On the derived category of a regular toric scheme

TL;DR

This paper proves that for regular toric schemes, the derived category of quasi-coherent sheaves can be constructed from twisted diagrams without gluing conditions, using model category techniques and explicit generators.

Contribution

It introduces a homotopy-theoretic approach to describe the derived category of regular toric schemes via twisted diagrams and model categories.

Findings

Derived category obtained from twisted diagrams by inverting homology isomorphisms.

Explicit finite set of weak generators for the derived category.

New proof of classical generation result for projective space.

Abstract

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes. A quasi-coherent sheaf on X gives rise, by taking sections over the covering sets, to a diagram of modules over the various coordinate rings. The resulting "twisted" diagram of modules satisfies a certain gluing condition, stating that the data is compatible with restriction to smaller open sets. In case X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category D(X) of quasi-coherent sheaves on X can be obtained from a category of twisted diagrams which do not necessarily satisfy any gluing condition by inverting maps which induce homology isomorphisms on hyper-derived inverse limits. Moreover, we given an explicit construction of a finite set of weak generators for the derived category. For example, if X is projective n-space then D(X) is generated by n+1 successive twists of the structure sheaf; the present paper gives a new homotopy-theoretic proof of this classical result. The approach taken uses the language of model categories, and the machinery of Bousfield-Hirschhorn colocalisation. 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 agree on intersections up to quasi-isomorphism only. In a second step it is shown that the homotopy category of homotopy sheaves is the derived category of X.