  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.
