Let $f: R\to S$ be a map between two commutative Noetherian rings. Let $G_0(R)=K_0(mod R)$ be the Grothendieck group of finite generated modules over $R$. It means $G_0(R)$ is the quotient of the free abelian group on all isomorphism class of finitely generated modules over $R$ by the subgroup generated by relations coming from short exact sequences.

If $fd_RS<\infty$, one can define a map $f^*:G_0(R)\to G_0(S)$ by: $$f^*([M]) = \sum_{i\geq0} (-1)^i [Tor_R^i(M,S)] $$

(for reference, see Section 7, Chapter 2 of Weibel's book on K-theory.

Now, if $R$ is not regular or $S$ is not a complete intersection in $R$, then having finite flat dimension is somewhat a miraculous condition. So my question is: Can a map $f^*$ be defined in a more general situation than for finite flat dimension maps?

EDIT: Let me elaborate a little bit because of some interesting answers and comments below (especially Clark's answer). The main motivation I have in mind is the case of $R$ being a hypersurface. Then most $R$- modules have infinite resolutions, but it is well known that their resolution is eventually periodic. So, even though they are not homologically finite, the modules can be homologically described with finite data (i.e. finite number of matrices).

The fact above is crucial in many results I know about hypersurfaces. For a random recent example, see here. In particular, in this situation one can define( at least when $S$ is finite $R$-module):

$$f^*([M])= [Tor^{2n}(M,S)] - [Tor^{2n-1}(M,S)]$$

for sufficiently big $n$.

So that's one of the reasons I wonder if there is more systematic map one can define.