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Register a new userUpper bounds for Extremal Betti Numbers of $t$-Spread Strongly Stable Ideals
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We study the extremal Betti numbers of the class of $t$--spread strongly stable ideals. More precisely, we determine the maximal number of admissible extremal Betti numbers for such ideals, and thereby we generalize the known results for $tin {1,2}$.
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Let $K$ be a field and let $S=K[x_1,dots,x_n]$ be a standard polynomial ring over a field $K$. We characterize the extremal Betti numbers, values as well positions, of a $t$-spread strongly stable ideal of $S$. Our approach is constructive. Indeed, given some positive integers $a_1,dots,a_r$ and some pairs of positive integers $(k_1,ell_1),dots,(k_r,ell_r)$, we are able to determine under which conditions there exist a $t$-spread strongly stable ideal $I$ of $S$ with $beta_{k_i, k_iell_i}(I)=a_i$, $i=1, ldots, r$, as extremal Betti numbers, and then to construct it.
Let $K$ be a field and $S = K[x_1,dots,x_n]$ be a polynomial ring over $K$. We discuss the behaviour of the extremal Betti numbers of the class of squarefree strongly stable ideals. More precisely, we give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of such a class of squarefree monomial ideals.
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 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 $b$.
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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