Contraherent cosheaves on schemes
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arXiv
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| Natura: | Preprint |
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2012
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| _version_ | 1866911296888242176 |
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| author | Positselski, Leonid |
| author_facet | Positselski, Leonid |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1209_2995 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | Contraherent cosheaves on schemes Positselski, Leonid Category Theory Algebraic Geometry 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. |
| title | Contraherent cosheaves on schemes |
| topic | Category Theory Algebraic Geometry |
| url | https://arxiv.org/abs/1209.2995 |