Derived cotorsion of bounded-below comodules for general coalgebras

Determine whether every bounded-below graded comodule over a general graded coalgebra is derived cotorsion, meaning that all right-derived functors \(R^nV\) of the right adjoint \(V\) to the extended comodule functor vanish.

Background

The paper proves that, under suitable P\mathcal{P}-algebra hypotheses, a bounded-below comodule is cotorsion, so V(M)=0V(M)=0. It then asks whether the stronger property of derived cotorsion holds, namely whether RnV(M)=0R^nV(M)=0 for every degree nn.

For the mod pp dual Steenrod algebra, the paper establishes derived cotorsion for bounded-below comodules, but it does not settle the corresponding question for general coalgebras. The issue is relevant because derived cotorsion is characterized topologically by contractibility of certain mapping spectra in the Steenrod-algebra case.

References

It is natural to ask whether it is also true that a bounded-below comodule is {\em derived} cotorsion, i.e., $R*V(M)\cong 0$. For general $\Gamma$, the author does not know the answer to that question.

— Ext-groups of graded comodules  (2609.26465 - Salch, 22 Sep 2026) in Section “\(V\) for a \(\mathcal{P}\)-algebra,” immediately after Proposition \ref{cotorsion vanishes on boundeds}

Surely a purely algebraic proof is possible, but the author does not know one.

— Ext-groups of graded comodules  (2609.26465 - Salch, 22 Sep 2026) in Immediately after Corollary \ref{top thm 1 cor 2}