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Register a new userRegularity of powers of Stanley-Reisner ideals of one-dimensional simplicial complexes
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Let $Delta$ be a one-dimensional simplicial complex. Let $I_Delta$ be the Stanley-Reisner ideal of $Delta$. We prove that for all $s ge 1$ and all intermediate ideals $J$ generated by $I_Delta^s$ and some minimal generators of $I_Delta^{(s)}$, we have $${rm reg} J = {rm reg} I_Delta^s = {rm reg} I_Delta^{(s)}.$$
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The goal of the present paper is the study of some algebraic invariants of Stanley-Reisner rings of Cohen-Macaulay simplicial complexes of dimension $d - 1$. We prove that the inequality $d leq mathrm{reg}(Delta) cdot mathrm{type}(Delta)$ holds for any $(d-1)$-dimensional Cohen-Macaulay simplicial complex $Delta$ satisfying $Delta=mathrm{core}(Delta)$, where $mathrm{reg}(Delta)$ (resp. $mathrm{type}(Delta)$) denotes the Castelnuovo-Mumford regularity (resp. Cohen-Macaulay type) of the Stanley-Reisner ring $Bbbk[Delta]$. Moreover, for any given integers $d,r,t$ satisfying $r,t geq 2$ and $r leq d leq rt$, we construct a Cohen-Macaulay simplicial complex $Delta(G)$ as an independent complex of a graph $G$ such that $dim(Delta(G))=d-1$, $mathrm{reg}(Delta(G))=r$ and $mathrm{type}(Delta(G))=t$.
In this article we associate to every lattice ideal $I_{L,rho}subset K[x_1,..., x_m]$ a cone $sigma $ and a graph $G_{sigma}$ with vertices the minimal generators of the Stanley-Reisner ideal of $sigma $. To every polynomial $F$ we assign a subgraph $G_{sigma}(F)$ of the graph $G_{sigma}$. Every expression of the radical of $I_{L,rho}$, as a radical of an ideal generated by some polynomials $F_1,..., F_s$ gives a spanning subgraph of $G_{sigma}$, the $cup_{i=1}^s G_{sigma}(F_i)$. This result provides a lower bound for the minimal number of generators of $I_{L,rho}$ and therefore improves the generalized Krulls principal ideal theorem for lattice ideals. But mainly it provides lower bounds for the binomial arithmetical rank and the $A$-homogeneous arithmetical rank of a lattice ideal. Finally we show, by a family of examples, that the bounds given are sharp.
In this article, we prove that for several classes of graphs, the Castelnuovo-Mumford regularity of symbolic powers of their edge ideals coincide with that of their ordinary powers.
In this article, we study the regularity of integral closure of powers of edge ideals. We obtain a lower bound for the regularity of integral closure of powers of edge ideals in terms of induced matching number of graphs. We prove that the regularity of integral closure of powers of edge ideals of graphs with at most two odd cycles is the same as the regularity of their powers.
We consider simplicial complexes admitting a free action by an abelian group. Specifically, we establish a refinement of the classic result of Hochster describing the local cohomology modules of the associated Stanley--Reisner ring, demonstrating that the topological structure of the free action extends to the algebraic setting. If the complex in question is also Buchsbaum, this new description allows for a specialization of Schenzels calculation of the Hilbert series of some of the rings Artinian reductions. In further application, we generalize to the Buchsbaum case the results of Stanley and Adin that provide a lower bound on the $h$-vector of a Cohen-Macaulay complex admitting a free action by a cyclic group of prime order.
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