Let $p$ be a prime number, let $k$ be a commutative ring in which $p=0$, and let $X = K( {\mathbb Z}/p {\mathbb Z}, n)$ be an Eilenberg-MacLane space. Let $F$ be the free $E_{\infty}$-algebra over $k$ generated by a class $\eta$ in (homological) degree $-n$. A result of Mandell asserts that there is a cofiber sequence of $E_{\infty}$-algebras over $k$ $$ F \stackrel{A}{\rightarrow} F \rightarrow C^{\ast}(X;k)$$ where $A$ is the ``Artin-Schreier'' map which carries $\eta$ to $\eta - P^0(\eta)$. In other words, as an $E_{\infty}$-algebra, the cochain complex $C^{\ast}(X;k)$ can be described by one generator (a class in degree $-n$) and one relation (the class should be fixed by $P^0$).

Is it possible to prove this result without explicitly computing the homotopy groups of the cofiber of $A$? Let's denote this cofiber by $R$. I can reduce to the problem of showing that $\pi_i R \simeq 0$ for $i > 0$ and that $\pi_0 R \simeq k$ (both of which are obvious consequences of Mandell's theorem). Is there some way to show this directly, without computing the other homotopy groups?