Interpretations for higher Tor functors

Question

Let's work in the category $R$-${\sf mod}$, for concreteness. I know that one can see the modules ${\rm Ext}^n_R(M,N)$ as modules of equivalence classes of $n$-extensions of $M$ by $N$ (Yoneda extensions), namely, exact sequences of the form $$0 \to N \to E_1 \to \cdots \to E_n \to M \to 0,$$with certain operations (more precisely, if one denotes such collection by ${\rm E}^n(M,N)$, there are natural isomorphisms ${\rm E}^n(M,N)\cong {\rm Ext}_R^n(M,N)$ for each $n$).

Is there anything similar for ${\rm Tor}^R_n(M,N)$?

I expect the answer to be highly non-trivial, for the following analogy: this business about $n$-extensions effectively gives us a way to describe the elements of ${\rm Ext}^n_R(M,N)$, but why should we expect any simple explanation for ${\rm Tor}$ if we cannot even describe the elements of ${\rm Tor}_0^R(M,N) = M\otimes_RN$ in general?

Apologies if by any chance this is a repeated question, a quick search on the website didn't show up anything here.

Answer 1

I don't know an answer for general rings/modules but the best approach is probably to write Tor in terms of Ext when you want a similar interpretation for Tor with exact sequences. For Artin algebras $R$ with duality $D$ (which include for example all quiver algebras $KQ/I$) and finitely generated modules $Y$ and $Z$ one then has $$\operatorname{Tor}_n^R(Y,Z)=D\operatorname{Ext}_R^n(Y,D(Z)).$$ So in this case Tor for $Y$ and $Z$ has an interpretation as a dual space of extensions between $Y$ and $D(Z)$. I think the same should work for general rings/modules with a duality having good properties.