Contraherent cosheaves on schemes
Abstract
Contraherent cosheaves are globalizations of contraadjusted or cotorsion modules over commutative rings obtained by gluing together over a scheme, with the colocalization functors HomR(S,-) used for the gluing (where S is the ring of functions on an affine open subscheme in SpecR). 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 Y 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.