I don't know if this is in SGA IV.5, but that's a good place to look for questions about Cech cohomology.

As I described here, the Cech cohomology with respect to a cover is the same as the sheaf cohomology in the sieve associated to that cover. If $\mathcal{U}$ is a cover of $X$, let $R$ be the category whose objects are maps $V \rightarrow X$ that factor through some object in $\mathcal{U}$. Then

$\check{H}^p(\mathcal{U}, F) = \varprojlim^{(p)}_{R} F = Ext^p(\mathbf{Z}_R, F)$

where $\varprojlim^{(p)}$ is the $p$-th derived functor $\varprojlim$. This can be calculated by taking a projective resolution of $\mathbf{Z}_R$. Here are two ways to do it:

$\displaystyle K_p = \sum_{i_1 < i_2 < \cdots < i_p} \mathbf{Z}_{U_{i_1} \cap \cdots \cap U_{i_p}}$

$\displaystyle L_p = \sum_{i_1, \ldots, i_p} \mathbf{Z}_{U_{i_1} \cap \cdots \cap U_{i_p}}$.`

One must check, of course, that these are indeed resolutions. (I don't have a slick explanation of why they are resolutions. The best I can do is to say that these complexes are associated via the Dold--Kan correspondence to simplicial resolutions of the final presheaf on $R$.) Taking $Hom$ into $F$ yields the two Cech complexes in question.

I don't know if this is in SGA IV.5, but that's a good place to look for questions about Cech cohomology.

As I described here, the Cech cohomology with respect to a cover is the same as the sheaf cohomology in the sieve associated to that cover. If $\mathcal{U}$ is a cover of $X$, let $R$ be the category whose objects are maps $V \rightarrow X$ that factor through some object in $\mathcal{U}$. Then

$\check{H}^p(\mathcal{U}, F) = \varprojlim^{(p)}_{R} F = Ext^p(\mathbf{Z}_R, F)$

where $\varprojlim^{(p)}$ is the $p$-th derived functor $\varprojlim$. This can be calculated by taking a projective resolution of $\mathbf{Z}_R$. Here are two ways to do it:

$\displaystyle K_p = \sum_{i_1 < i_2 < \cdots < i_p} \mathbf{Z}_{U_{i_1} \cap \cdots \cap U_{i_p}}$

$\displaystyle L_p = \sum_{i_1, \ldots, i_p} \mathbf{Z}_{U_{i_1} \cap \cdots \cap U_{i_p}}$.`

One must check, of course, that these are indeed resolutions. (I don't have a slick explanation of why they are resolutions. The best I can do is to say that these complexes are associated via the Dold--Kan correspondence to simplicial resolutions of the final presheaf on $R$.) Taking $Hom$ into $F$ yields the two Cech complexes in question.