 Hochschild cohomology

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 ${\mathrm{Hom}}_k(A^{\otimes n},M)$. Its differential measures compatibility with multiplication and the two module actions.

One has $H{H}^0(A,M)=\{m:am=ma\text{ for all }a\in A\}$ and $H{H}^1(A,M)={\mathrm{Der}}_k(A,M)/\mathrm{Inn}(A,M)$. 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.