Čech Complexes, Čech Homology and Cohomology

Abstract

Čech complexes are important tools in various fields of Mathematics, in particular in Algebraic Geometry and Commutative Algebra. In Commutative Algebra the Čech complex is known for its relation to local cohomology in the case when the underlying ring is Noetherian. Here we start with a general investigation of the construction of the Čech complex with respect to a sequence of elements $\underline{x}= x_1,\ldots , x_k$ of a commutative ring R. We investigate Čech homology and cohomology and prove the Ext-depth Tor-codepth sensitivity of the Čech complex as well as some inequalities. One of the new features here is the general assumption of a finite set of elements in a commutative ring R and unbounded R-complexes X.