I have the following topologically motivated question:
Let $R=k[v_1,\ldots,v_m]/I$ be a $k$-algebra for some field $k.$ And let $\widetilde{R}=R/J$ be the quotient algebra of $R$ generated by some linear ideal $J.$
I want to compute the following: $$ \operatorname{Tor}_R(\operatorname{Tor}_{k[v_1,\ldots,v_m]}(R,k),\widetilde{R}) $$
I assume the result to be closely related to the initial ring $R,$ since for trivial $J=0$ it's just $R.$ Perhaps there is some sort of "double Tor duality" kinda like Koszul resolution is self-dual?