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\begin{document}
\title{Structure within Bousfield Lattices}
\author{Luke Wolcott}
\date{}
\maketitle
This talk will attempt to incorporate things from all the previous talks. The following categories have come up:
\begin{itemize}
\item the stable homotopy category $\mc{S}$;
\item $D(R)$ for a commutative ring $R$;
\item $D(R)$ for a commutative Noetherian ring $R$;
\item $\mc{C}((kG)^*)$ (BCR);
\item the stable module category $stmod(kG)$;
\item $Stable(A_*)$ for $A_*$ the dual Steenrod algebra;
\item general tensored triangulated categories.
\end{itemize}
In fact these are all tensored triangulated categories, and all but the last are monogenic (i.e.\! generated by the unit object) \textit{axiomatic} stable homotopy categories (i.e.\! they are tensored-triangulated, have arbitrary set-indexed coproducts, and the unit $S$ is a small (i.e.\! $[S,\coprod X_\alpha] = \coprod [S,X_\alpha]$) weak generator (i.e.\! $\pi_*(X)=0$ iff $X=0$)). (We could add the equivariant stable homotopy category, but then we'd have to remove the word ``monogenic''.)
One can define a Bousfield lattice for any axiomatic stable homotopy category.
\begin{definition}
Recall that a \textit{thick} subcategory of a triangulated category of finite (=compact) objects is a subcategory that is full, triangulated, and closed under summands. A \textit{localizing} subcategory is one that is full, triangulated, and closed under coproducts.
\end{definition}
There are two equivalence relations we can use to define Bousfield classes: we can use homology or cohomology. Cohomology is a bit scary and icky, so we'll stick to homology. Note that in all the above categories we have a version of Brown representability, so all cohomology theories take the form $[X,E]_*$; for homology theories to get something covariant \textit{the most natural thing we can do} is have $[S,E\sm X]_*$.
\begin{definition}
Let $\mc{C}$ be a (monogenic) axiomatic stable homotopy category, $X$ an object. The \textit{(homological) Bousfield class} is given by $\brax{X} = \{W:X\sm W=0\} = \{W : X_*W = 0\}$ (by the weak generator assumption). These are called $X_*$-\textit{acyclic}. We say that $X$ and $Y$ are \textit{Bousfield equivalent} if $\brax{X}=\brax{Y}$. The collection of Bousfield classes is called the \textit{Bousfield lattice}. We define a partial order by reverse inclusion, $\brax{X}\leq \brax{Y}$ whenever $W\in \brax{Y} \Rightarrow W\in \brax{X}$. Under this partial order, the maximum element is $\brax{S}$ and the minimum element is $\brax{0}$.
\end{definition}
\begin{example}
Easy examples of spectra that are not homotopy equivalent but are nevertheless Bousfield equivalent include the facts that $\brax{H\Z}=\brax{S}$ and that $\brax{E} = \brax{E \vee E}$. The unit object $S$ always generates the thick subcategory of \textit{finite objects}, and the localizing category it generates is the entire category.
\end{example}
\begin{proposition}
Every $\brax{X}$ is a localizing subcategory.
\end{proposition}
\begin{proof}
Suppose $A\ra B\ra C$ is a triangle. Then $A\sm X\ra B\sm X \ra C\sm X$ is a triangle. So if two of these are trivial, then so is the third. Moreover, if $W_\alpha \in \brax{X}$ then $X\sm (\coprod W_\alpha) = \coprod (X\sm W_\alpha)=0$, so this is closed under coproducts.
\end{proof}
We have a \textit{join} (a/k/a least upper bound) operation coming from the fact that we're looking at a lattice; this is given by $\brax{X}\vee \brax{Y} = \brax{X\vee Y}$. We have a well-defined operation $\brax{X}\sm \brax{Y} = \brax{X\sm Y}$; this is less than both $\brax{X}$ and $\brax{Y}$, but in general it is \textit{not} the greatest lower bound. However, we'd like to define the \textit{meet} (a/k/a greatest lower bound) by $\brax{X} \bigwedge \brax{Y} = \vee_{\brax{W}\leq\brax{X}, \ \brax{W} \leq \brax{Y}} \brax{W}$, but this may not be defined! However, there \textit{is} in fact a \textit{set} of homological Bousfield classes, so this is legal. This is what gives us meets and joins, which makes us a lattice.
Within this lattice we can look at sublattices, in particular there's one where $\bigwedge$ \textit{is} the meet operation. This makes us into a distributive lattice, and inside of that we have a Boolean algebra. In spectra, every smashing localization corresponds to an object in this Boolean algebra, so studying it may be a way at getting at the telescope conjecture.
As we heard before, given an object $E$ we get a Bousfield localization functor $L_E$ that kills exactly $\brax{E}$, the $E$-acyclics. We might summarize this by saying that ``homological localization exists''. In fact, every localization functor is determined either by the things that it kills (a/k/a its acyclics) or its image (a/k/a its locals). Thus if $\brax{E}=\brax{F}$ then $L_E = L_F$. So we can think of the Bousfield lattice as exactly the set of ways in which we can homologically localize.
In the stable homotopy category, we have $\mc{F}$, the finite objects. We then have the \textit{type} stratification
\begin{equation*}
\cdots \subsetneq C_2 \subsetneq C_1 \subsetneq C_0 = \mc{F}.
\end{equation*}
Here $C_n = \brax{K(n-1)}\cap \mc{F}$. The \textit{class invariance theorem} tells us that if $X,Y\in \mc{F}$, then $\brax{X} \leq \brax{Y}$ iff $type(Y)\leq type(X)$. So we have
\begin{equation*}
\brax{0} \leq \brax{F(\infty)}= \brax{H\mathbb{F}_p} \leq \cdots \leq \brax{F(1)} \leq \brax{S} = \brax{F(0)}.
\end{equation*}
\vspace{10pt}
The reason we use homology instead of cohomology is that it's unknown in ZFC whether cohomological localization exists! (However,, it's known given an additional ``large cardinal'' set-theoretic axiom.) Moreover, nobody knows whether there's a \textit{set} of cohomological Bousfield classes.
It can be shown that
\begin{equation*}
\{ \textup{homological Bousfield classes}\} \subseteq \{\textup{cohomological Bousfield classes}\} \subseteq \{\textup{localizing subcategories}\},
\end{equation*}
but nobody knows whether any of these are strict containments or actual equalities or what.
\end{document}