  Matlis proved a lot of homological properties of the fraction field of an
integral domain. In this paper, we simplify and extend some of them from
1-dimensional (resp. rank one) cases to the higher dimensional (resp. finite
rank) cases. For example, we study the weakly co-torsion property of
Ext$(-,\sim)$, and use it to present splitting criteria. These are equipped
with several applications. For instance, we compute the projective dimension of
$\widehat{R}$ and present some non-noetherian versions of Grothendieck's
localization problem. We construct a new class of co-Hopfian modules and extend
Matlis' decomposability problem to higher ranks. In particular, this paper
deals with the basic properties of Matlis' quadric $(Q,Q/R,
\widehat{R},\overset{\sim}R).$
