The Stacks project

Example 60.19.1. Suppose that $A_*$ is any cosimplicial ring. Consider the cosimplicial module $M_*$ defined by the rule

\[ M_ n = \bigoplus \nolimits _{i = 0, ..., n} A_ n e_ i \]

For a map $f : [n] \to [m]$ define $M_*(f) : M_ n \to M_ m$ to be the unique $A_*(f)$-linear map which maps $e_ i$ to $e_{f(i)}$. We claim the identity on $M_*$ is homotopic to $0$. Namely, a homotopy is given by a map of cosimplicial modules

\[ h : M_* \longrightarrow \mathop{\mathrm{Hom}}\nolimits (\Delta [1], M_*) \]

see Section 60.16. For $j \in \{ 0, \ldots , n + 1\} $ we let $\alpha ^ n_ j : [n] \to [1]$ be the map defined by $\alpha ^ n_ j(i) = 0 \Leftrightarrow i < j$. Then $\Delta [1]_ n = \{ \alpha ^ n_0, \ldots , \alpha ^ n_{n + 1}\} $ and correspondingly $\mathop{\mathrm{Hom}}\nolimits (\Delta [1], M_*)_ n = \prod _{j = 0, \ldots , n + 1} M_ n$, see Simplicial, Sections 14.26 and 14.28. Instead of using this product representation, we think of an element in $\mathop{\mathrm{Hom}}\nolimits (\Delta [1], M_*)_ n$ as a function $\Delta [1]_ n \to M_ n$. Using this notation, we define $h$ in degree $n$ by the rule

\[ h_ n(e_ i)(\alpha ^ n_ j) = \left\{ \begin{matrix} e_{i} & \text{if} & i < j \\ 0 & \text{else} \end{matrix} \right. \]

We first check $h$ is a morphism of cosimplicial modules. Namely, for $f : [n] \to [m]$ we will show that
\begin{equation} \label{crystalline-equation-cosimplicial-morphism} h_ m \circ M_*(f) = \mathop{\mathrm{Hom}}\nolimits (\Delta [1], M_*)(f) \circ h_ n \end{equation}

The left hand side of ( evaluated at $e_ i$ and then in turn evaluated at $\alpha ^ m_ j$ is

\[ h_ m(e_{f(i)})(\alpha ^ m_ j) = \left\{ \begin{matrix} e_{f(i)} & \text{if} & f(i) < j \\ 0 & \text{else} \end{matrix} \right. \]

Note that $\alpha ^ m_ j \circ f = \alpha ^ n_{j'}$ where $0 \leq j' \leq n + 1$ is the unique index such that $f(i) < j$ if and only if $i < j'$. Thus the right hand side of ( evaluated at $e_ i$ and then in turn evaluated at $\alpha ^ m_ j$ is

\[ M_*(f)(h_ n(e_ i)(\alpha ^ m_ j \circ f) = M_*(f)(h_ n(e_ i)(\alpha ^ n_{j'})) = \left\{ \begin{matrix} e_{f(i)} & \text{if} & i < j' \\ 0 & \text{else} \end{matrix} \right. \]

It follows from our description of $j'$ that the two answers are equal. Hence $h$ is a map of cosimplicial modules. Let $0 : \Delta [0] \to \Delta [1]$ and $1 : \Delta [0] \to \Delta [1]$ be the obvious maps, and denote $ev_0, ev_1 : \mathop{\mathrm{Hom}}\nolimits (\Delta [1], M_*) \to M_*$ the corresponding evaluation maps. The reader verifies readily that the compositions

\[ ev_0 \circ h, ev_1 \circ h : M_* \longrightarrow M_* \]

are $0$ and $1$ respectively, whence $h$ is the desired homotopy between $0$ and $1$.

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