= Hochschild-Kostant-Rosenberg theorem
{c}
= HKR theorem
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{synonym}
= Hochschild-Kostant-Rosenberg
{c}
{synonym}
For the <polynomial ring> $A=k[X_1,\ldots,X_r]$ in <characteristic> zero, antisymmetrization identifies $HH^*(A,A)$ with $\bigwedge_A^*\operatorname{Der}_k(A)$. The map on a wedge of $p$ <derivations> is $\frac1{p!}\sum_{\sigma\in S_p}\operatorname{sgn}(\sigma)\prod_jD_{\sigma(j)}(a_j)$. The <Hochschild cup product> becomes the <exterior product>, and the left <Gerstenhaber bracket> becomes the left <Schouten-Nijenhuis bracket>. This assertion requires a smoothness hypothesis when generalized beyond <polynomial rings>.
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