Cohomology groups of integral domains and -algebras

Abstract

In this paper, we introduce a new homology theory devoted to the study of linear operators such as local mutipliers and band preserving operators. The idea is to study the vanishing homology problem. This enables us to characterize integral domains in which any local multiplier is a multiplier, which gives a partial answer to a problem posed by R. V. Kadison [J. Algebra 130, No.2, (1990) 494-509]. Finally, we solve the Wickstead problem [Compositio Math, 35(3) (1977), 225--238] for the class of Archimedean unital f-algebras.