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This is the same question as an earlier question of mine, except in a different category.

Let $Spt_{T(h)}^{fin}$ be the category of finite $T(h)$-local spectra. Let $K_0^\oplus(Spt_{T(h)}^{fin})$ be the direct-sum $K$-theory of the category. This is a nonunital algebra over $\mathbb Z[\Sigma, \Sigma^{-1}]$.

Question: Let $X \in Spt_{T(h)}^{fin}$. Then must the class $[X] \in K_0^\oplus(Spt_{T(h)}^{fin})$ be integral over $\mathbb Z[\Sigma, \Sigma^{-1}]$?

Notes:

  1. I suspect the answer is no in general, but the obstruction identified by Nick Kuhn in $K_0^\oplus(Spt^{fin})$ uses the action of the Steenrod algebra and doesn't work here.

  2. I'm also interested in the analogous questions where "$T(h)$" is replaced by "$K(h)$" and/or "finite" is replaced by "dualizable" or "in the thick subcategory generated by the sphere".

  3. The answer is yes if $h = 0$. I'd be interested to understand any other special case. For instance, when $p$ is large compared to $h$, the answer might be closer to "yes". For another instance, things might just be tractable when $h = 1$.

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    $\begingroup$ By "finite", do you mean "compact" ? If not, I don't get why the algebra is non-unital (to me finite means "built in finitely many steps from the sphere" ) $\endgroup$
    – Maxime Ramzi
    Commented Oct 1, 2022 at 8:39
  • $\begingroup$ @MaximeRamzi Yeah, I mean compact (though as mentioned in (2) above I'd also be interested to generalize to $X$ in the thick subcategory generated by the sphere). $\endgroup$
    – Tim Campion
    Commented Oct 1, 2022 at 13:19

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