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

Let ${\check{C}}_{\underset{‾}{x}}$ denote the \v{C}ech complex with respect to a system of elements $\underset{‾}{x}=x_1,\dots ,x_r$ of a commutative ring $R$. We construct a bounded complex ${\mathcal{L}}_{\underset{‾}{x}}$ of free $R$-modules and a quasi-isomorphism $$\mathcal{L}${\underline{x}} \stackrel{\sim}{\longrightarrow} \check{C}{\underline{x}}$andisomorphisms$$\mathcal{L}${\underline{x}} \otimes_R X \cong K^{\bullet}(\underline{x}-\underline{U}; X[\underline{U}^{-1}])$and$\operatorname{Hom}R($\mathcal{L}${\underline{x}},X) \cong K{\bullet}(\underline{x}-\underline{U};X[[\underline{U}]])$foran$R$-complex$X$\mathrm{.}Here$\underline{x} - \underline{U}$denotesthesequenceofelements$x_1-U_1,\ldots,x_r-U_r$inthepolynomialring$R[\underline{U}] = R[U_1,\ldots,U_r]$inthevariables$\underline{U}= U_1,\ldots,U_r$over$R$. Moreover $X[[\underset{‾}{U}]]$ denotes the formal power series complex of $X$ in $\underset{‾}{U}$ and $X[{\underset{‾}{U}}^{-1}]$ denotes the complex of inverse polynomials of $X$ in $\underset{‾}{U}$. Furthermore $K_{\bullet }(\underset{‾}{x}-\underset{‾}{U};X[[\underset{‾}{U}]])$ resp. $K^{\bullet }(\underset{‾}{x}-\underset{‾}{U};X[{\underset{‾}{U}}^{-1}])$ denotes the corresponding Koszul complex resp. the corresponding Koszul co-complex. In particular, there is a bounded $R$-free resolution of ${\check{C}}_{\underset{‾}{x}}$ by a certain Koszul complex. This has various consequences e.g. in the case when $\underset{‾}{x}$ is a weakly pro-regular sequence. Under this additional assumption it follows that the local cohomology $H_{\underset{‾}{x}R}^i(X)$ and the left derived functors of the completion ${\mathrm{\Lambda }}_i^{\underset{‾}{x}R}(X),i\in \mathbb{Z},$ is a certain Koszul cohomology and Koszul homology resp. This provides new approaches to the right derived functor of torsion and the left derived functor of completion with various applications.