The Hochschild cocycle corresponding to a long exact sequence

$(a_{0}, \cdots, a_{l+1})\Omega_{\beta}=\overline{a}_{0}\beta_{0}\overline{a}_{1}\beta_{1}\cdots\overline{a}_{t}\beta_{t}\overline{a}_{t+1}$ , for $a_{0},$ $\cdots,$ $a_{t+1}\in A$ , where $\overline{a}_{i}$ denotes the scalar multiplication by $a_{i}$ (on $Z_{i}$ ); note that all maps will be written on the right of the argument, thus the composition of $\beta_{0}$ : $Z_{0}\rightarrow Z_{1}$ , and $\beta_{1}$ : $Z_{1}\rightarrow Z_{2}$ is denoted by $\beta_{0}\beta_{1}$ . Given the exact sequence $E$ exhibited above, it clearly splits as a sequence of k-spaces, thus there are k-linear maps