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 $v_1otimes dots otimes v_k$ 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 $$tau(v_1otimes dots otimes v_k)= (-1)^{operatorname{deg}(v_k)(sum_1^{k-1} operatorname{deg}v_i)} v_k otimes v_1 otimes dots otimes v_{k-1}.$$

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.