6
$\begingroup$

Let $k$ be a perfect field of characteristic $p$. I heard that the Topological Hochschild homology of a smooth proper stable infinity category (or dg-category) is dualizable as a THH(k)-module spectrum in the infinity category of cyclotomic spectra. Does this follow from the fact that a smooth proper stable infinity category is dualizable in the category $\text{Cat}^{\text{perf}}_{\infty,k}$ ?

$\endgroup$

1 Answer 1

10
$\begingroup$

$\newcommand{\THH}{\mathrm{THH}} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\perf}{\mathrm{perf}} \newcommand{\Sp}{\mathrm{Sp}} \newcommand{\Mod}{\mathrm{Mod}}$ If you ask about dualizability in $\THH(k)$-modules in $\Sp$, it indeed follows from this together with the fact that $\THH: \Cat^{\perf}_{\infty,k}\to \Mod_{\THH(k)}(\Sp)$ is symmetric monoidal, hence preserves dualizability.

This symmetric monoidality follows from the following two properties of $\THH: \Cat^{\perf}_\infty\to \Sp$:

1- It is symmetric monoidal

2- It commutes with sifted colimits.

The latter ensures that if you do a relative tensor product (which is defined via a bar construction, which is a colimit of a simplicial object), you can do it before or after applying $\THH$.

The exact same argument works for modules over $\THH(k)$ in $\Sp^{BS^1}$.

If you ask, as you did, about modules in cyclotomic spectra, then this is still so, and this is still because $\THH$ has these two properties with values in cyclotomic spectra, but it is somehow not as easy to prove : for starters, I don't know of a convenient reference for $\Cat^\perf_\infty\to \mathrm{CycSp}$, although it is folklore that such a thing exists and is symmetric monoidal.

Once you have this, the rest follows in exactly the same way because colimits in $\mathrm{CycSp}$ are computed underlying, say, by corollary II.1.7. in Nikolaus-Scholze.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.