arXiv subjects

Thanh Vu

Publications and source records attributed to Thanh Vu.

78 records · Page 5Linked to original sources

arXiv2016

Linearity defect of edge ideals and Fröberg's theorem

Fröberg's classical theorem about edge ideals with $2$-linear resolution can be regarded as a classification of graphs whose edge ideals have linearity defect zero. Extending his theorem, we classify all graphs whose edge ideals have linearity defect at most $1$. Our characterization is independent of the characteristic of the base field: the graphs in question are exactly weakly chordal graphs with induced matching number at most $2$. The proof uses the theory of Betti splittings of monomial ideals due to Francisco, Hà, and Van Tuyl and the structure of weakly chordal graphs. Along the way, we compute the linearity defect of edge ideals of cycles and weakly chordal graphs. We are also able to recover and generalize previous results due to Dochtermann-Engström, Kimura, and Woodroofe on the projective dimension and Castelnuovo-Mumford regularity of edge ideals.

math.AC↗

arXiv2015

Regularity over homomorphisms and a Frobenius characterization of Koszul algebras

Let $R$ be a standard graded algebra over an $F$-finite field of characteristic $p > 0$. Let $ϕ:R\to R$ be the Frobenius endomorphism. For each finitely generated graded $R$-module $M$, let ${}^ϕ\!M$ be the abelian group $M$ with the $R$-module structure induced by the Frobenius endomorphism. The $R$-module ${}^ϕ\!M$ has a natural grading given by $\text{deg} x=j$ if $x\in M_{jp+i}$ for some $0\le i \le p-1$. In this paper, we prove that $R$ is Koszul if and only if there exists a non-zero finitely generated graded $R$-module $M$ such that $\text{reg}_R\,{}^ϕ\!M <\infty$. This result supplies another instance for the ability of the Frobenius in detecting homological properties, as exemplified by Kunz's famous regularity criterion. The main technical tool is the notion of Castelnuovo-Mumford regularity over certain homomorphisms between $\mathbb{N}$-graded algebras. The latter notion is a common generalization of the relative and absolute Castelnuovo-Mumford regularity of modules.

math.AC↗

arXiv2014

Koszul determinantal rings and $2\times e$ matrices of linear forms

Let $k$ be an algebraically closed field of characteristic $0$. Let $X$ be a $2\times e$ matrix of linear forms over a polynomial ring $k[\mathsf{x}_1, \ldots,\mathsf{x}_n]$ (where $e,n\ge 1$). We prove that the determinantal ring $R = k[\mathsf{x}_1,\ldots,\mathsf{x}_n]/I_2(X)$ is Koszul if and only if in the Kronecker-Weierstrass normal form of $X$, the largest length of a nilpotent block is at most twice the smallest length of a scroll block. As an application, we classify rational normal scrolls whose all section rings by natural coordinates are Koszul. This result settles a conjecture due to Conca.

math.AC↗

arXiv2013

The Koszul property of pinched Veronese varieties

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.

math.AC↗

arXiv2013

Periodicity of betti numbers of monomial curves

Let $K$ be an arbitrary field. Let $\a = (a_1< ... <a_n)$ be a sequence of positive integers. Let $C(\a)$ be the affine monomial curve in ${\mathbb A}^n$ parametrized by $t\to (t^{a_1}, ..., t^{a_n})$. Let $I(\a)$ be the defining ideal of $C(\a)$ in $K[x_1, ..., x_n]$. For each positive integer $j$, let $\a+j$ be the sequence $(a_1 + j, ..., a_n+j)$. In this paper, we prove the conjecture of Herzog and Srinivasan saying that the betti numbers of $I(\a + j)$ are eventually periodic in $j$ with period $a_n -a_1$. When $j$ is large enough, we describe the betti table for the closure of $C(\a+j)$ in $\PP^n$.

math.AC↗

arXiv2013

$N_6$ property for third Veronese embeddings

The rational homology groups of the matching complexes are closely related to the syzygies of the Veronese embeddings. In this paper we will prove the vanishing of certain rational homology groups of matching complexes, thus proving that the third Veronese embeddings satisfy the property $N_6$. This settles the Ottaviani-Paoletti conjecture for third Veronese embeddings. This result is optimal since $ν_3(\PP^n)$ does not satisfy the property $N_7$ for $n\ge 2$ as shown by Ottaviani-Paoletti in \cite{OP}.

math.AC↗