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  "body": "\n## Definition\n\nLet $k$ be a field of characteristic $0$ and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$ with identity.\nTensor products are over $k$.\nFor $n \\ge 0$ let $t \\colon A^{\\otimes (n+1)} \\to A^{\\otimes (n+1)}$ be the cyclic operator\n$$t(a_0 \\otimes a_1 \\otimes \\cdots \\otimes a_n) = (-1)^n \\, a_n \\otimes a_0 \\otimes \\cdots \\otimes a_{n-1},$$\nand let $C^\\lambda_n(A) = A^{\\otimes (n+1)} / \\im(1 - t)$.\nThe Hochschild boundary $b$ of the complex $C_\\bullet(A, A)$ of [[Hochschild homology|S5H153HFWC]] maps $\\im(1 - t)$ into $\\im(1 - t)$, so it induces a boundary on $C^\\lambda_\\bullet(A)$.\nThe *cyclic homology* of $A$ is $HC_n(A) = H_n(C^\\lambda_\\bullet(A), b)$, the homology of this Connes complex.\n\n## Remarks\n\nExample: in degree $0$, $C^\\lambda_0(A) = A$ and $b(a_0 \\otimes a_1) = a_0 a_1 - a_1 a_0$, so $HC_0(A) = A/[A, A] = HH_0(A)$.\nExample: for $A = k$, the space $C^\\lambda_n(k)$ is $k$ for $n$ even and $0$ for $n$ odd, since then $t = -1$ and $1 - t = 2$ is invertible.\nThe boundary vanishes on it, so $HC_n(k) = k$ for even $n \\ge 0$ and $HC_n(k) = 0$ for odd $n$.\nOver a general ground ring, cyclic homology is defined from Connes' bicomplex instead, and the two definitions agree when $k$ contains $\\Q$.\nThere is a long exact sequence of Connes relating the two theories:\n$$\\cdots \\to HH_n(A) \\to HC_n(A) \\to HC_{n-2}(A) \\to HH_{n-1}(A) \\to \\cdots.$$\n",
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