  Given a Serre class $\mathcal{S}$ of modules, we compare the containment of
the Koszul homology, Ext modules, Tor modules, local homology, and local
cohomology in $\mathcal{S}$ up to a given bound $s \geq 0$. As some
applications, we give a full characterization of noetherian local homology
modules. Further, we establish a comprehensive vanishing result which readily
leads to the formerly known descriptions of the numerical invariants width and
depth in terms of Koszul homology, local homology, and local cohomology. Also,
we immediately recover a few renowned vanishing criteria scattered about the
literature.
