On the derived category of a regular toric scheme |
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Authors: | Thomas Hüttemann |
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Institution: | 1.Pure Mathematics Research Centre,Queen’s University Belfast,Belfast,Northern Ireland, UK |
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Abstract: | Let X be a quasi-compact scheme, equipped with an open covering by affine schemes U
σ
= Spec A
σ
. A quasi-coherent sheaf on X gives rise, by taking sections over the U
σ
, to a diagram of modules over the coordinate rings A
σ
, indexed by the intersection poset Σ of the covering. If X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category of quasi-coherent
sheaves on X can be obtained from a category of Σop-diagrams of chain complexes of modules by inverting maps which induce homology isomorphisms on hyper-derived inverse limits.
Moreover, we show that there is a finite set of weak generators, one for each cone in the fan Σ. The approach taken uses the
machinery of Bousfield–Hirschhorn colocalisation of model categories. The first step is to characterise colocal objects; these turn out to be homotopy sheaves
in the sense that chain complexes over different open sets U
σ
agree on intersections up to quasi-isomorphism. In a second step it is shown that the homotopy category of homotopy sheaves
is equivalent to the derived category of X. |
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Keywords: | |
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