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

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