Čech (Co-) Complexes as Koszul Complexes and Applications

Abstract

In the main results of the paper it is shown that the Čech (co-) homology might be considered as an appropriate Koszul (co-) homology. Let $\check {C}_{\underline {x}}$ denote the Čech 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 ${\mathscr{L}}_{\underline {x}}$ of free R-modules and a quasi-isomorphism ${\mathscr{L}}_{\underline {x}} \overset {\sim }{\longrightarrow } \check {C}_{\underline {x}}$ and isomorphisms ${\mathscr{L}}_{\underline {x}} \otimes _{R} X \cong K^{\bullet }(\underline {x}-\underline {U}; X[\underline {U}^{-1}])$ and $\text {Hom}_{R}({\mathscr{L}}_{\underline {x}},X) \cong K_{\bullet }(\underline {x}-\underline {U};X[[\underline {U}]])$ for an R-complex X. Here $\underline {x} - \underline {U}$ denotes the sequence of elements x₁ − U₁,…,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 proregular 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 ${\Lambda }_{i}^{\underline {x} R}(X), i \in \mathbb {Z},$ are 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.