Boij-Söderberg and Veronese decompositions

Abstract

Boij-S\"oderberg theory has had a dramatic impact on commutative algebra. We determine explicit Boij-S\"oderberg coefficients for ideals with linear resolutions and illustrate how these arise from the usual Eliahou-Kervaire computations for Borel ideals. In addition, we explore a new numerical decomposition for resolutions based on a row-by-row approach; here, the coefficients of the Betti diagrams are not necessarily positive. Finally, we demonstrate how the Boij-S\"oderberg decomposition of an arbitrary homogeneous ideal with a pure resolution changes when multiplying the ideal by a homogeneous polynomial.