The graded Betti numbers of truncation of ideals in polynomial rings

Let $R=\mathbb{K}[x_1,\dots,x_n]$, a graded algebra $S=R/I$ satisfies $N_{k,p}$ if $I$ is generated in degree $k$, and the graded minimal resolution is linear the first $p$ steps, and the $k$-index of $S$ is the largest $p$ such that $S$ satisfies $N_{k,p}$. Eisenbud and Goto have shown that for any graded ring $R/I$, then $R/I_{\geq k}$, where $I_{\geq k}=I\cap M^k$ and $M=(x_1,\dots,x_n)$, has a $k$-linear resolution (satisfies $N_{k,p}$ for all $p$) if $k\gg0$. For a squarefree monomial ideal $I$, we are here interested in the ideal $I_k$ which is the squarefree part of $I_{\geq k}$. The ideal $I$ is, via Stanley-Reisner correspondence, associated to a simplicial complex $\Delta_I$. In this case, all Betti numbers of $R/I_k$ for $k>\min\{\text{deg}(u)\mid u\in I\}$, which of course is a much finer invariant than the index, can be determined from the Betti diagram of $R/I$ and the $f$-vector of $\Delta_I$. We compare our results with the corresponding statements for $I_{\ge k}$. (Here $I$ is an arbitrary graded ideal.) In this case we show that the Betti numbers of $R/I_{\ge k}$ can be determined from the Betti numbers of $R/I$ and the Hilbert series of $R/I_{\ge k}$

On the Eliahou and Villarreal conjecture about the projective dimension of co-chordal graphs

Let $I(G)$ be the edge ideal of a graph $G$ with $|V(G)|=n$ and $R=\mathbb{K}[x\mid x\in V(G)]$ be a polynomial ring in $n$ variables over a field $\mathbb{K}$. In this paper we are interested in a conjecture of Eliahou and Villarreal which states that $\text{pdim}(R/I(G))=\max_{1\leq i \leq n}\left\{\text{deg}_{G}(x_i)\right\}$ when $G$ is connected and co-chordal. We show that this conjecture is not true in general. In fact we show that the difference between $\text{pdim}(R/I(G))$ and $\max_{1\leq i \leq n}\left\{\text{deg}_{G}(x_i)\right\}$ is not necessarily bounded. For any graph $G$ we prove that $\max_{1\leq i \leq n}\left\{\text{deg}_{G}(x_i)\right\}\leq \text{pdim}(R/I(G))$. For a non-increasing sequence of positive integers $(d_1,d_2,\dots,d_q)$, we define the $(d_1,d_2,\dots,d_q)$-tree graphs. We show that the independence complex of the complements of these type of trees is vertex decomposable and quasi-forest. Finally we show that the conjecture is valid when the complement of $G$ is a $(d_1,d_2,\dots,d_q)$-tree or $G$ has a full-vertex. To our knowledge the results in this paper generalise all the existing classes of graphs for which the conjecture is true.Comment: 15 pages, comments are welcom