First-order associative deformation ($a\ast b=ab+\epsilon f(a,b),{\epsilon }^2=0$)

The order-$\epsilon$ part of associativity is $af(b,c)-f(ab,c)+f(a,bc)-f(a,b)c=0$, so $f$ is a Hochschild cocycle. A change of coordinates $T=1+\epsilon g$ changes $f$ to $f+\delta g$. Therefore equivalence classes of first-order associative deformations of a fixed underlying algebra are its second Hochschild cohomology. Unital deformations use normalized cochains.

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.