[Paper Review] The cone of Betti diagrams of bigraded artinian modules of codimension two

Mats Boij, Gunnar Fløystad|arXiv (Cornell University)|Jan 19, 2010

Algebraic structures and combinatorial models4 references3 citations

TL;DR

This paper characterizes the positive cone of bigraded Betti diagrams for artinian modules of codimension two over a bigraded polynomial ring. It shows that when the total degree differences $e_1$ and $e_2$ are relatively prime, the extremal rays of the cone are parametrized by order ideals in $\mathbb{N}^2$ within the region $e_1x + e_2y < (e_1-1)(e_2-1)$, with each such ideal corresponding to a unique pure resolution. The construction generalizes both the equivariant resolution and the monomial quotient resolution, providing a complete classification of extremal rays in this setting.

ABSTRACT

We describe the positive cone generated by bigraded Betti diagrams of artinian modules of codimension two, whose resolutions become pure of a given type when taking total degrees. If the differences of these total degrees, p and q, are relatively prime, the extremal rays are parametrised by order ideals in N^2 contained in the region px + qy &lt; (p-1)(q-1). We also consider some examples concerning artinian modules of codimension three.

Motivation & Objective

• To describe the positive cone generated by bigraded Betti diagrams of artinian modules of codimension two whose resolutions become pure upon taking total degrees.
• To classify the extremal rays of this cone when the total degree differences $e_1$ and $e_2$ are relatively prime.
• To generalize the construction of pure resolutions beyond the equivariant and monomial quotient resolutions, showing that all extremal rays arise from order ideals in a specific region of $\mathbb{N}^2$.
• To explore whether similar classification results extend to codimension three modules, particularly regarding finiteness and structure of extremal rays.

Proposed method

• The authors use multigraded Herzog-Kühl equations to constrain possible Betti diagrams of bigraded artinian modules.
• They define a region $R(e_1,e_2)$ in $\mathbb{N}^2$ bounded by $e_1x + e_2y < (e_1-1)(e_2-1)$, which contains all relevant order ideals.
• For each order ideal $\lambda \subset R(e_1,e_2)$, they construct a pure resolution of the form $S^{e_2} \leftarrow S^{e_2 + e_1} \leftarrow S^{e_1}$ with specific bigraded Betti numbers.
• They prove that the Betti diagrams $\beta_\lambda(\mathbf{a})$ for $\mathbf{a} \in \mathbb{Z}^2$ generate the extremal rays of the positive cone $P(e_1,e_2)$, using degeneration techniques and kernel analysis on general matrices.
• They verify that these diagrams correspond to actual resolutions by constructing explicit complexes via matrix degenerations and verifying the Herzog-Kühl equations.
• For codimension three, they analyze the cone $P(1,2,1)$ and show that not all diagrams in the relaxed cone $P'(1,2,1)$ arise from actual modules, indicating a richer structure beyond the codimension two case.

Experimental results

Research questions

• RQ1 Are there only finitely many extremal rays in the positive cone of Betti diagrams for codimension two bigraded artinian modules with fixed total degree differences?
• RQ2 Can the extremal rays of the positive cone of codimension two Betti diagrams be parametrized by combinatorial objects such as order ideals in $\mathbb{N}^2$?
• RQ3 Does the construction of pure resolutions via order ideals in $R(e_1,e_2)$ yield all possible extremal rays in the positive cone $P(e_1,e_2)$ when $\gcd(e_1,e_2) = 1$?
• RQ4 In the codimension three case, is the positive cone $P(\mathbf{e})$ isomorphic to the relaxed cone $P'(\mathbf{e})$, or are there diagrams in $P'(\mathbf{e})$ not realizable by actual modules?
• RQ5 Are the extremal rays in the codimension three case still generated by a finite set of translation classes, and does a unique maximal or minimal element exist in the poset of such rays?

Key findings

• When $e_1$ and $e_2$ are relatively prime, the extremal rays of the positive cone $P(e_1,e_2)$ are in bijection with order ideals $\lambda$ contained in the region $R(e_1,e_2) = \{ (x,y) \in \mathbb{N}^2 \mid e_1x + e_2y < (e_1-1)(e_2-1) \}$.
• The maximal order ideal corresponds to the equivariant resolution, and the empty order ideal corresponds to the resolution of a quotient of monomial ideals constructed in [1].
• Each order ideal $\lambda$ gives rise to a pure resolution of type $S^{e_2} \leftarrow S^{e_2 + e_1} \leftarrow S^{e_1}$, with bigraded Betti numbers determined by the shape of $\lambda$, and these generate the extremal rays of $P(e_1,e_2)$.
• In the codimension three case, the cone $P(1,2,1)$ contains diagrams not realizable by actual modules—e.g., a non-negative linear combination $[(2,1,0)+(0,2,1)+(1,0,2)-(1,1,1)]\beta$ satisfies the Herzog-Kühl equations but does not arise from a module, showing $P(1,2,1) \subsetneq P'(1,2,1)$.
• The Betti diagram $\alpha = [(2,1,0)+(2,0,1)+(1,2,0)+(0,2,1)+(1,0,2)+(0,1,2)-(1,1,1)]\beta$ is shown to be a Betti diagram of a resolution of an indecomposable artinian module and generates an extremal ray in $P(1,2,1)$, distinct from the equivariant diagram $\beta$.
• The existence of such diagrams that are not multiples of a single ray indicates that the structure of the positive cone in codimension three is more complex than in codimension two, with no known complete classification of extremal rays.