Poset Structures in Boij–Söderberg Theory

Abstract

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.Outline. In this paper, we first focus on the cone of Betti diagrams for S. In Section 2, we prove the reverse implications of Theorems 1.1 and 1.3. We then construct nonzero morphisms between modules with pure resolutions. Sections 3 and 4, respectively, address the forward directions of Theorems 1.1 and 1.3. We next address the cone of cohomology tables for P n−1 . In Section 5, we prove the reverse implications of Theorems 1.2 and 1.4. We then turn to the construction of nonzero morphisms between supernatural sheaves: Sections 6 and 7, respectively, address the forward directions of Theorems 1.2 and 1.4. Finally, we provide in Section 8 a brief discussion of how Theorem 1.1 has been applied in the study of Boij-Söderberg theory over other graded rings. We suggest the survey [ES10b] to the reader seeking additional background on Boij-Söderberg theory. The poset of degree sequencesLet M be a finitely generated graded S-module. The (i, j)th graded Betti number of M, denoted β i,j (M), is dim k Tor S i (k, M) j . The Betti diagram of M is a table, with rows indexed by Z and columns by 0, . . . , n, such that the entry in column i and row j is length of d, denoted ℓ(d), is the largest integer t such that d t is finite. Definition 2.1. A graded S-module M is said to have a pure resolution of type d if a minimal free resolution of M has the form 0 ← M ← S(−d 0 ) β 0,d 0 ← S(−d 1 ) β 1,d 1 ← · · · ← S(−d ℓ(d) ) β ℓ(d),d ℓ(d) ← 0.

Tensor complexes: multilinear free resolutions constructed from higher tensors

“…This result is proven in §7 and is related to [BEKS,Theorem 1.2], as the maps considered in that result are special cases of the above construction.…”

“…Remark 7.4. The proof of [BEKS,Theorem 1.2] can be reinterpreted in terms of Proposition 1.4 and Lemma 7.3. Namely, the pure resolutions of [BEKS,Theorem 1.2] are specializations of certain tensor complexes of the form a × (2, .…”

“…However, we note that Proposition 1.4 does not directly imply that result, as the above map of complexes could be null-homotopic. The essential step in the proof of [BEKS,Theorem 1.2] is checking that certain maps of complexes induce nonzero maps M(φ a ×b , w ) → M(φ a×b , w), which requires analyzing the detailed description of ν • provided by Lemma 7.3.…”

Equations and syzygies of some Kalman varieties

“…In the above proof, we know from general principles that the comparison maps F i → G i must be nonzero since both O 1,2,n and O 2,2,n are Cohen-Macaulay (see the proof of [BEKS,Proposition 2.3]). So one can deduce the required cancellations using just representation theory (at least in characteristic 0) without understanding the differentials.…”

Boij–Söderberg decompositions of lexicographic ideals

“…Although the theory is quite recent and has a lot of open problems, improvements and contributions to this theory are quite impressive. Cook in [5] and Berkesch, Erman, Kumini, and Sam in [1] discuss Boij-Söderberg theory in the perspective of poset structures. In [14], Nagel and Sturgeon examine the Boij-Söderberg decomposition of some ideals that are raised from some combinatorial objects.…”