First-order associative deformation
= First-order associative deformation
{title2=$a*b=ab+\varepsilon f(a,b),\quad\varepsilon^2=0$}
The order-$\varepsilon$ 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 $T=1+\varepsilon g$ 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.