It is well-known that cohomology has a richer structure than homology. However, so far, in practice, the use of cohomology in persistence setting has been limited to speeding up of barcode computations. Some of the recently introduced invariants, namely, persistent cup-length, persistent cup modules and persistent Steenrod modules, to some extent, fill this gap. When added to the standard persistence barcode, they lead to invariants that are more discriminative than the standard persistence barcode. In this work, we devise an $O(d n^4)$ algorithm for computing the persistent $k$-cup modules for all $k \in \{2, \dots, d\}$, where $d$ denotes the dimension of the filtered complex, and $n$ denotes its size. Moreover, we note that since the persistent cup length can be obtained as a byproduct of our computations, this leads to a faster algorithm for computing it.

翻译：众所周知,同族体结构比同族体结构更为丰富。 然而,迄今为止,在实际中,在持久性环境中对同族体学的使用仅限于加速条形码计算。最近引入的一些异同体,即持久性杯长、持久性杯底模块和持久性Steenrod模块,在某种程度上填补了这一空白。在加入标准持久性条形码时,它们会导致比标准持久性条形码更具有歧视性的异同体。在这项工作中,我们为计算所有恒定的美元 cup 模块设计了一个$O(d n<unk> 4)的算法,用于计算所有 $k $2,\ dots, d<unk> $, 其中美元表示过滤过的综合体的尺寸, 美元表示其大小。此外,我们注意到,由于持久性杯长度可以作为我们计算结果的副产品获得,因此计算速度更快。</s>