论文标题
离散的微型摩尔斯理论
Discrete Microlocal Morse Theory
论文作者
论文摘要
我们在有限的POSET和简单复合物的设置中建立了几个结果，将离散的Morse理论和微局部捆起来理论结合在一起。我们的主要工具是用Alexandrov拓扑结构的Poset上有界派生类别的计算方法描述。我们证明，有限POSET上的每条有界的滑轮络合物都承认了一种独特的（达到复合物的同构）最小的注射分辨率，并且我们提供了计算最小的注射式复合物以及几个有用函数的算法分辨率，以及衍生的分流类别之间的几个有用的功能。对于简单络合物上的常数捆，我们在使用这些算法的计算最小内注射分辨率的复杂性上给出了渐近紧密的界限。我们的主要结果是对有限孔的有界束带的离散微撑杆的新颖定义。我们详细介绍了离散微撑杆的几种基础性质，以及离散同源定理和摩尔斯的微元素概括。
We establish several results combining discrete Morse theory and microlocal sheaf theory in the setting of finite posets and simplicial complexes. Our primary tool is a computationally tractable description of the bounded derived category of sheaves on a poset with the Alexandrov topology. We prove that each bounded complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes) minimal injective resolution, and we provide algorithms for computing minimal injective resolution of an injective complex, as well as several useful functors between derived categories of sheaves. For the constant sheaf on a simplicial complex, we give asymptotically tight bounds on the complexity of computing the minimal injective resolution using those algorithms. Our main result is a novel definition of the discrete microsupport of a bounded complex of sheaves on a finite poset. We detail several foundational properties of the discrete microsupport, as well as a microlocal generalization of the discrete homological Morse theorem and Morse inequalities.
