  Very flat and contradjusted modules naturally arise in algebraic geometry in
the study of contraherent cosheaves over schemes. Here, we investigate the
structure and approximation properties of these modules over commutative
noetherian rings. Using an analogy between projective and flat Mittag-Leffler
modules on one hand, and very flat and locally very flat modules on the other,
we prove that each of the following statements are equivalent to the finiteness
of the Zariski spectrum Spec(R) of a noetherian domain R: (i) the class of all
very flat modules is covering, (ii) the class of all locally very flat modules
is precovering, and (iii) the class of all contraadjusted modules is
enveloping. We also prove an analog of Pontryagin's criterion for locally very
flat modules over Dedekind domains.
