Recursive strategy for decomposing Betti tables of complete intersections

In the spirit of Boij–Söderberg theory, we introduce a recursive decomposition algorithm for the Betti diagram of a complete intersection using the diagram of a complete intersection defined by a subset of the original generators. This alternative algorithm is the main tool that we use to investigate stability and compatibility of the Boij–Söderberg decompositions of related diagrams; indeed, when the biggest generating degree is sufficiently large, the alternative algorithm produces the Boij–Söderberg decomposition. We also provide a detailed analysis of the Boij–Söderberg decomposition for Betti diagrams of codimension four complete intersections where the largest generating degree satisfies the size condition.