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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 $mathrm{type}(Delta(G))=t$.
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 tha
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 hav
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
In this paper we determine the possible Hilbert functions of a Cohen-Macaulay local ring of dimension $d$, multiplicity $e$ and first Hilbert coefficient $e_1$ in the case $e_1 = e + 2$.
Let $(A,mathfrak{m})$ be a hypersurface ring with dimension $d$, and $M$ a MCM $A-$module with reduction no.2 and $mu(M)=2$ or $3$ then we have proved that depth$G(M)geq d-mu(M)+1$. If $e(A)=3$ and $mu(M)=4$ then in this case we have proved that dept