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  "body": "\n## Summary\n\nHochschild homology is a homology theory for [[associative algebras|3CTW1P4HBP]], with coefficients in a [[bimodule|EM7FHAFD3N]].\nIt is computed from the Hochschild complex, whose $n$-th term is $M \\otimes A^{\\otimes n}$.\nIn degree $0$ it gives $A/[A, A]$, the quotient of the algebra by its commutators.\nFor smooth commutative algebras in characteristic $0$ it recovers [[Kähler differential forms|7HCFARHG7D]], so it is a noncommutative analogue of differential forms.\n\n## Explanations\n\n### L1 human\n\nHochschild homology measures how far an algebra is from being commutative, and in which ways.\nIn the lowest degree it keeps an element $a$ only up to adding commutators $bc - cb$.\nFor the matrix algebra $M_n(k)$ this quotient is a single copy of $k$, detected by the trace.\nIn higher degrees it records relations among such commutators, built from tensor products of elements of the algebra.\nFor the polynomial ring $k[x]$ it is $k[x]$ in degree $0$, the forms $f(x)\\,dx$ in degree $1$, and zero above.\n",
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