Given a $k$-algebra $A$ with an associative multiplication on it $m:A\otimes A\to A$. It seems to be part of the mathematical folklore that the second Hochschild cohomology group ($HH^2(A,A)$) classifies infinitesimal first order deformations of $m$ up to equivalence.
I have seen the following argument several times: vanishing of the second cohomology group implies rigidity, i.e. they are isolated in the moduli space of associative algebras.
Here is how I think it works: I know that one can interpret the first order deformations as vectors in the Zariski tangent space and suppose that one quotients out the tangent space of orbits of a group action. Then, one sees that the obtained space is isomorphic to the cohomology group. The latter vanishes, which can be used to show that all curves through the point $(A,m)$ are locally in the orbit of $(A,m)$.
However, I am having a hard time making this rigorous or finding a good reference for it.