The derived category of $p$-complete abelian groups is comonadic over the derived category of $\mathbb F_p$-vector spaces?

Question

$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ext{Ext}$Let $p$ be a prime. The adjunction

$$\mathbb F_p \otimes_\mathbb{Z} (-) : \Mod(\mathbb Z) \rightleftarrows \Mod(\mathbb F_p) : U $$

descends (almost tautologically) to an adjunction on the Bousfield localization,

$$\mathbb F_p \otimes_\mathbb{Z} (-) : \Mod(\mathbb Z)^\wedge_p \rightleftarrows \Mod(\mathbb F_p) : U $$

I believe this adjunction is comonadic, and I believe this is well-known.

Question 1: What is a reference for the comonadicity of the above adjunction?

Question 2: Are there any examples of folks applying this comonadicity to the study of infinite $p$-complete abelian groups (or — almost equivalently — to infinite abelian $p$-groups)?

Notes:

• In Bousfield's language, the statement is that the $\mathbb F_p$-nilpotent completion of a $\mathbb Z$-module always coincides with its $\mathbb F_p$-localization. This is related to saying something about the associated Adams spectral sequence, although I think for general objects of $\Mod(\mathbb Z)^\wedge_p$ there will still be $\varprojlim^1$ issues with the spectral sequence.

• It follows, for example, from the main result of Baker–Lazarev that the above adjunction is comonadic at the unit $\mathbb Z_p$ (and hence at any compact object), but I'm not sure about general objects. Also I think the above result should be older.

• This doesn't seem to follow from Mathew's theory of descendable adjunctions — $\mathbb Z_p$ is not in the thick subcategory generated by $\mathbb F_p$.

• In case it's not obvious: in the above, I take an implicit $\infty$-convention and write $A = HA$ everywhere. $\Mod(R)$ is the stable $\infty$-category / dg-category of chain complexes of $R$-modules localized at quasi-isomorphism. $\Mod(\mathbb Z)^\wedge_p$ is the Bousfield localization of $\Mod(\mathbb Z)$ at $\mathbb F_p$. Alternatively, it is the stable $\infty$-category / dg-category of chain complexes of $p$-complete abelian groups localized at quasi-isomorphism. An abelian group $A$ is $p$-complete if it is a $\mathbb Z_{(p)}$-module and $\Ext^\ast(\mathbb Q, A) = 0$. For the comonadicity statement here, it's important to work with more than just the homotopy categories or just the hearts of these categories.

Answer 1

The comments are correct that because $\mathbb{F}_p \in D^{\mathrm{perf}}(\mathbb{Z})$ the Barr-Beck-Lurie theorem applies. But we can prove the analogous result for the residue field of any prime $\mathfrak{p}$ in any Noetherian ring $A$, without assuming $\mathfrak{p}$ is generated by a single regular element. That is, the derived extension of scalars functor from $\mathfrak{p}$-local, $\mathfrak{p}$-complete complexes to $D(\kappa(\mathfrak{p}))$ is comonadic.

We can assume wlog that $A$ is a local ring with maximal ideal $\mathfrak{p}$. Let $K$ be the koszul complex on a list generators for $\mathfrak{p}$. Then $K$ is dualizable and by Proposition 3.34 of "The Galois group of a stable homotopy theory" the map $K \to \kappa(\mathfrak{p})$ admits descent (in the strong sense of Akhil Mathew); in particular $D(K) \to D(\kappa(\mathfrak{p}))$ is comonadic. Dualizability of $K$ means that $D^{\mathfrak{p}-\mathrm{comp}}(A) \to D(K)$ preserves all limits, hence is comonadic by Barr-Beck-Lurie. The composite of comonadic functors is in general not comonadic, but when the first one preserves limits we can apply Barr-Beck-Lurie to see the composite is again comonadic.

For prior work here see Gunnar Carlsson's "Derived completions in stable homotopy theory" which shows the functor is pre-comonadic, or as you say it's comonadic at every object