**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.



Chromatic t-structures?
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.


