Relationship between classical and derived notions of completeness for graded R-modules

Determine the precise relationship between the following two notions of completeness for a left graded module over the Cartier–Dieudonné–Raynaud ring R: (1) classical completeness of the graded R-module with respect to the standard filtration Fil^n(M^i)=V^n M^i + d V^n M^{i-1} (i.e., the natural map M^i → lim_n M^i/Fil^n(M^i) is an isomorphism for each i), and (2) completeness of the corresponding object in the derived category DG(R) defined via Ekedahl’s completion functor M ↦ lim_n (R_n ⊗^L_R M), where R_n=R/(V^nR + dV^nR). Ascertain whether either notion implies the other, whether they are equivalent, or otherwise characterize their exact relationship.

Background

The paper introduces two completeness notions for graded modules over the Cartier–Dieudonné–Raynaud ring R. Definition 2.3 (“complete R-module”) defines a classical, module-theoretic completeness using the standard filtration Filn(Mi)=Vn Mi + dVn M{i-1}. Separately, Section 2.1 constructs Ekedahl’s completion functor on DG(R) using the pro-system R_n=R/(VnR + dVnR), giving a derived notion of completeness as objects in the essential image of this functor.

The authors note a lack of clarity about how these two notions relate. Clarifying whether one implies the other or if they coincide would sharpen the foundations of the homological algebra used throughout, particularly in analyzing de Rham–Witt complexes and coherent R-modules.

References

We warn the readers that for a left graded R-module M, it is unclear to us if being classically complete as a graded left $R$-module in \Cref{complete R-module} is related to it being complete when viewed as an object in $DG(R)$ in the sense of \Cref{completion on DGr(R)}.

— On de Rham--Witt Cohomology of Classifying Stacks  (2604.03062 - Li et al., 3 Apr 2026) in Warning, Section 2 (Left graded R-modules), subsection: Ekedahl’s completion functor