Boij–Söderberg theory and tensor complexes

The conjectures of M. Boij and J. Söderberg [3], proven by D. Eisenbud and F.-O. Schreyer [8] (see also [7, 4]), link the extremal properties of invariants of graded free resolutions of finitely generated modules over the polynomial ring S = k[x1, . . . , xn] with the Herzog–Huneke–Srinivasan Multiplicity Conjectures. Here k is any field and S has the standard Z-grading. In the course of their proof, Eisenbud and Schreyer introduce a groundbreaking relationship between the study of free resolutions over the S and the study of the cohomology of coherent sheaves on Pn−1 k , via a nonnegative pairing of their associated numerics. This pairing has recently been categorified through work of Eisenbud and Erman [6], further extending the reach of Boij–Söderberg theory to larger classes of derived objects. We now outline the main result of Boij–Söderberg theory for S. For simplicity, we restrict our attention to a graded S-module M that is of finite length; minor modifications yield the general situation. A minimal free resolution of M is an acyclic complex (F•, ∂•) such that H0(F•) = M , ∂i(Fi) ⊆ 〈x1, . . . , xn〉Fi−1 for each i, and Fi = ⊕ j∈Z S(−j)i,j . The ranks βi,j of the free modules are independent of the choice of resolution F• of M , and they are called the Betti numbers of M . We record these Betti numbers into a Betti table for M , denoted β(M). It appears to be a difficult question to classify which integer tables can be realized as the Betti table of a graded S-module. In a shift in perspective, Boij and Söderberg suggested that this task be approached up to scalar multiple. In other words, view β(M) ∈ ⊕n i=0 ⊕ j∈Z Q =: V and describe instead the cone of Betti tables