论文标题
通过间隔分解成员和持久模块的间隔分辨率进行近似
Approximation by interval-decomposables and interval resolutions of persistence modules
论文作者
论文摘要
在拓扑数据分析中，可以使用2D交换网格的表示理论来研究两参数的持久性，这是A型的两种Dynkin Quivers的张量。在先前的工作中，我们使用限制了间隔近似，以与Mobius Inversion一起进行间隔的基本角度进行限制。在这项工作中，我们考虑使用间隔分辨率的同源近似值，并表明间隔分辨率的全局尺寸是有限posets的有限尺寸，并且等于Auslander-Reiten的间隔尺寸的最大值是间隔表示的最大值。实际上，对于后一种平等，我们在有限维代数和分辨率的情况下获得了一般公式。此外，在交换阶梯的情况下，通过适当的间隔近似修改，我们提供了一个链接两个近似概念的公式。
In topological data analysis, two-parameter persistence can be studied using the representation theory of the 2d commutative grid, the tensor product of two Dynkin quivers of type A. In a previous work, we defined interval approximations using restrictions to essential vertices of intervals together with Mobius inversion. In this work, we consider homological approximations using interval resolutions, and show that the interval resolution global dimension is finite for finite posets and that it is equal to the maximum of the interval dimensions of the Auslander-Reiten translates of the interval representations. In fact, for the latter equality, we obtained a general formula in the setting of finite-dimensional algebras and resolutions relative to a generator-cogenerator. Furthermore, in the commutative ladder case, by a suitable modification of our interval approximation, we provide a formula linking the two conceptions of approximation.
