What is $f_!$ in the context of commutative rings? Given a morphism of schemes $f:X \to Y$ there is a functor $f_!:Sh(X) \to Sh(Y)$ where
$$
f_!\mathcal{F}(U) = \{ s \in \mathcal{F}(f^{-1}(U)) : f:\text{supp}(s) \to U \text{ is proper} \}
$$
How do I repackage this in the setting of commutative rings? I know that $f_*:\text{Mod}(S) \to \text{Mod}(R)$ is just composition of the $S$-action with the morphism $f:R \to S$, but I'm not sure how to interpret the support and properness in this setting.
 A: I hope this answer isn't too sloppy, but I think you're mixing two different worlds here: There's the 'analytic world', where you consider sheaves of vector spaces over spaces like manifolds or etale sites over schemes, and there's the 'algebraic world', where you consider coherent sheaves over complex manifolds or schemes. The construction of $f_!$ as you describe it belongs to the analytic world, and - to my knowledge at least, please correct me if I'm wrong - can't be made sense of in the algebraic context (although it often makes sense to put $f_!:=f_\ast$ for coherent sheaves, see below). In any case, I think one should first be clear about what kinds of sheaves one considers.
Edit If you restrict to quasi-compact, separated morphisms between quasi-compact schemes (perhaps I forgot some more assumptions), one puts $f_!=f_\ast$ (which makes sense since the main point is to have a pushforward which commutes with sums, and in the context of schemes, quasi-compactness rather than properness ensures this) and shows the existence of a derived six-functor formalism as KReiser hinted at below. But still, I am not aware of a general six-functor-formalism for coherent sheaves attached to any morphism of schemes, especially none in which one puts $f_!\neq f_\ast$. 
