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What would be a source for an explicit derivation – for $E$ an $E_\infty$-ring and $X$ a spectrum – that the classical $E$-Adams tower corresponding to the canonical $E$-Adams resolution $N(X \wedge E^{\wedge^\bullet})$ is equivalently the Tot-tower of the cosimplicial spectrum $X \wedge E^{\wedge^\bullet}$? That’s not immediate, is it?
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