Let $A = Sym^*(V)$ be a polynomial ring. The Koszul complex
$\cdots \to \wedge^2 V \otimes A \to V \otimes A \to A$
gives a resolution of the residue field $k$, so for any $A$-module $M$, the homology of
$\cdots \to \wedge^2 V \otimes M \to V \otimes M \to M$
can be interpreted as $Tor^A_*(M, k)$.
I recently came across the following variant of this complex (and in particular, its homology) where the exterior powers are replaced by tensor powers, so you have something like
$\cdots \to V^{\otimes 3} \otimes M \to V^{\otimes 2} \otimes M \to V \otimes M \to M$
(the differentials are defined a similar way: take an alternating sum of all possible ways to multiply a copy $V$ with $M$) and would like to know if there is some meaning to the homology groups.
I know of one special case: let $B = Sym^*(V) / Sym^2(V)$, i.e., the polynomial ring modulo the ideal generated by all degree $2$ polynomials. Then the minimal free resolution of the residue field over $B$ is
$\cdots \to V^{\otimes 3} \otimes B \to V^{\otimes 2} \otimes B \to V \otimes B \to B$
so that if $M$ were a module over $B$, I could interpret it as $Tor^B_*(M,k)$.
