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

Abstract

Let $\check{C}_{\underline{x}}$ denote the \v{C}ech complex with respect to a system 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 $\mathcal{L}_{\underline{x}} \otimes_R X \cong K^{\bullet}(\underline{x}-\underline{U}; X[\underline{U}^{-1}])$ and $\operatorname{Hom}_R(\mathcal{L}_{\underline{x}},X) \cong K_{\bullet}(\underline{x}-\underline{U};X[[\underline{U}]])$ 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 inverse polynomials 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 the corresponding Koszul complex resp. the corresponding Koszul co-complex. In particular, there is a bounded $R$-free resolution of $\check{C}_{\underline{x}}$ by a certain Koszul complex. This has various consequences e.g. in the case when $\underline{x}$ is a weakly pro-regular sequence. 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 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.