= Hochschild cohomology of a polynomial algebra in one variable
{c}
{title2=$HH^0(k[x])=k[x],\quad HH^1(k[x])=k[x]\partial_x,\quad HH^{q\ge2}(k[x])=0$}
Over a <field>, the <bimodule> resolution $0\to k[x]\otimes k[x]\xrightarrow{x\otimes1-1\otimes x}k[x]\otimes k[x]\to k[x]\to0$ is exact: multiplication identifies the quotient by $x\otimes1-1\otimes x$, 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> $f(x)\partial_x$.
Back to article page
