论文标题
加权简单同源性的计算框架
A computational framework for weighted simplicial homology
论文作者
论文摘要
我们为扭转生成器提供了自下而上的构造，用于在离散估值环上加权复合物的加权同源性$ r = \ mathbb {f} [[π]] $。这是通过从$ n $ th骨骼的经典同源性开始的基础建筑群的基础，并在残留场$ \ mathbb {f} $中的系数，然后将其提起，然后将其提升为具有$ r $系数的加权同源性。使用后者，在$ n+1 $和$ n $尺寸的简单之间建立了双线，其权重比提供了$π$ - 工程学的指数，该指数在$ r $ $ $ $ $ $的结构定理的结构定理中生成了每个扭转。我们提出算法通过将扭转计算降低到$ r $的残基字段中的归一化，并描述我们实现的Python软件包，该软件包利用了此减少的优势并有效地执行了计算。
We provide a bottom up construction of torsion generators for weighted homology of a weighted complex over a discrete valuation ring $R=\mathbb{F}[[π]]$. This is achieved by starting from a basis for classical homology of the $n$-th skeleton for the underlying complex with coefficients in the residue field $\mathbb{F}$ and then lifting it to a basis for the weighted homology with coefficients in the ring $R$. Using the latter, a bijection is established between $n+1$ and $n$ dimensional simplices whose weight ratios provide the exponents of the $π$-monomials that generate each torsion summand in the structure theorem of the weighted homology modules over $R$. We present algorithms that subsume the torsion computation by reducing it to normalization over the residue field of $R$, and describe a Python package we implemented that takes advantage of this reduction and performs the computation efficiently.
