Hochschild cohomology of a polynomial algebra in one variable ($H{H}^0(k[x])=k[x],H{H}^1(k[x])=k[x]{\partial }_x,H{H}^{q\ge 2}(k[x])=0$)

Over a field, the bimodule resolution $0\to k[x]\otimes k[x]\stackrel{x\otimes 1-1\otimes x}{\to }k[x]\otimes k[x]\to k[x]\to 0$ is exact: multiplication identifies the quotient by $x\otimes 1-1\otimes x$, and that element is a non-zero-divisor in the two-variable polynomial ring. Applying the bimodule Hom functor gives zero differential, proving the displayed Hochschild cohomology. Degree-one classes are the derivations $f(x){\partial }_x$.