  Contraherent cosheaves are globalizations of contraadjusted or cotorsion
modules over commutative rings obtained by gluing together over a scheme, with
the colocalization functors $\operatorname{Hom}_R(S,{-})$ used for the gluing
(where $S$ is the ring of functions on an affine open subscheme in
$\operatorname{Spec}R$). The category of contraherent cosheaves over a scheme
is a Quillen exact category with exact functors of infinite product. Over a
quasi-compact semi-separated scheme or a Noetherian scheme of finite Krull
dimension (in a different version - over any locally Noetherian scheme), it
also has enough projectives. We construct the derived co-contra correspondence
over a scheme in two forms. The "naive" one is an equivalence of the
conventional derived categories of quasi-coherent sheaves and contraherent
cosheaves, valid over any quasi-compact semi-separated scheme. The more
sophisticated version is an equivalence between the coderived category of
quasi-coherent sheaves and the contraderived category of contraherent cosheaves
over a Noetherian scheme with a dualizing complex. The former point of view
allows us to obtain an explicit construction of the Lipman-Neeman extraordinary
inverse image functor $f^!$ for a morphism of quasi-compact semi-separated
schemes $f\colon Y\to X$. The latter approach provides an expanded version of
the covariant Serre-Grothendieck duality theory and leads to the
Hartshorne-Deligne extraordinary inverse image functor $f^!$ (which we denote
by $f^+$) for a morphism of finite type $f$ between Noetherian schemes. We also
construct a derived semico-semicontra correspondence, mounting the "naive"
version along the fibers on top of the one depending on a dualizing complex on
the base of a flat fibration. Noncommutative analogues of Noetherian stacks,
affine Noetherian formal schemes, and ind-affine ind-schemes are briefly
discussed in the appendices.
