Combinatorial aspects of virtually Cohen-Macaulay sheaves
Combinatorial aspects of virtually Cohen-Macaulay sheaves
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虚拟 Cohen-Macaulay 滑轮的组合方面
DOI：
10.1112/tlm3.12036
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Yang, Jay
中科院分区：
文献类型：
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作者：
Berkesch, Christine;Klein, Patricia;Loper, Michael C.;Yang, Jay
当研究光滑投射环面簇的考克斯环上的分次模时，通常有两种标准类型的分解来收集信息：自由分解和其层化的向量丛分解。每种方法都有自己的挑战。有一些几何信息，自由的决议未能编码，而矢量束决议可以抵制使用代数和组合技术的研究。最近，Berkesch，Erman和Smith介绍了虚拟分辨率，它捕获所需的几何信息，也适合代数和组合研究。虚拟分辨率的理论包括虚拟科恩-麦考利属性的概念，尽管用于评估哪些模块是虚拟科恩-麦考利的工具直到最近才开始开发。在这篇文章中，我们继续在两个相关的方式这个研究计划。第一个是，当是一个产品的射影空间，我们产生了一个大的新类的虚拟Cohen-Macaulay斯坦利-Reisner环，我们表明是虚拟Cohen-Macaulay通过明确的建设适当的虚拟决议反映了潜在的组合结构。第二个是，对于任意光滑的投射复曲面簇，我们开发了用于评估虚拟Cohen-Macaulay性质的同调工具。其中一些工具提供了排除标准，另一些则是产生适当简短的虚拟决议的建设性方法。我们还使用这些工具来建立数学上，几何上和虚拟的Cohen-Macaulay属性之间的关系。
When studying a graded moduleover the Cox ring of a smooth projective toric variety, there are two standard types of resolutions commonly used to glean information: free resolutions ofand vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study. The theory of virtual resolutions includes a notion of a virtually Cohen–Macaulay property, though tools for assessing which modules are virtually Cohen–Macaulay have only recently started to be developed. In this article, we continue this research program in two related ways. The first is that, whenis a product of projective spaces, we produce a large new class of virtually Cohen–Macaulay Stanley–Reisner rings, which we show to be virtually Cohen–Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety, we develop homological tools for assessing the virtual Cohen–Macaulay property. Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. We also use these tools to establish relationships among the arithmetically, geometrically and virtually Cohen–Macaulay properties.