We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and develop an efficient algorithm to compute the homology of a complex of presentations. To deal with inputs that are not given in terms of presentations, we give an efficient algorithm to compute a presentation of a morphism of persistence modules. This allows us to compute persistent (co)homology of instances giving rise to complexes of non-free modules. Our methods lead to a new efficient algorithm for computing the persistent homology of simplicial towers and they enable efficient algorithms to compute the persistent homology of cosheaves over simplicial towers and cohomology of persistent sheaves on simplicial complexes. We also show that we can compute the cohomology of persistent sheaves over arbitrary finite posets by reducing the computation to a computation over simplicial complexes.

翻译：我们将持续算法（视为计算自由持续模或分次模复形同调的方法）扩展至非自由模的复形。通过用表示来代替持续模，我们开发了一种高效算法来计算表示复形的同调。针对非以表示形式给出的输入，我们提出了一种高效算法来计算持续模态射的表示。这使得我们能够计算引发非自由模复形的实例的持续（上）同调。我们的方法为计算单纯塔的持续同调提供了一种新高效算法，并实现了计算单纯塔上余层持续同调及单纯复形上持续层上同调的高效算法。我们还证明，可以通过将计算简化为单纯复形上的计算，来计算任意有限偏序集上持续层的上同调。