Chromatic t-structures?

Question

Questions: Fix a prime $p$ and $n \in \mathbb N_{\geq 1}$.

1. Does the category $Sp_{K(n)}$ of $K(n)$-local spectra admit a nontrivial $t$-structure?

By "nontrivial", I simply mean that $\{0\} \subsetneq Sp_{K(n),\geq 0} \subsetneq Sp_{K(n)}$.

2. Does $Sp_{K(n)}$ admit a nontrivial monoidal $t$-structure?

"Monoidal" means that (1) $\mathbb S_{K(n)} \in Sp_{K(n),\geq 0}$ (where $\mathbb S_{K(n)}$ is the $K(n)$-local sphere) and (2) $Sp_{K(n),\geq 0}$ is closed under the $K(n)$-local smash product. (Evidently I am using homological, rather than cohomological, indexing.)

As usual, the corresponding $T(n)$-local questions are also interesting, though presumably harder. For context, I'd also be interested in hearing about the $E(n)$-local or $T(0) \vee \dots \vee T(n)$-local versions of these questions.

In the above, feel free to interpret "the category $Sp_{K(n)}$" as either "the triangulated category $Sp_{K(n)}$" or as "the stable $\infty$-category $Sp_{K(n)}$" -- whichever is most comfortable.

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There's an easier question which has a negative answer: for $n \in \mathbb N_{\geq 1}$, the category $Mod_{K(n)}$ of $K(n)$-module spectra does not admit a nontrivial $t$-structure. For every object of $Mod_{K(n)}$ is a coproduct of shifts of $K(n)$. So if $0 \neq X \in Mod_{K(n),\geq 0}$, then there is a retract $\Sigma^k K(n)$ of $X$ which is a shift of $K(n)$, so that $\Sigma^k K(n) \in Mod_{K(n),\geq 0}$. Then because $K(n)$ is periodic, every object $Y \in Mod_{K(n)}$ is a coproduct of nonnegative shifts of $\Sigma^k K(n) \in Mod_{K(n),\geq 0}$, and so $Y \in Mod_{K(n),\geq 0}$.

But of course, the category $Sp_{K(n)}$ is much more complicated than the category $Mod_{K(n)}$.

When $n = 0$ (so that $K(n) = H \mathbb Q$) or $n = \infty$ (so that $K(n) = H \mathbb F_p$), $Mod_{K(n)}$ does admit a monoidal $t$-structure given by usual connectivity, and $Sp_{K(n)}$ inherits a monoidal $t$-structure by pullback along the free functor $Sp_{K(n)} \to Mod_{K(n)}$ (which is an equivalence for $n = 0$, of course). I don't think these cases shed much light on the case $n \in \mathbb N_{\geq 1}$, though.

Answer 1

To expand on Tim's answer, the arguments generalize to show that $Sp_{K(n)}$ admits no non-trivial t-structures in general.

The crucial ingredient is that $Sp_{K(n)}$ has no non-trivial localising or colocalising subcategories, see 7.5 in Hovey-Strickland. Thus, to finish the argument it is enough to show that any subcategory $C \subseteq Sp_{K(n)}$ which is closed under limits is in fact colocalising, ie. it's also closed under suspension.

If $X$ is any $K(n)$-local spectrum, then it is well-known (see 7.10 in the aforementioned book) that it can be written as a limit $X \simeq lim \ X \wedge F_{i}$ of its smash products with type n generalized Moore spectra. If $X \in C$, then some desuspension of $X \wedge F_{i}$ is contained in $C$ as well, but as Tim observed these spectra are periodic and so $\Sigma^{n} X \wedge F_{i} \in C$ for all $n \in \mathbb{Z}$. It follows that $\Sigma^{n} X \in C$, ending the argument.

Answer 2

The second question turns out to have a surprisingly easy negative answer. This is depressing on two counts: both that the answer is negative and that it's so easy.

Suppose that $Sp_{K(n)}$ has a $t$-structure such that $\mathbb S_{K(n)} \in Sp_{K(n),\geq 0}$. Let $F(n)$ be any finite type-$n$ $p$-local spectrum. Then for some $k \geq 0$ we have that $\Sigma^k F(n)$ is in the closure of $\mathbb S_{(p)}$ under finite colimits in the category $Sp_{(p)}$ of $p$-local spectra. Therefore, $\Sigma^k F(n)_{K(n)} \in Sp_{K(n),\geq 0}$. But we also have $\Sigma^k F(n)_{K(n)} \simeq T(n)_{K(n)}$, which is a periodic spectrum. Thus $\Sigma^l F(n)_{K(n)} \in Sp_{K(n),\geq 0}$ for all $l \in \mathbb Z$. Since $F(n)$ was an arbitrary finite type-$n$ spectrum, we see that all $K(n)$-localizations of finite type-$n$ spectra are in $Sp_{K(n),\geq 0}$.

Now I'm pretty sure that every object of $Sp_{K(n)}$ is a colimit of $K(n)$-localizations of finite type-$n$ spectra. It follows that every object is in $Sp_{K(n),\geq 0}$ and the $t$-structure is trivial. But I can't find a reference for this fact at the moment, so here's an alternate argument. It's at least the case that $\mathbb \Sigma^l \mathbb S_{K(n)}$ is a (sequential) colimit of $K(n)$-localizations of finite type-$n$ spectra for all $l \in \mathbb Z$, and so $\Sigma^l \mathbb S_{K(n)} \in Sp_{K(n),\geq 0}$ for all $l \in \mathbb Z$. If the $t$-structure is monoidal, it follows that $Sp_{K(n),\geq 0}$ is closed under desuspension, i.e. $Sp_{K(n),\geq 0} \subseteq Sp_{K(n)}$ is a a stable subcategory. This kind of $t$-structure is not very interesting, and anyway I believe that since $Sp_{K(n)}$ doesn't admit any nontrivial Bousfield localizations, it doesn't admit any such $t$-structures which are nontrivial either.