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，上同调比同调具有更丰富的结构。然而，迄今为止，在实际应用中，上同调在持续同调设置中的使用仅限于加速条形码计算。最近引入的一些不变量，即持续杯长度、持续杯模和持续斯廷罗德模，在一定程度上填补了这一空白。当将它们添加到标准持续同调条形码中时，得到的比标准持续同调条形码更具区分性的不变量。在本文中，我们设计了一种$O(d n^4)$算法，用于计算所有$k \in \{2, \dots, d\}$的持续$k$杯模，其中$d$表示过滤复形的维数，$n$表示其大小。此外，我们注意到，由于持续杯长度可以作为我们计算的副产品获得，这导致了一种更快的计算持续杯长度的算法。