Homogeneous spectral sets and local-global methods in Banach algebras (Q1090896)

The search for topological characterizations of algebraic properties of commutative Banach algebras has often led to the calculation of cohomology groups of its spectrum. In this paper, a tool is developed which permits the application of sheaf cohomological methods to such problems by ''lifting'' sequences of complex analytic manifolds to sequences of spectral sets and cohomology groups. This is made possible by a definition of sheaf of an algebra over its spectrum, and by the fact that in lifting exact sequences of Lie groups and homogeneous spaces to sequences of spectral sets, exactness is preserved locally.