Narrow Results

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

We classify the Hilbert polynomial of a local ring $(R,𝔪)$ satisfying ${𝔪}^4=0$ which admits a doubly-infinite eventually linear resolution $C$ which is “partially” complete — that is, for which $H^i{Hom}_R(C,R)$ vanishes for all $\left|i\right|\gg 0$. As a corollary to our main result, we show that an ${𝔪}^4=0$ local ring can admit certain classes of asymmetric partially complete resolutions only if its Hilbert polynomial is symmetric. Moreover, we show that the Betti sequence associated to an eventually linear partially complete resolution over an ${𝔪}^4=0$ local ring cannot be periodic of period two or three.

When ${𝒞}$ is a chordal clutter in the sense of Woodroofe or Emtander, we show that the complement clutter is edgewise strongly shellable. When ${𝒞}$ is indeed a finite simple graph, we provide additional characterization of chordal graphs from the point of view of strong shellability. In particular, the generic graph $G_T$ of a tree is shown to be bi-strongly shellable.

A projective variety in a projective space is said to be $p$-linear if it is $p$-regular and has no defining equation of degree $<p$. It is well known that $2$-linear varieties are exactly varieties of minimal degree. In this paper, we study $3$-linear varieties of codimension $2$. We classify all smooth $3$-linear varieties of codimension 2. There are six kinds of such varieties. Also, we provide some nonconic singular $3$-linear varieties of codimension $2$.

Let $K$ be a field with $\mathrm{\text{char}}(K)=0$ and $E=K〈e_1,\dots ,e_n〉$ an exterior algebra over $K$ with a standard grading $\mathrm{\text{deg}}{e}_i=1$. Let $R=E/J$ be a graded algebra, where $J$ is a graded ideal in $E$. In this paper, we study universally Koszul and initially Koszul properties of $R$ and find classes of ideals $J$ which characterize such properties of $R$. As applications, we classify arrangements whose Orlik–Solomon algebras are universally Koszul or initially Koszul. These results are related to a long-standing question of Shelton–Yuzvinsky [B. Shelton and S. Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc.56 (1997) 477–490].

On the one hand, Ohsugi and Hibi characterized the edge ring of a finite connected simple graph with a $2$-linear resolution. On the other hand, Hibi, Matsuda and the author conjectured that the edge ring of a finite connected simple graph with a $q$-linear resolution, where $q\ge 3$, is a hypersurface and proved the case $q=3$. In this paper, we solve this conjecture for the case of finite connected simple bipartite graphs.

Let ${𝒟}=(V({𝒟}),E({𝒟}))$ be a weighted oriented graph and $I({𝒟})$ denote the corresponding edge ideal. In this paper, we give a combinatorial characterization of $I({𝒟})$ which has a linear resolution. As a consequence, we prove that if $I({𝒟})$ is the edge ideal of a weighted oriented graph ${𝒟}$, then $I({𝒟})$ has a linear resolution if and only if all powers of $I({𝒟})$ have a linear resolution. Also, we prove that if ${𝒟}$ is a weighted oriented graph and $w(x)>1$ for all $x\in V({𝒟})$, then $I({𝒟})$ has a linear resolution if and only if all powers of $I({𝒟})$ have linear quotients. We provide a lower bound for the regularity of powers of edge ideals of weighted oriented graphs in terms of induced matching. Finally, we obtain a general upper bound for the regularity of edge ideals of weighted oriented graphs.

The concept of the weakly polymatroidal ideal, which generalizes both the polymatroidal ideal and the prestable ideal, is introduced. A fundamental fact is that every weakly polymatroidal ideal has a linear resolution. One of the typical examples of weakly polymatroidal ideals arises from finite partially ordered sets. We associate each weakly polymatroidal ideal with a finite sequence, alled the polymatroidal sequence, which will be useful for the computation of graded Betti numbers of weakly polymatroidal ideals as well as for the construction of weakly polymatroidal ideals.

In this paper, we study the notion of chordality and cycles in clutters from a commutative algebraic point of view. The corresponding concept of chordality in commutative algebra is having a linear resolution. We mainly consider the generalization of chordality proposed by Bigdeli et al. in 2017 and the concept of cycles introduced by Cannon and Faridi in 2013, and study their interrelations and algebraic interpretations. In particular, we investigate the relationship between chordality and having linear quotients in some classes of clutters. Also, we show that if $\mathcal{C}$ is a clutter such that 〈$\mathcal{C}$〉 is a vertex decomposable simplicial complex or I($\overline{\mathcal{C}}$) is squarefree stable, then $\mathcal{C}$ is chordal.

Let V be a k–vector space with basis e₁,…,e_n and let E be the exterior algebra over V. For any subset σ = {i₁,…, i_d} of {1,…, n} with i₁ < i₂ < … < i_d we call e_σ = e_i1 ∧…∧e_id a monomial of degree d and we denote the set of all monomials of degree d by M_d. We order the monomials lexicographically so that e₁ > e₂ > … > e_n. Then a lexsegment ideal is an ideal generated by a subset of M_d of the form L(u,v) = {w ∈ M_d : u ≥ w ≥ v}, where u, v ∈ M_d and u ≥ v. We describe all lexsegment ideals with linear resolution in the exterior algebra. Then we study the vanishing and non vanishing of reduced simplicial cohomology groups of a simplicial complex Δ and of certain subcomplexes of Δ with coefficients in a field k. Finally we give an idea of the applicative aspects of our results.