[Paper Review] Face ring multiplicity via CM-connectivity sequences

Isabella Novik, Ed Swartz|ArXiv.org|Jun 10, 2006

Commutative Algebra and Its Applications23 references4 citations

TL;DR

This paper verifies the multiplicity conjecture of Herzog, Huneke, and Srinivasan for face rings of matroid complexes, one- and two-dimensional simplicial complexes, and Gorenstein complexes of dimension at most four. It establishes the lower bound for doubly Cohen-Macaulay complexes with 1-skeleton connectivity at most codimension plus one and for $d$-Cohen-Macaulay complexes of dimension $d-1$, using a novel interpretation of Betti number shifts via Cohen-Macaulay connectivity of simplicial skeletons.

ABSTRACT

The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at most four. The lower bound part of this conjecture is also established for the face rings of all doubly Cohen-Macaulay complexes whose 1-skeleton's connectivity does not exceed the codimension plus one as well as for all (d-1)-dimensional d-Cohen-Macaulay complexes. The main ingredient of the proofs is a new interpretation of the minimal shifts in the resolution of the face ring via the Cohen-Macaulay connectivity of the skeletons of the complex.

Motivation & Objective

• To prove the multiplicity conjecture of Herzog, Huneke, and Srinivasan for face rings of specific classes of simplicial complexes.
• To establish the lower bound of the conjecture for doubly Cohen-Macaulay complexes with bounded 1-skeleton connectivity and for $d$-Cohen-Macaulay complexes of dimension $d-1$.
• To introduce and apply a new interpretation of minimal and maximal shifts in free resolutions via the Cohen-Macaulay connectivity of simplicial complex skeletons.
• To connect algebraic invariants of Stanley-Reisner rings to topological and combinatorial properties of simplicial complexes.

Proposed method

• Use of Hochster’s formula to relate Betti numbers to reduced homology of induced subcomplexes.
• Definition of minimal and maximal shifts $m_i$ and $M_i$ in terms of homology non-vanishing over induced subcomplexes of size $|W|$.
• Introduction of the CM-connectivity sequence as a new invariant to interpret shifts in resolutions.
• Application of the Upper Bound Theorem for Gorenstein* complexes and Dehn-Sommerville relations to bound face numbers.
• Use of Turán’s theorem to bound the number of edges in flag complexes with no 5-clique.
• Case analysis based on the values of $q_j = ext{codim}({f k}[ ext{link}_ au riangle]) + 1$ for faces $\tau$ to analyze resolution purity and multiplicity bounds.

Experimental results

Research questions

• RQ1 Does the multiplicity conjecture hold for the face ring of a matroid complex?
• RQ2 Can the upper and lower bounds of the multiplicity conjecture be established for 1- and 2-dimensional simplicial complexes?
• RQ3 Does the multiplicity conjecture hold for Gorenstein complexes of dimension 3 and 4?
• RQ4 Under what conditions does the lower bound of the multiplicity conjecture hold for doubly Cohen-Macaulay complexes?
• RQ5 Can the minimal and maximal shifts in the resolution of a face ring be interpreted via the Cohen-Macaulay connectivity of the complex’s skeletons?

Key findings

• The multiplicity conjecture is verified for face rings of matroid complexes, including all pure complexes with pure induced subcomplexes.
• The upper bound of the multiplicity conjecture holds for all 1- and 2-dimensional simplicial complexes.
• The upper bound is established for Gorenstein complexes of dimension 3 and 4, with equality in the upper bound only if the complex is 2-neighborly and $q_3 - 1 = 4$, which leads to a contradiction under the assumptions.
• The lower bound of the multiplicity conjecture holds for 3- and 4-dimensional Gorenstein* complexes, and equality implies a pure resolution.
• For $d$-Cohen-Macaulay complexes of dimension $d-1$, the multiplicity conjecture holds if $n - d = 4$ and $d \leq 22$, or if $M_1 \geq \lfloor d/2 \rfloor + 1$.
• The proof relies on showing that the product of $M_i$'s divided by $c!$ exceeds the number of top-dimensional faces, using bounds from the Upper Bound Theorem and Dehn-Sommerville relations.