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 ${\mathrm{Hom}}_R(S,-)$ used for the gluing (where $S$ is the ring of functions on an affine open subscheme in $\mathrm{Spec}R$). 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 $f:Y\to X$. 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.

arXiv commentLaTeX 2e with xy-pic and mathx fonts; 470 pages, 62 commutative diagrams. v.18: all the material of old version v6 now included; v.19: Introduction expanded, Preface added; v.22: Remark 2.3.1 and Example 2.3.2 inserted, Theorems 7.1.1(c) and 7.2.2(a) became more general, new Section 8.4 inserted; v.25: this is the final arXiv version, the publisher's version will be more complete

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