Derived functors of product and limit in the category of comodules over the dual Steenrod algebra

Abstract: In the 2000s, Sadofsky constructed a spectral sequence which converges to the mod $p$ homology groups of a homotopy limit of a sequence of spectra. The input for this spectral sequence is the derived functors of sequential limit in the category of graded comodules over the dual Steenrod algebra. Since then, there has not been an identification of those derived functors in more familiar or computable terms. Consequently there have been no calculations using Sadofsky's spectral sequence except in cases where these derived functors are trivial in positive cohomological degrees. In this paper, we prove that the input for the Sadofsky spectral sequence is simply the local cohomology of the Steenrod algebra, taken with appropriate (quite computable) coefficients. This turns out to require both some formal results, like some general results on torsion theories and local cohomology of noncommutative non-Noetherian rings, and some decidedly non-formal results, like a 1985 theorem of Steve Mitchell on some very specific duality properties of the Steenrod algebra not shared by most finite-type Hopf algebras. Along the way there are a few results of independent interest, such as an identification of the category of graded $A_*$-comodules with the full subcategory of graded $A$-modules which are torsion in an appropriate sense.