arXiv subjects
Publications and source records attributed to Naoki Terai.
The lemma given by Schmitt and Vogel is an important tool in the study of arithmetical rank of squarefree monomial ideals. In this paper, we give a Schmitt-Vogel type lemma for reductions as an analogous result.
We study $h$-vectors of simplicial complexes which satisfy Serre's condition ($S_r$). We say that a simplicial complex $Δ$ satisfies Serre's condition ($S_r$) if $\tilde H_i(\lk_Δ(F);K)=0$ for all faces $F \in Δ$ and for all $i < \min \{r-1,\dim \lk_Δ(F)\}$, where $\lk_Δ(F)$ is the link of $Δ$ with respect to $F$ and where $\tilde H_i(Δ;K)$ is the reduced homology groups of $Δ$ over a field $K$. The main result of this paper is that if $Δ$ satisfies Serre's condition ($S_r$) then (i) $h_k(Δ)$ is non-negative for $k =0,1,...,r$ and (ii) $\sum_{k\geq r}h_k(Δ)$ is non-negative.
We consider a class of graphs $G$ such that the height of the edge ideal $I(G)$ is half of the number $\sharp V(G)$ of the vertices. We give Cohen-Macaulay criteria for such graphs.
We show that the Stanley-Reisner ideal of the one-dimensional simplicial complex whose diagram is an $n$-gon is always a set-theoretic complete intersection in any positive characteristic.
In this paper, we prove that the Stanley--Reisner ideal of any connected simplicial complex of dimension $\ge 2$ that is locally complete intersection is a complete intersection ideal. As an application, we show that the Stanley--Reisner ideal whose powers are Buchsbaum is a complete intersection ideal.
When a cone is added to a simplicial complex $Δ$ over one of its faces, we investigate the relation between the arithmetical ranks of the Stanley-Reisner ideals of the original simplicial complex and the new simplicial complex $Δ'$. In particular, we show that the arithmetical rank of the Stanley-Reisner ideal of $Δ'$ equals the projective dimension of the Stanley-Reisner ring of $Δ'$ if the corresponding equality holds for $Δ$.
We prove that certain class of Stanley--Reisner rings having sufficiently large multiplicities are Cohen--Macaulay using Alexander duality.
Algebraic and combinatorial properties of a monomial ideal and its radical are compared.
In this paper, we study non-Cohen--Macaulay Buchsbaum Stanley--Reisner rings with linear free resolution. In particular, for given integers $c$, $d$, $q$ with $c \ge 1$, $2 \le q \le d$, we give an upper bound $h_{c,d,q}$ on the dimension of the unique non-vanishing homology $\widetilde{H}_{q-2}(Δ;k)$ of a $d$-dimensional Buchsbaum ring $k[Δ]$ with $q$-linear resolution and codimension $c$. Also, we discuss about existence for such Buchsbaum rings with $\dim_k \widetilde{H}_{q-2}(Δ;k) = h$ for any $h$ with $0 \le h \le h_{c,d,q}$, and prove an existence theorem in the case of $q=d=3$ using the notion of Cohen--Macaulay linear cover. On the other hand, we introduce the notion of Buchsbaum Stanley--Reisner rings with minimal multiplicity of type $q$, which extends the notion of Buchsbaum rings with minimal multiplicity defined by Goto. As an application, we give many examples of Buchsbaum Stanley--Reisner rings with $q$-linear resolution.