Contraherent cosheaves on schemes
Leonid Positselski
TL;DR
This work develops contraherent cosheaves as covariant duals to quasi-coherent sheaves on schemes, establishing a robust co-contra framework that parallels comodules/contramodules in algebra. It constructs two principal derived-equivalence frameworks—coderived-contraderived (via dualizing complexes) and the nàıve, fiberwise form—linking the derived categories of quasi-coherent sheaves with contraherent cosheaves and enabling explicit functors like Lipman–Neeman f^! and Hartshorne–Deligne f^+. The text also develops the calculus of tensor and Hom operations in this setting, investigates locally contraherent and coflasque variants, and proves gluing, base-change, and projection-formula results that underpin global dualities. The framework extends to non-semi-separated schemes, noncommutative analogues, and Noetherian formal schemes, offering a broad, adaptable approach to co-contra dualities with potential applications to D–Ω duality and p-adic Hodge theory. Overall, it provides a comprehensive foundation for co-contra correspondences on schemes and opens avenues for further extension and applications in noncommutative and formal-geometric contexts.
Abstract
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.