The Stacks project

Lemma 20.23.6. Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ be an open covering. Assume $I$ comes equipped with a total ordering. The map $c \circ \pi $ is homotopic to the identity on $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$. In particular the inclusion map $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ is a homotopy equivalence.

Proof. For any multi-index $(i_0, \ldots , i_ p) \in I^{p + 1}$ there exists a unique permutation $\sigma : \{ 0, \ldots , p\} \to \{ 0, \ldots , p\} $ such that

\[ i_{\sigma (0)} \leq i_{\sigma (1)} \leq \ldots \leq i_{\sigma (p)} \quad \text{and} \quad \sigma (j) < \sigma (j + 1) \quad \text{if} \quad i_{\sigma (j)} = i_{\sigma (j + 1)}. \]

We denote this permutation $\sigma = \sigma ^{i_0 \ldots i_ p}$.

For any permutation $\sigma : \{ 0, \ldots , p\} \to \{ 0, \ldots , p\} $ and any $a$, $0 \leq a \leq p$ we denote $\sigma _ a$ the permutation of $\{ 0, \ldots , p\} $ such that

\[ \sigma _ a(j) = \left\{ \begin{matrix} \sigma (j) & \text{if} & 0 \leq j < a, \\ \min \{ j' \mid j' > \sigma _ a(j - 1), j' \not= \sigma (k), \forall k < a\} & \text{if} & a \leq j \end{matrix} \right. \]

So if $p = 3$ and $\sigma $, $\tau $ are given by

\[ \begin{matrix} \text{id} & 0 & 1 & 2 & 3 \\ \sigma & 3 & 2 & 1 & 0 \end{matrix} \quad \text{and} \quad \begin{matrix} \text{id} & 0 & 1 & 2 & 3 \\ \tau & 3 & 0 & 2 & 1 \end{matrix} \]

then we have

\[ \begin{matrix} \text{id} & 0 & 1 & 2 & 3 \\ \sigma _0 & 0 & 1 & 2 & 3 \\ \sigma _1 & 3 & 0 & 1 & 2 \\ \sigma _2 & 3 & 2 & 0 & 1 \\ \sigma _3 & 3 & 2 & 1 & 0 \\ \end{matrix} \quad \text{and} \quad \begin{matrix} \text{id} & 0 & 1 & 2 & 3 \\ \tau _0 & 0 & 1 & 2 & 3 \\ \tau _1 & 3 & 0 & 1 & 2 \\ \tau _2 & 3 & 0 & 1 & 2 \\ \tau _3 & 3 & 0 & 2 & 1 \\ \end{matrix} \]

It is clear that always $\sigma _0 = \text{id}$ and $\sigma _ p = \sigma $.

Having introduced this notation we define for $s \in \check{\mathcal{C}}^{p + 1}(\mathcal{U}, \mathcal{F})$ the element $h(s) \in \check{\mathcal{C}}^ p(\mathcal{U}, \mathcal{F})$ to be the element with components

20.23.6.1
\begin{equation} \label{cohomology-equation-first-homotopy} h(s)_{i_0\ldots i_ p} = \sum \nolimits _{0 \leq a \leq p} (-1)^ a \text{sign}(\sigma _ a) s_{i_{\sigma (0)} \ldots i_{\sigma (a)} i_{\sigma _ a(a)} \ldots i_{\sigma _ a(p)}} \end{equation}

where $\sigma = \sigma ^{i_0 \ldots i_ p}$. The index $i_{\sigma (a)}$ occurs twice in $i_{\sigma (0)} \ldots i_{\sigma (a)} i_{\sigma _ a(a)} \ldots i_{\sigma _ a(p)}$ once in the first group of $a + 1$ indices and once in the second group of $p - a + 1$ indices since $\sigma _ a(j) = \sigma (a)$ for some $j \geq a$ by definition of $\sigma _ a$. Hence the sum makes sense since each of the elements $s_{i_{\sigma (0)} \ldots i_{\sigma (a)} i_{\sigma _ a(a)} \ldots i_{\sigma _ a(p)}}$ is defined over the open $U_{i_0 \ldots i_ p}$. Note also that for $a = 0$ we get $s_{i_0 \ldots i_ p}$ and for $a = p$ we get $(-1)^ p \text{sign}(\sigma ) s_{i_{\sigma (0)} \ldots i_{\sigma (p)}}$.

We claim that

\[ (dh + hd)(s)_{i_0 \ldots i_ p} = s_{i_0 \ldots i_ p} - \text{sign}(\sigma ) s_{i_{\sigma (0)} \ldots i_{\sigma (p)}} \]

where $\sigma = \sigma ^{i_0 \ldots i_ p}$. We omit the verification of this claim. (There is a PARI/gp script called first-homotopy.gp in the stacks-project subdirectory scripts which can be used to check finitely many instances of this claim. We wrote this script to make sure the signs are correct.) Write

\[ \kappa : \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \]

for the operator given by the rule

\[ \kappa (s)_{i_0 \ldots i_ p} = \text{sign}(\sigma ^{i_0 \ldots i_ p}) s_{i_{\sigma (0)} \ldots i_{\sigma (p)}}. \]

The claim above implies that $\kappa $ is a morphism of complexes and that $\kappa $ is homotopic to the identity map of the Čech complex. This does not immediately imply the lemma since the image of the operator $\kappa $ is not the alternating subcomplex. Namely, the image of $\kappa $ is the “semi-alternating” complex $\check{\mathcal{C}}_{semi\text{-}alt}^ p(\mathcal{U}, \mathcal{F})$ where $s$ is a $p$-cochain of this complex if and only if

\[ s_{i_0 \ldots i_ p} = \text{sign}(\sigma ) s_{i_{\sigma (0)} \ldots i_{\sigma (p)}} \]

for any $(i_0, \ldots , i_ p) \in I^{p + 1}$ with $\sigma = \sigma ^{i_0 \ldots i_ p}$. We introduce yet another variant Čech complex, namely the semi-ordered Čech complex defined by

\[ \check{\mathcal{C}}_{semi\text{-}ord}^ p(\mathcal{U}, \mathcal{F}) = \prod \nolimits _{i_0 \leq i_1 \leq \ldots \leq i_ p} \mathcal{F}(U_{i_0 \ldots i_ p}) \]

It is easy to see that Equation (20.9.0.1) also defines a differential and hence that we get a complex. It is also clear (analogous to Lemma 20.23.4) that the projection map

\[ \check{\mathcal{C}}_{semi\text{-}alt}^\bullet (\mathcal{U}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}_{semi\text{-}ord}^\bullet (\mathcal{U}, \mathcal{F}) \]

is an isomorphism of complexes.

Hence the Lemma follows if we can show that the obvious inclusion map

\[ \check{\mathcal{C}}_{ord}^ p(\mathcal{U}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}_{semi\text{-}ord}^ p(\mathcal{U}, \mathcal{F}) \]

is a homotopy equivalence. To see this we use the homotopy

20.23.6.2
\begin{equation} \label{cohomology-equation-second-homotopy} h(s)_{i_0 \ldots i_ p} = \left\{ \begin{matrix} 0 & \text{if} & i_0 < i_1 < \ldots < i_ p \\ (-1)^ a s_{i_0 \ldots i_{a - 1} i_ a i_ a i_{a + 1} \ldots i_ p} & \text{if} & i_0 < i_1 < \ldots < i_{a - 1} < i_ a = i_{a + 1} \end{matrix} \right. \end{equation}

We claim that

\[ (dh + hd)(s)_{i_0 \ldots i_ p} = \left\{ \begin{matrix} 0 & \text{if} & i_0 < i_1 < \ldots < i_ p \\ s_{i_0 \ldots i_ p} & \text{else} & \end{matrix} \right. \]

We omit the verification. (There is a PARI/gp script called second-homotopy.gp in the stacks-project subdirectory scripts which can be used to check finitely many instances of this claim. We wrote this script to make sure the signs are correct.) The claim clearly shows that the composition

\[ \check{\mathcal{C}}_{semi\text{-}ord}^\bullet (\mathcal{U}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}_{ord}^\bullet (\mathcal{U}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}_{semi\text{-}ord}^\bullet (\mathcal{U}, \mathcal{F}) \]

of the projection with the natural inclusion is homotopic to the identity map as desired. $\square$


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