An algebraic construction for integral Cech cohomology Original Research Article

We present a cohomology construction for a pair consisting of a commutative algebra and a space associated to the algebra; the space has as points some subcollection of the saturated multiplicative sets of the algebra, and topology a generalized version of the Zariski topology. We show that these cohomology modules coincide with the integral Cech cohomology modules of a compact Hausdorff space in a special case. The theory of compact ringed spaces is an essential tool in the approach taken. We indicate briefly how this approach simplifies the proof of a long established connection between a cohomology construction for a commutative algebra and the Cech cohomology of the maximal ideal space of a commutative Gelfand ring (of which the real Cech cohomology of a compact Hausdorff space is a special case).