 Second-order obstruction to an associative deformation

Expanding associativity for $a\ast b=ab+tf(a,b)+t^2{\mu }_2(a,b)$ modulo $t^3$ gives the displayed equation. If $\delta f=0$, its right side is a degree-three Hochschild cocycle. Its class in $H{H}^3(R,R)$ must vanish to extend the first-order formal associative deformation to second order. A nonzero class in $H{H}^2$ therefore need not integrate to a full formal associative deformation.