[Paper Review] The structure of the Boij-Söderberg posets

David Cook|arXiv (Cornell University)|Jun 10, 2010

Commutative Algebra and Its Applications2 references4 citations

TL;DR

This paper investigates the combinatorial structure of Boij-Söderberg posets, which are central to the theory of Betti diagrams of graded modules. It proves these posets are bounded complete lattices and that their order complexes are vertex-decomposable, hence Cohen-Macaulay and squarefree glicci, using a recursive atom ordering based on lexicographic ordering of atoms.

ABSTRACT

Boij and Söderberg made a pair of conjectures, which were subsequently proven by Eisenbud and Schreyer and then extended by Boij and Söderberg, about the structure of Betti diagrams of Graded modules. In the theory, a particular family of posets, and their associated order complexes, play an integral role. We explore the structure of this family. In particular, we show the posets are bounded complete lattices and the order complexes are vertex-decomposable, hence Cohen-Macaulay and squarefree glicci.

Motivation & Objective

• To analyze the intrinsic combinatorial structure of Boij-Söderberg posets, which are fundamental in the classification of Betti diagrams of graded modules.
• To determine whether these posets possess lattice-theoretic properties such as being bounded and complete.
• To investigate the topological properties of their associated order complexes, particularly vertex-decomposability and Cohen-Macaulayness.
• To establish a recursive atom ordering for the posets to enable structural and topological conclusions about their order complexes.
• To clarify the algebraic and combinatorial significance of these posets in the broader context of Boij-Söderberg theory and module theory.

Proposed method

• Define the Boij-Söderberg poset $\Pi_{\underline{d},\overline{d}}$ as the set of strictly increasing sequences between $\underline{d}$ and $\overline{d}$ in $\mathbb{Z}^{p+1}$, with componentwise partial order.
• Prove the poset is a bounded complete lattice by showing every subset has a meet and join, and that unique minimal and maximal elements exist.
• Construct a recursive atom ordering by ordering atoms lexicographically from smallest to largest, satisfying the conditions of Björner and Wachs.
• Use the recursive atom ordering to deduce that the order complex $\Delta(\Pi_{\underline{d},\overline{d}})$ is vertex-decomposable.
• Apply known results from combinatorial commutative algebra to conclude that vertex-decomposable complexes are Cohen-Macaulay and squarefree glicci.
• Verify the recursive structure via induction on the number of differing positions between $\underline{d}$ and $\overline{d}$, and analyze covering relations and joins of atoms.

Experimental results

Research questions

• RQ1 Are the Boij-Söderberg posets bounded complete lattices?
• RQ2 Does the order complex of a Boij-Söderberg poset admit a recursive atom ordering?
• RQ3 Is the order complex of a Boij-Söderberg poset vertex-decomposable, and what are the implications for its topological invariants?
• RQ4 Can the recursive atom ordering be explicitly constructed using lexicographic ordering of atoms?
• RQ5 Do the topological properties of the order complex (e.g., Cohen-Macaulayness) persist under vertex deletion, and is the resulting complex still a Boij-Söderberg poset?

Key findings

• The Boij-Söderberg posets $\Pi_{\underline{d},\overline{d}}$ are bounded complete lattices, as every subset has a meet and a join, and they possess unique minimal and maximal elements.
• The posets admit a recursive atom ordering when atoms are ordered lexicographically from smallest to largest, as established by Theorem 3.1.
• The order complex of each Boij-Söderberg poset is vertex-decomposable, as a consequence of the recursive atom ordering.
• Vertex-decomposable order complexes are Cohen-Macaulay and squarefree glicci, as stated in Corollary 3.2.
• The deletion of any shedding vertex from a Boij-Söderberg poset results in a poset that is not itself a Boij-Söderberg poset, as shown in Remark 3.3.
• The topological and combinatorial structure of the order complex is preserved under the recursive decomposition process, confirming its shellability and homological properties.