arXiv subjects

Yuji Muta

Publications and source records attributed to Yuji Muta.

6 recordsLinked to original sources

arXiv2026

Comparisons between ordinary and symbolic powers of edge ideals with respect to regularity and depth

In this paper, we investigate the difference between ordinary and symbolic powers of edge ideals on the Castelnuovo-Mumford regularity and the depth. As a main theorem, we prove that the regularities of ordinary and symbolic powers coincide for edge ideals of simplicial graphs, as a partial result of Minh's conjecture. For the depth, we first show that, for each $k\geq2$, the inequality $\operatorname{depth} S/I^{(k)}\geq\operatorname{depth} S/I^{k}$ does not hold for squarefree monomial ideals in general. On the other hand, we prove that it holds for edge ideals when $k=2,3$. We also study the symbolic-ordinary discrepancy module $I(G)^{(k)}/I(G)^{k}$ of the edge ideal $I(G)$ of a graph $G$ and give a graph-theoretic formula for its Krull dimension in terms of induced odd cycles of $G$, thereby answering a question and a problem posed by Ha and Minh.

math.AC↗

arXiv2026

Symbolic Rees algebras of complementary edge ideals

Let $G$ be a finite simple graph on $[n]$ and let $I_c(G)$ denote its complementary edge ideal in the polynomial ring $S = K[x_1,\dots,x_n]$. We give a combinatorial description, in terms of the structure of $G$, of the minimal generators of the symbolic Rees algebra $\mathcal{R}_s(I_c(G)) = \bigoplus_{k \geq 0} I_c(G)^{(k)} t^k$, and show that this algebra is generated in degree at most $6$. Moreover, we completely determine the minimal generators of $\mathcal{R}_{s}(I_{c}(G))$ in graph-theoretic terms. We then study in more detail the homological invariants of the symbolic powers $I_c(G)^{(k)}$ for the classes of cycle graphs and complete multipartite graphs. For theses families, we study the behavior of the symbolic depth function $k\mapsto\operatorname{depth} S/I_c(G)^{(k)}$, we obtain the limit depth of the symbolic powers and the Waldschmidt constant of $I_c(G)$, and further prove that all the symbolic powers $I_c(G)^{(k)}$ are componentwise linear.

math.AC↗

arXiv2025

Algebraic study on rooted products of graphs and multi-clique corona graphs

In this paper, we study rooted products of graphs from the perspective of combinatorial commutative algebra. For edge ideals, we introduce the 2-Cohen-Macaulayness with respect to a vertex and use it to investigate when edge ideals of rooted products of graphs are Cohen-Macaulay. Moreover, we completely determine when attaching a graph on at most six vertices to a given graph as rooted products, yields a Cohen-Macaulay edge ideal. Also, we define mulit-clique corona graphs as a generalization of clique-corona graphs and multi-whisker graphs. We prove that multi-clique corona graphs are vertex decomposable and hence sequentially Cohen-Macaulay. Also, we give formulas for the projective dimension and the Castelnuovo-Mumford regularity.

math.AC↗

arXiv2025

The Serre depth of Stanley-Reisner rings and the depth of their symbolic powers

We investigate an invariant, called the Serre depth, from the perspective of combinatorial commutative algebra. In this paper, we establish several properties of an analogue of the depth of Stanley-Reisner rings. In particular, we relate the Serre depth both to the minimal free resolution of a Stanley-Reisner ring and to that of its Alexander dual. Also, we establish an analogue of a known result that describes the depth of Stanley-Reisner rings in terms of skeletons. Moreover, we study the Serre depth for $(S_{2})$ and the depth on the symbolic powers of Stanley--Reisner ideals. It had been an open question whether the depth of the symbolic powers of Stanley-Reisner ideals satisfies a non-increasing property, but Nguyen and Trung provided a negative answer. We construct an example that the Serre depth for $(S_{2})$ and the depth do not satisfy this property and its second symbolic power is Cohen-Macaulay. Moreover, we prove that the sequence of the Serre depth for $(S_{2})$ on the symbolic powers is convergent and that its limit coincides with the minimum value. Finally, we study the Serre depth on edge and cover ideals. Whether the depth on symbolic powers of edge ideals satisfies a non-increasing property has remained an open question. We address a related problem and show that the Serre depth for $(S_{2})$ on edge ideals of any well-covered graph satisfies a non-increasing property. In addition, we prove that the Serre depth for $(S_{2})$ on the cover ideals of any graph also satisfies a non-increasing property. Moreover, we determine the Serre depth on edge ideals of very well-covered graphs.

math.AC↗

arXiv2025

On minimal free resolutions of the cover ideals of clique-whiskered graphs

We explicitly construct a minimal free resolution of the cover ideals of clique-whiskered graphs. In particular, Cohen--Macaulay chordal graphs, clique corona graphs, and Cohen--Macaulay Cameron--Walker graphs are examples of clique-whiskered graphs. We also introduce multi-clique-whiskered graphs as a generalization of both clique-whiskered graphs and multi-whisker graphs. We prove that multi-clique-whiskered graphs are vertex decomposable and hence sequentially Cohen--Macaulay. Moreover, we provide formulas for the projective dimension and the Castelnuovo--Mumford regularity of their edge ideals. Finally, we construct minimal free resolutions of the cover ideals of both multi-clique-whiskered graphs and very well-covered graphs.

math.AC↗

arXiv2025

The v-numbers of Stanley-Reisner ideals from the viewpoint of Alexander dual complexes

We express the v-number of the Stanley-Reisner ideal in terms of its Alexander dual complex and prove that the v-number of a cover ideal is just two less than the initial degree of the its syzygy module. We give some relation between the v-number of the Stanley-Reisner ideal and the Serre-depth of the quotient ring of the second symbolic power of the Stanley-Reisner ideal of its Alexander dual. We also show that the v-number of the Stanley-Reisner ideal of a 2-pure simplicial complex is equal to the dimension of its Stanley-Reisner ring.

math.AC↗