Lemma 60.16.1. Let $A_*$ be a cosimplicial ring. Let $\varphi _*, \psi _* : K_* \to M_*$ be homomorphisms of cosimplicial $A_*$-modules.

If $\varphi _*$ and $\psi _*$ are homotopic, then

\[ \varphi _* \otimes 1, \psi _* \otimes 1 : K_* \otimes _{A_*} L_* \longrightarrow M_* \otimes _{A_*} L_* \]are homotopic for any cosimplicial $A_*$-module $L_*$.

If $\varphi _*$ and $\psi _*$ are homotopic, then

\[ \wedge ^ i(\varphi _*), \wedge ^ i(\psi _*) : \wedge ^ i(K_*) \longrightarrow \wedge ^ i(M_*) \]are homotopic.

If $\varphi _*$ and $\psi _*$ are homotopic, and $A_* \to B_*$ is a homomorphism of cosimplicial rings, then

\[ \varphi _* \otimes 1, \psi _* \otimes 1 : K_* \otimes _{A_*} B_* \longrightarrow M_* \otimes _{A_*} B_* \]are homotopic as homomorphisms of cosimplicial $B_*$-modules.

If $I_* \subset A_*$ is a cosimplicial ideal, then the induced maps

\[ \varphi ^\wedge _*, \psi ^\wedge _* : K_*^\wedge \longrightarrow M_*^\wedge \]between completions are homotopic.

Add more here as needed, for example symmetric powers.

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