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Computation of cohomology of Eilenberg-Maclane spaces

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Spf{Spf}$Background: If $E$ is a complex-oriented spectrum, then $E^*(K(\mathbb{Z}/p^k,1))$ sits inside a long exact ...