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Let $\check{C}_{\underline{x}}$ denote the \v{C}ech complex with respect to asystem of elements $\underline{x} = x_1,\ldots,x_r$ of a commutative ring $R$.We construct a bounded complex $\mathcal{L}_{\underline{x}}$ of free$R$-modules and a quasi-isomorphism $\mathcal{L}_{\underline{x}}\stackrel{\sim}{\longrightarrow} \check{C}_{\underline{x}}$ and isomorphisms$\mathcal{L}_{\underline{x}} \otimes_R X \congK^{\bullet}(\underline{x}-\underline{U}; X[\underline{U}^{-1}])$ and$\operatorname{Hom}_R(\mathcal{L}_{\underline{x}},X) \congK_{\bullet}(\underline{x}-\underline{U};X[[\underline{U}]])$ for an $R$-complex$X$. Here $\underline{x} - \underline{U}$ denotes the sequence of elements$x_1-U_1,\ldots,x_r-U_r$ in the polynomial ring $R[\underline{U}] =R[U_1,\ldots,U_r]$ in the variables $\underline{U}= U_1,\ldots,U_r$ over $R$.Moreover $X[[\underline{U}]]$ denotes the formal power series complex of $X$ in$\underline{U}$ and $X[\underline{U}^{-1}]$ denotes the complex of inversepolynomials of $X$ in $\underline{U}$. Furthermore$K_{\bullet}(\underline{x}-\underline{U};X[[\underline{U}]])$ resp.$K^{\bullet}(\underline{x}-\underline{U}; X[\underline{U}^{-1}])$ denotes thecorresponding Koszul complex resp. the corresponding Koszul co-complex. Inparticular, there is a bounded $R$-free resolution of$\check{C}_{\underline{x}}$ by a certain Koszul complex. This has variousconsequences e.g. in the case when $\underline{x}$ is a weakly pro-regularsequence. Under this additional assumption it follows that the local cohomology$H^i_{\underline{x} R}(X)$ and the left derived functors of the completion$\Lambda_i^{\underline{x} R}(X), i \in \mathbb{Z},$ is a certain Koszulcohomology and Koszul homology resp. This provides new approaches to the rightderived functor of torsion and the left derived functor of completion withvarious applications.

\v{C}ech (co-) complexes as Koszul complexes and applications