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Full faithfulness of the pullback functor c*: GrD(Z_) → D(B)

Determine whether the pullback functor c*: GrD(Z_)→D(B) associated to the map B→BG_{m, Z_} is fully faithful beyond the discrete graded Z-module case, or characterize precisely the largest subcategory on which c* is fully faithful.

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Background

To relate graded objects to quasi-coherent sheaves on the quotient stack by the adic unit circle, the authors consider a pullback functor c*. They prove full faithfulness on the discrete graded subcategory but leave the general case unresolved.

Understanding the exact scope of full faithfulness for c* would clarify the connection between graded algebraic data and sheaves on the analytic stack B.

References

We don't know if $c*$ is fully faithful (it probably isn't), but at least it's fully faithful when restricted to the full sub-$\infty$-category $\mathrm{Gr}D(\mathbb{Z})$ spanned by the discrete graded $\mathbb{Z}$-modules.

— Derived $q$-Hodge complexes and refined $\operatorname{TC}^-$ (2410.23115 - Meyer et al., 30 Oct 2024) in Graded objects and sheaves on B, Section 5.2 (par:GradedAdicII)