arXiv subjects
Publications and source records attributed to Lizhong Chu.
In this note, we compute depth of the 3-path ideal of square of a path and show that the 3-path ideal I3(P 2 n) of square of a path graph is Cohen-Macaulay if and only if n = 3 or 4. Also, we consider the limit behavior of depth of powers of the 3-path ideal of square of a path.
This paper investigates the v-number of various classes of monomial ideals. First, we considers the relationship between the v-number and the regularity of the mixed product ideal $I$, proving that $\mathrm{v}(I) \leq \mathrm{reg}(S/I)$. Next, we investigate an open conjecture on the v-number: if a monomial ideal $I$ has linear powers, then for all $k \geq 1$, $\mathrm{v}(I^k) = α(I)k - 1.$ We prove that if a monomial ideal $I$ with linear powers is a homogeneous square-free ideal and ($k \geq 1$) has no embedded associated primes, then $\mathrm{v}(I^k) = α(I)k - 1.$ We have also drawn some conclusions about the k-th power of the graph.Additionally, we calculate the v-number of various powers of edge ideals(including ordinary power ,square-free powers, symbolic powers). Finally, we propose a conjecture that the v-number of ordinary powers of line graph is equal to the v-number of square-free powers.
Let $Δ$ be an one-dimensional simplicial complex on $\{1,2,\ldots,s\}$ and $S$ the polynomial ring $K[x_1,\ldots,x_s]$ over a field $K$. The explicit formula for $a_0(S/I_Δ^n)$ is presented when $\mathrm{girth}(Δ)\geq 4$. If $\mathrm{girth}(Δ)=3$ we characterize the simplicial complexes $Δ$ for which $a_0(S/I_Δ^n)=3n-1$ or $3n-2$.
For any ideal $I$ in a Noetherian local ring or any graded ideal $I$ in a standard graded $K$-algebra over a field $K$, we introduce the socle module $\mathrm{Soc}(I)$, whose graded components give us the socle of the powers of $I$. It is observed that $\mathrm{Soc}(I)$ is a finitely generated module over the fiber cone of $I$. In the case that $S$ is the polynomial ring and all powers of $I\subseteq S$ have linear resolution, we define the module $\mathrm{Soc}^*(I)$ which is a module over the Rees ring of $I$. For the edge ideal of a graph and for classes of polymatroidal ideals we study the module structure of their socle modules.
We express the Partial regularities and $a^*$-invariants of a Borel type ideal in terms of its irredundant irreducible decomposition. In addition we consider the behaviours of those invariants under intersections and sums.