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Let $E=E_n$ be Morava $E$-theory of height $n$. In previous work Devinatz andHopkins introduced the $K(n)$-local $E_n$-Adams spectral sequence and showedthat, under certain conditions, the $E_2$-term of this spectral sequence can beidentified with continuous group cohomology. We work with the category of$L$-complete $E_*E$-comodules, and show that in a number of cases the$E_2$-term of the above spectral sequence can be computed by a relative Extgroup in this category. We give suitable conditions for when we can identifythis Ext group with continuous group cohomology.

The $E_2$-term of the $K(n)$-local $E_n$-Adams spectral sequence