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Guangjun Zhu

Publications and source records attributed to Guangjun Zhu.

47 records · Page 3Linked to original sources
arXiv2019

On the fiber cone of monomial ideals

We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $I\subset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for interesting classes of height $2$ monomial ideals $I\subset K[x,y]$, which are generated by $4$ elements. For these classes of ideals we also show that $F(I)$ is Cohen--Macaulay if and only if the defining ideal $J$ of $F(I)$ is generated by at most 3 elements. In all the cases a minimal set of generators of $J$ is determined.

math.AC↗
arXiv2019

Regularity of powers of edge ideals of vertex-weighted oriented unicyclic graphs

In this paper we provide some exact formulas for the regularity of powers of edge ideals of vertex-weighted oriented cycles and vertex-weighted unicyclic graphs. These formulas are functions of the weight of vertices and the number of edges. We also give some examples to show that these formulas are related to direction selection and the weight of vertices.

math.AC↗
arXiv2019

Projective dimension and regularity of powers of edge ideals of vertex-weighted rooted forests

In this paper we provide some exact formulas for projective dimension and the regularity of powers of edge ideals of vertex-weighted rooted forests. These formulas are functions of the weight of the vertices and the number of edges. We also give some examples to show that these formulas are related to direction selection and the assumptions about "rooted" forest such that $w(x)\geq 2$ if $d(x)\neq 1$ cannot be dropped.

math.AC↗
arXiv2018

The relevance of Freiman's theorem for combinatorial commutative algebra

Freiman's theorem gives a lower bound for the cardinality of the doubling of a finite set in $\RR^n$. In this paper we give an interpretation of his theorem for monomial ideals and their fiber cones. We call a quasi-equigenerated monomial ideal a Freiman ideal, if the set of its exponent vectors achieves Freiman's lower bound for its doubling. Algebraic characterizations of Freiman ideals are given, and finite simple graphs are classified whose edge ideals or matroidal ideals of its cycle matroids are Freiman ideals.

math.AC↗
arXiv2017

Freiman ideals

In this paper we study the Freiman inequality for the minimal number of generators of the square of an equigenerated monomial ideal. Such an ideal is called a Freiman ideal if equality holds in the Freiman inequality. We classify all Freiman ideals of maximal height, the Freiman ideals of certain classes of principal Borel ideals, the Hibi ideals which are Freiman, and classes of Veronese type ideals which are Freiman.

math.AC↗
arXiv2016

Depth and Stanley depth of the path ideal associated to an $n$-cyclic graph

We compute the depth and Stanley depth for the quotient ring of the path ideal of length $3$ associated to a $n$-cyclic graph, given some precise formulas for depth when $n\not\equiv 1\,(\mbox{mod}\ 4)$, tight bounds when $n\equiv 1\,(\mbox{mod}\ 4)$ and for Stanley depth when $n\equiv 0,3\,(\mbox{mod}\ 4)$, tight bounds when $n\equiv 1,2\,(\mbox{mod}\ 4)$. Also, we give some formulas for depth and Stanley depth of a quotient of the path ideals of length $n-1$ and $n$.

math.AC↗
arXiv2016

Projective dimension and regularity of the path ideal of the line graph

By generalizing the notion of the path ideal of a graph, we study some algebraic properties of some path ideals associated to a line graph. We show that the quotient ring of these ideals are always sequentially Cohen-Macaulay and also provide some exact formulas for the projective dimension and the regularity of these ideals. As some consequences, we give some exact formulas for the depth of these ideals.

math.AC↗
arXiv2015

Cartan maps and projective modules

Let $R$ be a commutative ring, $π$ be a finite group, $Rπ$ be the group ring of $π$ over $R$. Theorem 1. If $R$ is a commutative artinian ring and $π$ is a finite group. Then the Cartan map $c:K_0(Rπ)\to G_0(Rπ)$ is injective. Theorem 2. Suppose that $R$ is a Dedekind domain with $\fn{char}R=p>0$ and $π$ is a $p$-group. Then every finitely generated projective $Rπ$-module is isomorphic to $F \oplus cA$ where $F$ is a free module and $cA$ is a projective ideal of $Rπ$. Moreover, $R$ is a principal ideal domain if and only if every finitely generated projective $Rπ$-module is isomorphic to a free module. Theorem 3. Let $R$ be a commutative noetherian ring with total quotient ring $K$, $A$ be an $R$-algebra which is a finitely generated $R$-projective module. Suppose that $I$ is an ideal of $R$ such that $R/I$ is artinian. Let $\{cM_1,\ldots,cM_n\}$ be the set of all maximal ideals of $R$ containing $I$. Assume that the Cartan map $c_i: K_0(A/cM_iA)\to G_0(A/cM_iA)$ is injective for all $1\le i\le n$. If $P$ and $Q$ are finitely generated $A$-projective modules with $KP\simeq KQ$, then $P/IP\simeq Q/IQ$.

math.GR↗