}
\section{Other localized categories} \label{sect-other}
\vspace{22pt}

In this section we will consider several other localized categories.  In each case, let $L_Z:\mathcal{S}\rightarrow \mathcal{S}$ denote the localization functor that annihilates $\<Z\>$, and let $\mathcal{L}_Z$ denote the category of $L_Z$-locals.

\subsection{The $H\mathbb{F}_p$-local category} \label{hfp} The Eilenberg-MacLane spectrum ${H\bf_p}$ is a skew field object in $\mathcal{S}$; in fact, every skew field object in $\mathcal{S}$ is isomorphic to either ${H\bf_p}$ or a $K(n)$.  Unlike the $\<K(n)\>$, it is not complemented; for example, $\<I\>\leq \<{H\bf_p}\>$ but $I\wedge {H\bf_p}=0$.  So $\<{H\bf_p}\>\in\mathsf{DL}\backslash\mathsf{BA}$.  Hovey and Palmieri~\cite{[HP]} have conjectured several results about the collection of classes less than $\<{H\bf_p}\>$ in $\mathsf{BL}(\mathcal{S})$.  The telescope conjectures and Bousfield lattice of $\mathcal{L}_{H\bf_p}$ are quite simple.

\begin{theorem} In $\mathcal{L}_{{H\bf_p}}$, all of ${\sf LTC1_n}$, ${\sf LTC2_n}$, ${\sf LTC3_n}$ hold for all $n$. \label{LTCs-hfp}
\end{theorem}

\begin{pf} For all $n$, $K(n)\wedge {H\bf_p} = 0$ and $T(n)\wedge {H\bf_p} = 0$, by~\cite[p.~16]{[HP]}.  Therefore $LK(n)=0=LT(n)$ and ${\sf LTC1_n}$ holds for all $n$.  Note that ${\it l}_n = {\it l}_n^f$ is the zero functor for all $n$.  \end{pf}

In order to discuss the ${\sf GSC}$ and ${\sf SDGSC}$ in this category, we must classify the smashing localizations.  We will do this by using what we know about the Bousfield lattice.

\begin{proposition} \label{bl-hfp} The Bousfield lattice of $\mathcal{L}_{H\bf_p}$ is the two-element lattice $\{\<0\>, \<{H\bf_p}\>\}$.
\end{proposition}

\begin{pf} This follows immediately from Proposition~\ref{sfobl} because ${H\bf_p}$ is a skew field object in $\mathcal{S}$.
\end{pf}

Recall that every smashing localization gives a pair of complemented classes in $\mathsf{BA}\subseteq \mathsf{BL}$.  Thus in $\mathcal{L}_{H\bf_p}$ there are exactly two smashing localizations, the trivial ones given by smashing with zero and with the unit.

\begin{proposition} In $\mathcal{L}_{H\bf_p}$, the ${\sf GSC}$ fails but the ${\sf SDGSC}$ holds. \label{gsc-hfp}
\end{proposition}

\begin{pf} The identity functor is smashing, and is localization away from $0$, which is compact and strongly dualizable.  By~\cite[Cor.~B.~13]{[HovS]}, there are no nonzero compact objects in $\mathcal{L}_{H\bf_p}$.  So the zero functor, which is localization away from $LS^0$, is generated by a strongly dualizable object but not a compact one. \end{pf}

One application of this Bousfield lattice calculation is to the question of classifying localizing subcategories.  Every Bousfield class is a localizing subcategory.  In Conjecture 9.1 of~\cite{[HP]}, Hovey and Palmieri conjecture the converse holds, in the $p$-local stable homotopy category.  The original conjecture is still open, but the question can be asked in any well-generated tensor triangulated category.  For example, in a stratified category every localizing subcategory is a Bousfield class.  The question is interesting, since in general localizing subcategories are hard to classify.  In many cases, including $\mathcal{S}$, it is not even known if there is a set of localizing subcategories.  Recently Stevenson~\cite{[stevenson-counterexample]} found the first counterexample, in an algebraic setting: in the derived category of an absolutely flat ring that is not semi-artinian, there are localizing subcategories that are not Bousfield classes.  Now we show that $\mathcal{L}_{H\bf_p}$ provides another counterexample.

\begin{proposition} \label{loc-not-bc} In $\mathcal{L}_{H\bf_p}$ there are localizing subcategories that are not Bousfield classes.
\end{proposition}

\begin{pf} The following counterexample was suggested to us by Mark Hovey.  The Bousfield lattice of $\mathcal{L}_{H\bf_p}$ has only two elements: $\<0\> = \mathcal{L}_{H\bf_p}$ and $\boldone = \{0\}$.  It suffices to find a proper nonzero localizing subcategory in $\mathcal{L}_{H\bf_p}$.

Consider the Moore spectrum $M(p)$, defined by the triangle $S^0 \stackrel{p}{\rightarrow} S^0 \rightarrow M(p)$; this spectrum is ${H\bf_p}$-local.  Consider the following full subcategory in $\mathcal{L}_{H\bf_p}$.

$$\mathcal{A} = \{ X\in \mathcal{L}_{H\bf_p}\;|\; [X, M(p)]_n=0 \mbox{ for all } n\in\mathbb{Z}\}.$$

This is a localizing subcategory, called the cohomological Bousfield class of $M(p)$ and denoted $\<M(p)^*\>$ in~\cite{[HovCBC]}.  The spectrum ${H\bf_p}$ is a ring object, hence local with respect to itself.  As mentioned in Section~\ref{sect-harm}, it is known that ${H\bf_p}$ is dissonant and $M(p)$ is harmonic, so $[{H\bf_p}, M(p)]_n=0$ for all $n$, and ${H\bf_p}\in \mathcal{A}$.  On the other hand, the identity on $M(p)$ is nonzero, so $M(p)\notin \mathcal{A}$.  This shows that $\mathcal{A}$ is a localizing subcategory that is not a Bousfield class.

Another example comes from $Z=L_{H\bf_p}(BP)$.  Clearly $Z\notin \<Z^*\>$.  But $BP$ is also harmonic, so $[{H\bf_p}, BP]_n = 0$ and $[{H\bf_p}, Z]_n=0$ for all $n$, and ${H\bf_p}\in \<Z^*\>$.  Since $Z\in \<M(p)^*\>$, we know that $\<M(p)^*\> \neq \<Z^*\>$. \end{pf}

Both these counterexamples are cohomological Bousfield classes.  It would be interesting to find a localizing subcategory in $\mathcal{L}_{H\bf_p}$ that is not a cohomological Bousfield class, or show there are none.  Also, it is not clear what, if anything, the previous proposition might tell us about the original conjecture in $\mathcal{S}$.  For example, as localizing subcategories in $\mathcal{S}$, we have~\cite[3.3]{[HovCBC]} that $\<M(p)^*\> = \<I\>$.

\subsection{ The $I$-local category}

Recall by $I$ we mean the Brown-Comenetz dual of the sphere spectrum.  It is a rare example of a nonzero spectrum that squares to zero.  Hovey and Palmieri~\cite[Lemma~7.8]{[HP]} conjecture that $\<I\>$ is minimal in $\mathsf{BL}(\mathcal{S})$.

\begin{theorem} \label{i-loc-tc} On $\mathcal{L}_I$, for all $n$ we have that ${\sf LTC1_n}$, ${\sf LTC2_n}$, and ${\sf LTC3_n}$ all hold.
\end{theorem}

\begin{pf} By Lemma $7.1(c)$ of~\cite{[HP]}, $T(n)\wedge I=0$ for all $n$, so $LT(n)=0$.  Since $K(i)$ is a $BP$-module, and $BP\wedge I=0$ by~\cite[Cor.~B.11]{[HovS]}, we also have $K(n)\wedge I=0$ for all $n$.  Therefore $\<LT(n)\> = \<0\>=\<LK(n)\>$ for all $n$, and the rest follows from Theorem~\ref{LTCs}. \end{pf}

\begin{proposition} \label{bl-i} The Bousfield lattice of $\mathcal{L}_I$ is the two-element lattice $\{\<0\>, \<L_IS^0\>\}$.
\end{proposition}

\begin{pf} By~\cite[7.1(c)]{[HP]}, $\<I\><\<{H\bf_p}\>$.  Then Proposition~\ref{lattice-map} implies that there is a surjective lattice map from $\mathsf{BL}(\mathcal{L}_{H\bf_p}) = \{\<0\>, \<{H\bf_p}\>\}$ onto $\mathsf{BL}(\mathcal{L}_I)$.  Note that by Lemma~\ref{compose}, $\<L_I{H\bf_p}\> = \<L_IL_{H\bf_p} S^0\> = \<L_IS^0\>$.

It remains to show that $\<L_IS^0\>\neq \<0\>$.  But any $X$ in $\mathcal{S}$ with $X\wedge I\neq 0$ in $\mathcal{S}$ will have $L_IX\neq 0$ and $L_IX\notin \<L_IS^0\>$ in $\mathsf{BL}(\mathcal{L}_I)$; this is because $L_IX\wedge_{\mathcal{L}_I}L_IS^0 = L_I(L_IX \wedge_{\mathcal{S}} L_IS^0) = L_I(X\wedge_{\mathcal{S}} S^0) = L_I(X)$.  For example, $F(n)\wedge I\neq 0$ for all $n$~\cite[7.1(e)]{[HP]}.   \end{pf}

\begin{corollary} In $\mathcal{L}_I$, the ${\sf GSC}$ fails but the ${\sf SDGSC}$ holds. \label{gsc-i}
\end{corollary}

\begin{pf} Corollary B.13 of~\cite{[HovS]} also shows that $\mathcal{L}_I$ has no nonzero compacts, so the proof is the same as for $\mathcal{L}_{H\bf_p}$. \end{pf}

In~\cite[Conj.~3.10]{[Hov95a]}, Hovey states the {Dichotomy Conjecture}: In $\mathcal{S}$ every spectrum has either a finite local or a finite acyclic.  In~\cite{[HP]} the authors discuss several equivalent formulations, and some implications.  We briefly point out a relationship between this conjecture and Proposition~\ref{bl-i}.

\begin{proposition} If the Dichotomy Conjecture holds, then the cardinality of $\mathsf{BL}(\mathcal{L}_I)$ is at most two.
\end{proposition}

\begin{pf} Lemma 7.8 of~\cite{[HP]} shows that if the Dichotomy Conjecture holds, then $\<I\>$ is minimal among nonzero classes in $\mathsf{BL}(\mathcal{S})$.  This is the case if and only if $a\<I\>$ is maximal among non-top classes in $\mathsf{BL}(\mathcal{S})$, where $a(-)$ denotes the complementation operation first studied by Bousfield~\cite{[Bou79b]}.  Let $a\<I\>{\uparrow}$ denote the sublattice $\{\<X\>\;|\; \<X\>\geq a\<I\>\}\subseteq \mathsf{BL}(\mathcal{S})$.  In~\cite[Prop.~3.2]{[preprint]} we show that there is a surjective lattice map from $a\<I\>{\uparrow}$ onto $\mathsf{BL}(\mathcal{L}_I)$.  Thus, if the Dichotomy Conjecture holds, $a\<I\>{\uparrow}$ has cardinality two and $\mathsf{BL}(\mathcal{L}_I)$ has cardinality at most two.
\end{pf}

As for classifying localizing subcategories of $\mathcal{L}_I$, or at least perhaps finding a proper nonzero localizing subcategory, we must get around the fact that so many spectra are $I$-acyclic.  We know that $LF(n)\neq 0$ for all $n$, however ${\sf loc}(LF(n))$ is the acyclics of ${\it l}_{n-1}^f:\mathcal{L}_I\rightarrow \mathcal{L}_I$ and Theorem~\ref{i-loc-tc} shows that ${\it l}_n^f=0$ for all $n$.  Thus ${\sf loc}(LF(n))={\sf loc}(LS^0)$ in $\mathcal{L}_I$ for each $n$.

\subsection{The $BP$-local category}

\begin{theorem} \label{LTCs-bp} On $\mathcal{L}_{BP}$, for all $n$ we have that ${\sf LTC1_n}$, ${\sf LTC2_n}$, and ${\sf LTC3_n}$ all hold.
\end{theorem}

\begin{pf} We will show that ${\sf LTC2_n}$ holds for all $n$, and the rest follows from Theorem~\ref{LTCs}.  Since each $K(i)$ is a $BP$-module spectrum, $\<K(i)\>\leq \<BP\>$, and since $K(i)$ is local with respect to itself this implies that $K(i)$ is $BP$-local.  Furthermore, this implies $\<E(n)\>\leq \<BP\>$, so from Lemma~\ref{compose} $L_n=L_nL=LL_n$ as functors on $\mathcal{S}$.

We claim that $L_n:\mathcal{L}_{BP}\rightarrow \mathcal{L}_{BP}$, taking $LY\mapsto L_nLY=L_nY$, is a smashing localization functor on $\mathcal{L}_{BP}$.  We have $L_n(LY) = L(L_nY) = L(L_nS^0\wedge_\mathcal{S} Y) = L(LL_nS^0\wedge_\mathcal{S} LY) = L(L_nLS^0\wedge_\mathcal{S} LY) = (L_nLS^0)\wedge_{\mathcal{L}_{BP}} (LY)$.  This shows that on $\mathcal{L}_{BP}$, the localization functor $L_n$ is also given by smashing with the localization of the unit, $L_nLS^0$, and thus is smashing.

Since both $L_n$ and ${\it l}_n$ are localization functors on $\mathcal{L}_{BP}$ that annihilate $\<K(0)\vee\cdots\vee K(n)\> = \<LK(0)\vee\cdots\vee LK(n)\>$, they are isomorphic.

On $\mathcal{S}$, the natural map $L_n^f X\rightarrow L_n X$ is a ${BP}$-equivalence~\cite[Thm.~2.7(iii)]{[RavLaTC]}.  This means that $LL_n^f X = LL_n X$ for all objects $X$ in $\mathcal{S}$, in particular for all ${BP}$-local objects.  Therefore $LL_n^f = L_n = {\it l}_n$ is a smashing localization functor on $\mathcal{L}_{BP}$.  The acyclics are $\<LL_n^f(LS^0)\> = \<LL_n^f S^0\> = \<LT(0)\vee\cdots \vee LT(n)\>$.  These are the same acyclics as for ${\it l}_n^f$, and so we conclude that ${\it l}_n^f$ and ${\it l}_n$ are isomorphic, and the natural map ${\it l}_n^f X\rightarrow {\it l}_n X$ is an isomorphism.
\end{pf}

\begin{proposition} The ${\sf GSC}$ fails in $\mathcal{L}_{BP}$. \label{gsc-bp}
\end{proposition}

\begin{pf}  The proof of the last theorem showed that $L_n:\mathcal{L}_{BP}\rightarrow \mathcal{L}_{BP}$ is a (different) smashing localization for each $n$.  However, by~\cite[Cor.~B.13]{[HovS]} the category $\mathcal{L}_{BP}$ has no nonzero compact objects.
\end{pf}

Note that the ${\sf SDGSC}$ could still hold, since all the smashing localizations we have identified on $\mathcal{L}_{BP}$ are of the form $L_n = {\it l}_n = {\it l}_n^f$, so are generated by strongly dualizable objects.  The question of finding any other smashing localizations on $\mathcal{L}_{BP}$ is probably at least as hard as doing so on $\mathcal{S}$, in light of Proposition~\ref{implications}.

All of $\<E(n)\>$, $\<K(n)\>$, $\<{H\bf_p}\>$, and $\<I\>$ are ``small" in $\mathsf{BL}(\mathcal{S})$, so by Lemma~\ref{compose} it is not surprising that the Bousfield lattices of their localized categories are not very large; this is not true of $\<BP\>$ in $\mathsf{BL}(\mathcal{S})$.  We have the following bounds on the Bousfield lattice of the local category.

\begin{proposition} The Bousfield lattice of $\mathcal{L}_{BP}$ has $2^{\aleph_0}\leq |\mathsf{BL}(\mathcal{L}_{BP})|\leq 2^{2^{\aleph_0}}$. \label{bl-bp}
\end{proposition}

\begin{pf} The second inequality is Corollary~\ref{bl-size}. Since $\<K(i)\>\leq \<{BP}\>$ for all $i$, we have $\<Q\> = \<\bigvee_{i\geq 0}K(i)\>\leq \<{BP}\>$, and so by Propositions~\ref{lattice-map} and~\ref{jb} we have $|\mathsf{BL}(\mathcal{L}_{BP})|\geq |\mathsf{BL}(\mathcal{H})| =2^{\aleph_0}$.
\end{pf}

\subsection{The $F(n)$-local category}  We conclude with a short discussion of the $F(n)$-local category.

Any smashing localization $L:\mathcal{S}\rightarrow \mathcal{S}$ gives a splitting of the Bousfield lattice $$\mathsf{BL}(\mathcal{S})\stackrel{\sim}{\longrightarrow} \mathsf{BL}(\mathcal{L}_{LS^0})\times \mathsf{BL}(\mathcal{L}_{CS^0}),$$ where $\<X\>\mapsto (\<X\wedge LS^0\>, \<X\wedge CS^0\>)$.  See~\cite[Prop.~6.12]{[IK]} or~\cite[Thm.~5.14]{[preprint]} for more details.  If we take $L=L_n^f:\mathcal{S}\rightarrow\mathcal{S}$, then we have $\<LS^0\> = \<T(0)\vee\cdots\vee T(n)\>$ and $\<CS^0\> = \<F(n+1)\>$.  Of course, the relationship between $\mathcal{L}_{T(0)\vee\cdots\vee T(n)}$ and $\mathcal{L}_{E(n)}$ of Subsection~\ref{sect-en} is immediately related to the original ${\sf TC1_n}$ in $\mathcal{S}$.  However, this suggests that $\mathcal{L}_{F(n)}$ is worth investigating further.

By Lemma~\ref{calcs}, in $\mathsf{BL}(\mathcal{S})$ there is a chain
$$\<S^0\> = \<F(0)\>\geq \<F(1)\> \geq \<F(2)\>\geq\cdots,$$ and by Lemma~\ref{compose} this gives a chain of lattice surjections
$$\mathsf{BL}(\mathcal{S}) = \mathsf{BL}(\mathcal{L}_{F(0)}) \twoheadrightarrow \mathsf{BL}(\mathcal{L}_{F(1)}) \twoheadrightarrow \mathsf{BL}(\mathcal{L}_{F(2)})\twoheadrightarrow\cdots.$$

From the above observations, we expect $\mathsf{BL}(\mathcal{L}_{F(n)})$ to be about as complicated as $\mathsf{BL}(\mathcal{S})$.  For example, $F(n)\wedge I\neq 0$ for all $n$, and so $L_{F(n)}I$ is a square-zero object in $\mathcal{L}_{F(n)}$.  This means that, unlike in most of the localized categories discussed throughout this paper, we know that $\mathsf{BA}(\mathcal{L}_{F(n)})\neq \mathsf{BL}(\mathcal{L}_{F(n)})$.

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\end{document}

redpajama_set_name: RedPajamaArXiv


text:
Acute bronchial asthma in emergency care in a city of Pernambuco during the COVID-19 pandemic: Case report
Thyago de Oliveira Afonso Universidade Federal de Pernambuco https://orcid.org/0000-0001-7616-9011
Samuel Lopes dos Santos Universidade Federal do Piauí https://orcid.org/0000-0003-3375-9171
Gustavo Baroni Araújo Universidade Estadual de Londrina https://orcid.org/0000-0002-3162-7477
Igor de Oliveira Carvalho Universidade Federal de Goiás https://orcid.org/0000-0002-7487-8073
Guilherme Dantas Borges Universidade de Vassouras https://orcid.org/0000-0002-3415-8725
Caio Victor Muniz de Lima Barros Centro Universitário UniFacid Wyden https://orcid.org/0000-0001-8052-2543
Simone Barroso de Carvalho Universidade Federal do Piauí https://orcid.org/0000-0001-7428-8420
Elvira Marques da Luz Dias Universidade Brasil https://orcid.org/0000-0001-7966-0877
Marília Girão de Oliveira Machado Universidade de Fortaleza https://orcid.org/0000-0001-7163-9061
Rosanna Rafena Ribeiro Barbosa Associação de Ensino Superior do Piauí https://orcid.org/0000-0003-1777-2245
Ana Izabel Aparecida Vieira Universidade de Vassouras https://orcid.org/0000-0001-6617-7499
Luziane Cardoso Costa Escola Superior da Amazônia https://orcid.org/0000-0003-4579-8223
Leandro Luiz da Silva Loures Universidade Federal de Juiz de Fora https://orcid.org/0000-0001-8863-8858
Bernardo da Luz Barbosa Universidade de Vassouras https://orcid.org/0000-0002-3866-4887
Gustavo Martins de Araújo Porto Universidade de Vassouras https://orcid.org/0000-0003-2930-3534
Isadora Nunes Delaia Universidade de Vassouras https://orcid.org/0000-0002-0336-848X
Erica Patrícia Dias de Souza UNIPOS https://orcid.org/0000-0001-7675-902X
Mariana Teixeira da Silva Universidade Federal do Piauí https://orcid.org/0000-0002-1904-0921
Camila Leanne Teixeira Coelho de Sousa Universidade Federal do Maranhão https://orcid.org/0000-0002-2525-0335
Ana de Cássia Ivo dos Santos Universidade Federal do Piauí https://orcid.org/0000-0001-5549-1263
Francisca Agda Oliveira Dias Instituto Dom Bosco https://orcid.org/0000-0002-1618-5180
Asthma; Symptom flare up; Therapeutics; Ambulatory care; Coronavirus infections.
Bronchial asthma is a chronic disease of the airways, heterogeneous and of unknown etiology. In Brazil, decompensated bronchial asthma is the fourth leading cause of hospitalizations, with a higher incidence in the Northeast and Southeast regions, with descending mortality rates. The most common symptoms and signs are dyspnea, wheezing, retrosternal chest tightness and cough, resulting from the inflammatory process, with bronchial hyperreactivity and bronchoconstriction. This article aims to alert health professionals about the early diagnosis and treatment of exacerbated asthma in an emergency in the pandemic context of COVID-19, through this case report. Also dealing with the differential diagnosis in respiratory symptoms for COVID-19 with assessment of epidemiological risks and radiography, not very sensitive and specific for COVID-19, but widely available. This case report is about a 19-year-old Caucasian woman, single, born in Caruaru-PE, with complaints of low intensity chest discomfort, dyspnea and fatigue for three days, presenting exacerbation of symptoms in the last four hours, with previous diagnosis of asthma since childhood, currently with treatment interrupted due to fear of worsening for COVID-19 and waking up at night every day during the last month. After therapeutic conduct and clinical improvement, the patient was discharged with instructions on continuity of oral corticosteroids, return to bronchial asthma control therapy, need for follow-up in Primary Health Care (PHC) and symptoms of severity for bronchial asthma and COVID- 19 that indicate the need for emergency care.
Boulet, L. P. et al. (2019). The global initiative for asthma (GINA): 25 years later. European Respiratory Journal, 54(2).
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Brasil. Ministério da Saúde. (2020). Orientações para manejo de pacientes com COVID-19. https://portalarquivos.saude.gov.br/images/pdf/2020/J une/18/Covid19-Orientac--o--esManejoPacientes.pdf.
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AFONSO, T. de O.; SANTOS, S. L. dos; ARAÚJO, G. B.; CARVALHO, I. de O.; BORGES, G. D.; BARROS, C. V. M. de L.; CARVALHO, S. B. de; DIAS, E. M. da L.; MACHADO, M. G. de O.; BARBOSA, R. R. R.; VIEIRA, A. I. A.; COSTA, L. C.; LOURES, L. L. da S.; BARBOSA, B. da L.; PORTO, G. M. de A.; DELAIA, I. N.; SOUZA, E. P. D. de; SILVA, M. T. da; SOUSA, C. L. T. C. de; SANTOS, A. de C. I. dos; DIAS, F. A. O. Acute bronchial asthma in emergency care in a city of Pernambuco during the COVID-19 pandemic: Case report. Research, Society and Development, [S. l.], v. 10, n. 5, p. e52910514959, 2021. DOI: 10.33448/rsd-v10i5.14959. Disponível em: https://www.rsdjournal.org/index.php/rsd/article/view/14959. Acesso em: 29 jan. 2023.
Copyright (c) 2021 Thyago de Oliveira Afonso; Samuel Lopes dos Santos; Gustavo Baroni Araújo; Igor de Oliveira Carvalho; Guilherme Dantas Borges; Caio Victor Muniz de Lima Barros; Simone Barroso de Carvalho; Elvira Marques da Luz Dias; Marília Girão de Oliveira Machado; Rosanna Rafena Ribeiro Barbosa; Ana Izabel Aparecida Vieira; Luziane Cardoso Costa; Leandro Luiz da Silva Loures; Bernardo da Luz Barbosa; Gustavo Martins de Araújo Porto; Isadora Nunes Delaia; Erica Patrícia Dias de Souza; Mariana Teixeira da Silva; Camila Leanne Teixeira Coelho de Sousa; Ana de Cássia Ivo dos Santos; Francisca Agda Oliveira Dias

redpajama_set_name: RedPajamaCommonCrawl


Implementing Advanced Cisco ASA Security (SASAA) v2.1 is an instructor-led course that provides an updated training with labs. The Labs focus on the key features of the Cisco ASA (covering up to the ASA 9.5.1 release). The goal of the course is to be able to implement the key features of the Cisco ASA, including Cisco ASA Firepower Services (including Firepower v6.0), ASA Cloud Web Security, ASA Identity Firewall, ASA Clustering and the Virtual ASA (ASAv).


Coastal prickly pear is native to southern California and Baja California, where it grows in coastal sage scrub and chaparral habitats. It generally grows in dense clumps spreading several feet wide. The branches are made up of oval-shaped flat segments (pads) up to 8 ½ inches long, and covered in clusters of yellowish spines. The flowers are yellow or red and bloom in the spring. The edible fruit is purplish red, reaching up to 2 inches long. This is an important plant for wildlife.
The fruits are collected in June and July and rolled, brushed, or shaken in an agave fiber carrying net to remove spines and then eaten fresh or dried. When green, the fruits are cooked like a pumpkin. The young green pads are boiled or fried. The blossom and seeds are also eaten. After the seeds are winnowed out, they are stored to be made into flour. The cactus spine is used in tattooing.


text:
Games People Play: The Evolution of Video Games
Added by the History Center
Time: July 2, 2011 at 10am to September 10, 2011 at 5pm
Location: Orange County Regional History Center
Street: 65 East Central Blvd.
City/Town: Orlando
Website or Map: http://www.thehistorycenter.o…
Event Type: exhibit, museum, family
Organized By: Orange County Regional History Center
Starting with the pinball machine, journey through time to discover how electronic games have evolved, from the development of the first computer games and rise and fall of the arcade, to the advanced game consoles of today. Games People Play: The Evolution of Video Gamesoffers a look 'inside the box,' highlighting the dramatic changes in size versus power as it marks key developments in game and console technology and illustrates the relevance and
sociological influences of these 'toys.' See and experience various gaming systems through the decades including the Atari 2600, Sega Genesis, Super Nintendo, Nintendo 64, and more. Included in general admission.
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