Cyclic homology

Definition

Let $k$ be a field of characteristic $0$ and let $A$ be an associative algebra over $k$ with identity. Tensor products are over $k$. For $n \ge 0$ let $t \colon A^{\otimes (n+1)} \to A^{\otimes (n+1)}$ be the cyclic operator $$t(a_0 \otimes a_1 \otimes \cdots \otimes a_n) = (-1)^n \, a_n \otimes a_0 \otimes \cdots \otimes a_{n-1},$$ and let $C^\lambda_n(A) = A^{\otimes (n+1)} / \im(1 - t)$. The Hochschild boundary $b$ of the complex $C_\bullet(A, A)$ of Hochschild homology maps $\im(1 - t)$ into $\im(1 - t)$, so it induces a boundary on $C^\lambda_\bullet(A)$. The cyclic homology of $A$ is $HC_n(A) = H_n(C^\lambda_\bullet(A), b)$, the homology of this Connes complex.