Poset structures in Boij-Söderberg theory

Boij-Söderberg theory is the study of two cones: the cone of Betti diagrams of standard graded minimal free resolutions over a polynomial ring and the cone of cohomology tables of coherent sheaves over projective space. We provide a new interpretation of these partial orders in terms of the existence of nonzero homomorphisms, for both the general and equivariant constructions. These results provide new insights into the families of modules and sheaves at the heart of Boij-Söderberg theory: Cohen-Macaulay modules with pure resolutions and supernatural sheaves. In addition, they suggest the naturality of these partial orders and provide tools for extending Boij-Söderberg theory to other graded rings and projective varieties.