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  "body": "\n## Summary\n\nCyclic homology is a homology theory for [[associative algebras|3CTW1P4HBP]] introduced by Alain Connes and, independently, by Boris Tsygan.\nIt refines [[Hochschild homology|S5H153HFWC]] by taking into account the cyclic rotation of tensor factors.\nFor smooth commutative algebras in characteristic $0$ it is computed from de Rham cohomology.\nSo it plays the role of de Rham theory in noncommutative geometry.\nIt is also the target of trace maps from [[algebraic K-theory|5GVFZWMWC8]].\n\n## Explanations\n\n### L1 human\n\nHochschild homology is built from strings $a_0 \\otimes a_1 \\otimes \\cdots \\otimes a_n$ of elements of an algebra.\nCyclic homology treats such a string as written around a circle, so that rotating it gives the same object up to a sign.\nThis extra symmetry makes cyclic homology closer to topology and to de Rham cohomology.\nFor the ground field $k$ itself, cyclic homology is $k$ in every even degree and $0$ in odd degrees.\nConnes used it to define index-type invariants for noncommutative spaces.\n",
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