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We introduce a recursive decomposition algorithm for the Betti diagram of acomplete intersection using the diagram of a complete intersection defined by asubset of the original generators. This alternative algorithm is the main toolthat we use to investigate stability and compatibility of the Boij-Soederbergdecompositions of related diagrams; indeed, when the biggest generating degreeis sufficiently large, the alternative algorithm produces the Boij-Soederbergdecomposition. We also provide a detailed analysis of the Boij-Soederbergdecomposition for Betti diagrams of codimension four complete intersectionswhere the largest generating degree satisfies the size condition.

Recursive strategy for decomposing Betti tables of complete intersections