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We study homological properties of random quadratic monomial ideals in a polynomial ring $R = {mathbb K}[x_1, dots x_n]$, utilizing methods from the Erd{o}s-R{e}nyi model of random graphs. Here for a graph $G sim G(n, p)$ we consider the `coedge ideal $I_G$ corresponding to the missing edges of $G$, and study Betti numbers of $R/I_G$ as $n$ tends to infinity. Our main results involve fixing the edge probability $p = p(n)$ so that asymptotically almost surely the Krull dimension of $R/I_G$ is fixed. Under these conditions we establish various properties regarding the Betti table of $R/I_G$, including sharp bounds on regularity and projective dimension, and distribution of nonzero normalized Betti numbers. These results extend work of Erman and Yang, who studied such ideals in the context of conjectured phenomena in the nonvanishing of asymptotic syzygies. Along the way we establish results regarding subcomplexes of random clique complexes as well as notions of higher-dimensional vertex $k$-connectivity that may be of independent interest.
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We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.
Let $G$ be a finite simple graph on the vertex set $V(G) = {x_1, ldots, x_n}$ and $I(G) subset K[V(G)]$ its edge ideal, where $K[V(G)]$ is the polynomial ring in $x_1, ldots, x_n$ over a field $K$ with each ${rm deg} x_i = 1$ and where $I(G)$ is generated by those squarefree quadratic monomials $x_ix_j$ for which ${x_i, x_j}$ is an edge of $G$. In the present paper, given integers $1 leq a leq r$ and $s geq 1$, the existence of a finite connected simple graph $G = G(a, r, d)$ with ${rm im}(G) = a$, ${rm reg}(R/I(G)) = r$ and ${rm deg} h_{K[V(G)]/I(G)} (lambda) = s$, where ${rm im}(G)$ is the induced matching number of $G$ and where $h_{K[V(G)]/I(G)} (lambda)$ is the $h$-polynomial of $K[V(G)]/I(G)$.
We study the equality of the extremal Betti numbers of the binomial edge ideal $J_G$ and those of its initial ideal ${rm in}(J_G)$ of a closed graph $G$. We prove that in some cases there is an unique extremal Betti number for ${rm in}(J_G)$ and as a consequence there is an unique extremal Betti number for $J_G$ and these extremal Betti numbers are equal
Let $R = mathbb{K}[x_1, ldots, x_n]$ and $I subset R$ be a homogeneous ideal. In this article, we first obtain certain sufficient conditions for the subadditivity of $R/I$. As a consequence, we prove that if $I$ is generated by homogeneous complete intersection, then subadditivity holds for $R/I$. We then study a conjecture of Avramov, Conca and Iyengar on subadditivity, when $I$ is a monomial ideal with $R/I$ Koszul. We identify several classes of edge ideals of graphs $G$ such that the subadditivity holds for $R/I(G)$. We then study the strand connectivity of edge ideals and obtain several classes of graphs whose edge ideals are strand connected. Finally, we compute upper bounds for multigraded Betti numbers of several classes of edge ideals.
Let $G$ be a simple graph on the vertex set $[n]$ and $J_G$ be the corresponding binomial edge ideal. Let $G=v*H$ be the cone of $v$ on $H$. In this article, we compute all the Betti numbers of $J_G$ in terms of Betti number of $J_H$ and as a consequence, we get the Betti diagram of wheel graph. Also, we study Cohen-Macaulay defect of $S/J_G$ in terms of Cohen-Macaulay defect of $S_H/J_H$ and using this we construct a graph with Cohen-Macaulay defect $q$ for any $qgeq 1$. We obtain the depth of binomial edge ideal of join of graphs. Also, we prove that for any pair $(r,b)$ of positive integers with $1leq b< r$, there exists a connected graph $G$ such that $reg(S/J_G)=r$ and the number of extremal Betti number of $S/J_G$ is $b$.
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