= Hochschild cohomology
{c}
{title2=$HH^n(R,M)$}
{wiki}
For an associative $k$-<algebra> $A$ and an $A$-<bimodule> $M$, <Hochschild cohomology> is the <cohomology> of the <Hochschild cochain complex>, whose degree-$n$ term is $\operatorname{Hom}_k(A^{\otimes n},M)$. Its differential measures compatibility with multiplication and the two <module> actions.
One has $HH^0(A,M)=\{m:am=ma\text{ for all }a\in A\}$ and $HH^1(A,M)=\operatorname{Der}_k(A,M)/\operatorname{Inn}(A,M)$. Thus degree zero detects central elements, and degree one detects <derivations> modulo the <inner derivations>. Degree two classifies suitable square-zero extensions and describes first-order deformations of multiplication, making this a useful invariant of algebra structure.
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