Triple cohomology of Lie–Rinehart algebras and the canonical class of associative algebras

We introduce a bicomplex which computes the triple cohomology of Lie–Rinehart algebras. We prove that the triple cohomology is isomorphic to the Rinehart cohomology provided the Lie–Rinehart algebra is projective over the corresponding commutative algebra. As an application we construct a canonical class in the third dimensional cohomology corresponding to an associative algebra and extend Sridharanʹs result on almost commutative algebras.