The integral presheaf Čech complex is a flat resolution of the constant presheaf of integers.

Lemma 21.10.5. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$ 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 $\mathcal{C}$. Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$. The chain complex

$\mathbf{Z}_{\mathcal{U}, \bullet } \otimes _{p, \mathbf{Z}} \mathcal{O}$

is exact in positive degrees. Here $\mathbf{Z}_{\mathcal{U}, \bullet }$ is the chain complex of Lemma 21.10.3, and the tensor product is over the constant presheaf of rings with value $\mathbf{Z}$.

Proof. Let $V$ be an object of $\mathcal{C}$. In the proof of Lemma 21.10.4 we saw that $\mathbf{Z}_{\mathcal{U}, \bullet }(V)$ is isomorphic as a complex to a direct sum of complexes which are homotopic to $\mathbf{Z}$ placed in degree zero. Hence also $\mathbf{Z}_{\mathcal{U}, \bullet }(V) \otimes _\mathbf {Z} \mathcal{O}(V)$ is isomorphic as a complex to a direct sum of complexes which are homotopic to $\mathcal{O}(V)$ placed in degree zero. Or you can use Modules on Sites, Lemma 18.28.10, which applies since the presheaves $\mathbf{Z}_{\mathcal{U}, i}$ are flat, and the proof of Lemma 21.10.4 shows that $H_0(\mathbf{Z}_{\mathcal{U}, \bullet })$ is a flat presheaf also. $\square$

Comment #2045 by Joe Berner on

Suggested slogan: The integral presheaf Cech complex is a flat resolution of the constant presheaf of integers

Comment #3606 by Laurent Moret-Bailly on

In the last sentence of the statement, "cochain complex" should read "chain complex".

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).