Ext-groups of graded comodules

Abstract

Let $Γ$ be a coalgebra over a field. There is a well-known covariant embedding of the category of graded $Γ$-comodules into the category of graded modules over the dual algebra $Γ^*$. We show that this covariant embedding preserves Ext-groups in a range of motivating cases, including in some cases like Steenrod algebras where the covariant embedding has otherwise bad homological properties and fails to preserve Ext-groups on the underlying ungraded categories. As an application, we prove the old conjecture in stable homotopy theory that there are no nonzero projectives in the category of graded comodules over the dual Steenrod algebra. As another application, we carry out a study of the right adjoint to the extended comodule functor, including a topological interpretation of this right adjoint. This yields generalizations of formulas of Margolis and Lin characterizing maps from Eilenberg-Mac Lane spectra into finite-type bounded-below spectra; our generalizations drop the "finite-type" hypothesis.