Cocycles in the relative Hochschild cohomology

Let $E$ be a long exact sequence of modules over an algebra $A$ admitting a contracting homotopy over a subalgebra $B\subset A$. In settings of the relative homological algebra, the Hochschild cocycle corresponding to the sequence $E$ is described explicitly. The “absolute case” has been earlier investigated by V. Dlab and C. M. Ringel. Being applied to that case, the present investigation gives a shorter arguments than that of Dlab–Ringel's work.