[Paper Review] The Koszul property of pinched Veronese varieties

Thanh Vu|arXiv (Cornell University)|Sep 12, 2013

Commutative Algebra and Its Applications10 references3 citations

TL;DR

This paper establishes that the semigroup ring $K[\Gamma]$, where $\Gamma$ is a pinched Veronese variety obtained by removing a single lattice point $\mathbf{a}$ from the set of degree-$d$ monomials in $n$ variables, is Koszul unless $d \geq 3$ and $\mathbf{a} = (0,\dots,0,2,d-2)$ or its permutations. The proof uses homological methods, specifically bounding the regularity of the residue field via simplicial homology of open chain complexes, and applies a theorem of Avramov and Peeva to conclude Koszulness when regularity is finite.

ABSTRACT

Let $K$ be an arbitrary field. Let $n,d \ge 2$ be positive integers. Let $V(n,d)$ be the set of all lattice points $\mathbf b = (b_1, ..., b_n)$ in ${\mathbb N}^n$ such that $\sum_{i=1}^n b_i = d$. Let $Γ= V(n,d) \setminus \{ \mathbf a \}$ for some element $\mathbf a \in V(n,d)$. In this paper we prove that the semigroup ring $K[Γ]$ is Koszul unless $d \ge 3$ and ${\mathbf a} = (0, ...,0, 2, d-2)$ or one of its permutations. This generalizes results of Caviglia, Conca, and Tancer.

Motivation & Objective

• To determine the Koszul property of semigroup rings $K[\Gamma]$ where $\Gamma = V(n,d) \setminus \{\mathbf{a}\}$ for $\mathbf{a} \in V(n,d)$.
• To generalize prior results by Caviglia, Conca, and Tancer on the Koszul property of pinched Veronese algebras.
• To identify the precise exceptional case where $K[\Gamma]$ fails to be Koszul, namely when $d \geq 3$ and $\mathbf{a} = (0,\dots,0,2,d-2)$ or its permutations.
• To establish a new large family of Koszul algebras through a novel homological method based on simplicial complexes of open chains.

Proposed method

• Use of the Laudal-Sletsjoe theorem to express Betti numbers of the residue field $K$ over $R = K[\Gamma]$ as dimensions of reduced homology groups of simplicial complexes $\Gamma_\lambda$.
• Definition of $\Gamma_\lambda$ as the simplicial complex of open chains from $0$ to $\lambda$ with links in $\Gamma$, which are pure of dimension $|\lambda| - 2$.
• Application of Avramov and Peeva's theorem that $R$ is Koszul if $\operatorname{reg}_R K$ is finite, reducing the problem to bounding regularity via homology.
• Comparison of $\Gamma_\lambda$ with the full Veronese complex $\Delta_\lambda$, which is known to have at most top-dimensional homology.
• Use of downward induction on the degree $|\lambda|$ and Mayer-Vietoris sequences to show that $\Gamma_\lambda$ has no homology in dimensions $\leq n-6$ for $|\lambda| \geq 7$.
• Leveraging combinatorial lemmas on simplicial complexes with specific facet patterns to prove trivial or vanishing homology in low degrees.

Experimental results

Research questions

• RQ1 Under what conditions is the semigroup ring $K[\Gamma]$ associated to a pinched Veronese variety Koszul?
• RQ2 Is there a complete classification of the exceptional cases where $K[\Gamma]$ fails to be Koszul?
• RQ3 Can the Koszul property be established for $K[\Gamma]$ when $\Gamma$ is $2$-full, i.e., $\Gamma + \Gamma = V(n,2d)$?
• RQ4 How does the homology of the simplicial complex $\Gamma_\lambda$ of open chains relate to the regularity of the residue field $K$ over $R = K[\Gamma]$?
• RQ5 What is the role of the specific exceptional point $\mathbf{a} = (0,\dots,0,2,d-2)$ in obstructing the Koszul property?

Key findings

• The algebra $K[\Gamma]$ is Koszul unless $d \geq 3$ and $\mathbf{a} = (0,\dots,0,2,d-2)$ or one of its permutations.
• For $|\lambda| \geq 7$, the simplicial complex $\Gamma_\lambda$ has no homology in dimensions $\leq n-6$, implying $\operatorname{reg}_R K \leq 5$.
• The regularity of the residue field $K$ over $R = K[\Gamma]$ is bounded by 5 when $\Gamma$ is $2$-full and $\mathbf{a}$ is not the exceptional point.
• The proof establishes that $K[\Gamma]$ is Koszul via the finiteness of $\operatorname{reg}_R K$, relying on Avramov and Peeva's criterion.
• The exceptional case arises because the cubic relation $y^3 - xz^2$ becomes minimal and non-linear in the resolution, preventing a linear resolution.
• The method generalizes and unifies prior results by Caviglia, Conca, and Tancer, providing a new framework for proving Koszulness in toric rings.