Lemma 21.9.3 (03AS)—The Stacks project

Lemma 21.9.3. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ exist in $\mathcal{C}$. Consider the chain complex $\mathbf{Z}_{\mathcal{U}, \bullet }$ of abelian presheaves

\[ \ldots \to \bigoplus _{i_0i_1i_2} \mathbf{Z}_{U_{i_0} \times _ U U_{i_1} \times _ U U_{i_2}} \to \bigoplus _{i_0i_1} \mathbf{Z}_{U_{i_0} \times _ U U_{i_1}} \to \bigoplus _{i_0} \mathbf{Z}_{U_{i_0}} \to 0 \to \ldots \]

where the last nonzero term is placed in degree $0$ and where the map

\[ \mathbf{Z}_{U_{i_0} \times _ U \ldots \times _ u U_{i_{p + 1}}} \longrightarrow \mathbf{Z}_{U_{i_0} \times _ U \ldots \widehat{U_{i_ j}} \ldots \times _ U U_{i_{p + 1}}} \]

is given by $(-1)^ j$ times the canonical map. Then there is an isomorphism

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_{\mathcal{U}, \bullet }, \mathcal{F}) = \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \]

functorial in $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{PAb}(\mathcal{C}))$.

Proof. This is a tautology based on the fact that

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}( \bigoplus _{i_0 \ldots i_ p} \mathbf{Z}_{U_{i_0} \times _ U \ldots \times _ U U_{i_ p}}, \mathcal{F}) & = \prod _{i_0 \ldots i_ p} \mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}( \mathbf{Z}_{U_{i_0} \times _ U \ldots \times _ U U_{i_ p}}, \mathcal{F}) \\ & = \prod _{i_0 \ldots i_ p} \mathcal{F}(U_{i_0} \times _ U \ldots \times _ U U_{i_ p}) \end{align*}

see Modules on Sites, Lemma 18.4.2. $\square$

The Stacks project

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