Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.

在某些条件下，Koszul 复形可用于计算由偏序集索引的向量空间值函子的相对 Betti 图，而无需显式计算全局最小相对分辨率。在此类函子的相对同调代数中，自由函子被任意函子族替换。相对贝蒂图以最小相对分辨率对这些函子的重数进行编码。在本文中，我们提供了对所选函子族进行分级可得到显式 Koszul 复合体的条件，其同源维数是相对 Betti 图，从而给出了计算这些数值描述符的方案。