# A cyclic-like chain complex

Let $$V$$ be a $$\mathbb{Z}$$-graded vector space over an algebraically closed field $$k$$ of characteristic zero.

Let $$\overline{TV}= \bigoplus_{n=1}^{\infty} V^{\otimes n}$$ be the reduced tensor algebra. The degree of an element $$v_1\otimes \dots \otimes v_k$$ is defined as $$\sum_i \operatorname{deg}(v_i)$$.

Suppose that $$d$$ is a differential of degree $$-1$$ on $$\overline{TV}$$ which respects the (graded) Leibniz rule.

Let $$\tau: \overline{TV} \to \overline{TV}$$ be the cyclic permutation operator defined on pure tensors by $$\tau(v_1\otimes \dots \otimes v_k)= (-1)^{\operatorname{deg}(v_k)(\sum_1^{k-1} \operatorname{deg}v_i)} v_k \otimes v_1 \otimes \dots \otimes v_{k-1}.$$

We can consider the module of co-invariants $$\overline{TV}^{\tau}:= \overline{TV}/ (1- \tau)$$, and observe that $$d$$ passes to the quotient.

Question: What does the homology of $$(\overline{TV}^{\tau}, d)$$ compute? If $$d=0$$, this seems to agree with cyclic homology of $$\overline{TV}$$, but I don't see what it is in general. I assume that this is something well known.