Cohomology for Operator Algebras: The Mayer–Vietoris Sequence

For an inclusion B⊆A of operator algebras we introduce the B-nil cohomology groupsHn(A∣B), which can be easier to compute than the standard cohomology groupsHn(A). We then develop two long exact sequences, the first linkingHn(A∣B),Hn(A) andHn(B), the second corresponding to the Mayer–Vietoris sequence of algebraic topology. These sequences provide powerful computational tools, as illustrated by the examples in the last two sections of the paper.