On the derived category of a regular toric scheme
On the derived category of a regular toric scheme
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关于正则复曲面方案的派生范畴
DOI：
10.1007/s10711-009-9389-7
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发表时间：
2008
影响因子：
0.5
通讯作者：
T. Hüttemann
中科院分区：
文献类型：
--
作者：
T. Hüttemann
设X是一个拟紧概型，它具有仿射概型Uσ = Spec Aσ的开覆盖. X上的拟凝聚层通过在Uσ上取截面，得到坐标环Aσ上的模图，由覆盖的交偏序集下标。若X是任意交换环上的正则环面概型，我们证明了X上的拟凝聚层的无界导范畴可以由模的链复形的拓扑图范畴通过在超导逆极限上诱导同调同构的逆映射得到.此外，我们还证明了存在一个有限的弱生成元集，每个锥都有一个弱生成元。所采取的方法使用的机器的Bousfield-Hirschhorn共定位模型类别。第一步是确定同域对象;在不同开集Uσ上的链复形在交上一致直到准同构的意义上，这些对象是同伦层。第二步证明了同伦层的同伦范畴等价于X的导出范畴。
Let X be a quasi-compact scheme, equipped with an open covering by affine schemes Uσ = Spec Aσ. A quasi-coherent sheaf on X gives rise, by taking sections over the Uσ, to a diagram of modules over the coordinate rings Aσ, 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 Uσ 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.