The Stacks project

Lemma 36.9.2. In Situation 36.9.1. Let $M$ be an $A$-module and denote $\mathcal{F}$ the associated $\mathcal{O}_ X$-module. Then there is a canonical isomorphism of complexes

\[ \mathop{\mathrm{colim}}\nolimits _ e \mathop{\mathrm{Hom}}\nolimits _ A(I^\bullet (f_1^ e, \ldots , f_ r^ e), M) \longrightarrow \check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F}) \]

functorial in $M$.

Proof. Recall that the alternating Čech complex is the subcomplex of the usual Čech complex given by alternating cochains, see Cohomology, Section 20.23. As usual we view a $p$-cochain in $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ as an alternating function $s$ on $\{ 1, \ldots , r\} ^{p + 1}$ whose value $s_{i_0\ldots i_ p}$ at $(i_0, \ldots , i_ p)$ lies in $M_{f_{i_0}\ldots f_{i_ p}} = \mathcal{F}(U_{i_0\ldots i_ p})$. On the other hand, a $p$-cochain $t$ in $\mathop{\mathrm{Hom}}\nolimits _ A(I^\bullet (f_1^ e, \ldots , f_ r^ e), M)$ is given by a map $t : \wedge ^{p + 1}(A^{\oplus r}) \to M$. Write $[i] \in A^{\oplus r}$ for the $i$th basis element and write

\[ [i_0, \ldots , i_ p] = [i_0] \wedge \ldots \wedge [i_ p] \in \wedge ^{p + 1}(A^{\oplus r}) \]

Then we send $t$ as above to $s$ with

\[ s_{i_0\ldots i_ p} = \frac{t([i_0, \ldots , i_ p])}{f_{i_0}^ e\ldots f_{i_ p}^ e} \]

It is clear that $s$ so defined is an alternating cochain. The construction of this map is compatible with the transition maps of the system as the transition map

\[ I^\bullet (f_1^ e, \ldots , f_ r^ e) \leftarrow I^\bullet (f_1^{e + 1}, \ldots , f_ r^{e + 1}), \]

of the (36.9.0.1) sends $[i_0, \ldots , i_ p]$ to $f_{i_0}\ldots f_{i_ p}[i_0, \ldots , i_ p]$. It is clear from the description of the localizations $M_{f_{i_0}\ldots f_{i_ p}}$ in Algebra, Lemma 10.9.9 that these maps define an isomorphism of cochain modules in degree $p$ in the limit. To finish the proof we have to show that the map is compatible with differentials. To see this recall that

\begin{align*} d(s)_{i_0\ldots i_{p + 1}} & = \sum \nolimits _{j = 0}^{p + 1} (-1)^ j s_{i_0\ldots \hat i_ j \ldots i_ p} \\ & = \sum \nolimits _{j = 0}^{p + 1} (-1)^ j \frac{t([i_0, \ldots , \hat i_ j, \ldots i_{p + 1}])}{f_{i_0}^ e\ldots \hat f_{i_ j}^ e \ldots f_{i_{p + 1}}^ e} \end{align*}

On the other hand, we have

\begin{align*} \frac{d(t)([i_0, \ldots , i_{p + 1}])}{f_{i_0}^ e\ldots f_{i_{p + 1}}^ e} & = \frac{t(d[i_0, \ldots , i_{p + 1}])}{f_{i_0}^ e\ldots f_{i_{p + 1}}^ e} \\ & = \frac{\sum _ j (-1)^ j f_{i_ j}^ e t([i_0, \ldots , \hat i_ j, \ldots i_{p + 1}])}{f_{i_0}^ e \ldots f_{i_{p + 1}}^ e} \end{align*}

The two formulas agree by inspection. $\square$


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