Related papers: Betti numbers of skeletons

Let $\Delta$ be a stable simplicial complex on $n$ vertexes. Over an arbitrary base field $K$, the symmetric algebraic shifted complex $\Delta^s$ of $\Delta$ is defined. It is proved that the Betti numbers of the Stanley-Reisner ideals in…

Let X be a k-dimensional simplicial complex such that the (k-j-2)-dimensional homology of the links of all j-dimensional simplices in X vanishes. An upper bound is given on the (k-1)-th Betti number of X. Examples based on sum complexes…

The Buchsbaum-Eisenbud-Horrocks Conjecture predicts that if M is a non-zero module of finite length and finite projective dimension over a local ring R of dimension d, then the i-th Betti number of M is at least d choose i. This conjecture…

We introduce a squarefree monomial ideal associated to the set of domino tilings of a $2\times n$ rectangle and proceed to study the associated minimal free resolution. In this paper, we use results of Dalili and Kummini to show that the…

We present a simple method to obtain information regarding the connectivity of the 1-skeleta of a wide family of simplicial complexes through bounds for the Castelnuovo-Mumford regularity of their Stanley-Reisner rings. In this way we…

The main goal of this paper is to size up the minimal graded free resolution of a homogeneous ideal in terms of its generating degrees. By and large, this is too ambitious an objective. As understood, sizing up means looking closely at the…

This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out…

Let $(M,d,\mathfrak{m})$ be a noncompact $\texttt{RCD}(0,N)$ space with $N\in\mathbb{N}_+$ and $\text{supp}\mathfrak{m}=M$. We prove that if the first Betti number of $M$ equals $N-1$, then $(M,d,\mathfrak{m})$ is either a flat Riemannian…

Let $M(d,\chi)$ with $(d,\chi)=1$ be the moduli space of semistable sheaves on $\mathbb{P}^2$ supported on curves of degree $d$ and with Euler characteristic $\chi$. The cohomology ring $H^*(M(d,\chi),\mathbb{Z})$ of $M(d,\chi)$ is…

When a cone is added to a simplicial complex $\Delta$ over one of its faces, we investigate the relation between the arithmetical ranks of the Stanley-Reisner ideals of the original simplicial complex and the new simplicial complex…

In this thesis we investigate certain types of monomial ideals of polynomial rings over fields. We are interested in minimal free resolutions of these ideals (or equivalently the quotients of the polynomial ring by the ideals) considered as…

We find a combinatorial formula which computes the first cotangent cohomology module of Stanley-Reisner rings associated to matroids. For arbitrary simplicial complexes we provide upper bounds for the dimensions of the multigraded…

A recent result of Eisenbud-Schreyer and Boij-S\"oderberg proves that the Betti diagram of any graded module decomposes as a positive rational linear combination of pure diagrams. When does this numerical decomposition correspond to an…

We consider the multiparameter random simplicial complex on a vertex set $\{ 1,\dots,n \}$, which is parameterized by multiple connectivity probabilities. Our key results concern the topology of this complex of dimensions higher than the…

Let $S$ be a polynomial ring and let $I \subseteq S$ be a monomial ideal. In this short note, we propose the conjecture that the Betti poset of $I$ determines the Stanley projective dimension of $S/I$ or $I$. Our main result is that this…

Let $I_1,\dots,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_1 \cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the…

The Upper Bound Theorem for convex polytopes implies that the $p$-th Betti number of the \v{C}ech complex of any set of $N$ points in $\mathbb R^d$ and any radius satisfies $\beta_{p} = O(N^{m})$, with $m = \min \{ p+1, \lceil d/2 \rceil…

Given any order ideal $U$ consisting of color-squarefree monomials involving variables with $d$ colors, we associate to it a balanced $(d-1)$-dimensional simplicial complex $\Delta_{\mathrm{bal}}(U)$ that we call a balanced squeezed…

Let $R:= \Bbbk[x_1,\ldots,x_{n}]$ be a polynomial ring over a field $\Bbbk$, $I \subset R$ be a homogeneous ideal with respect to a weight vector $\omega = (\omega_1,\ldots,\omega_n) \in (\mathbb{Z}^+)^n$, and denote by $d$ the Krull…

Let $X\subset\mathbb{P}^{n+e}$ be any $n$-dimensional closed subscheme. We are mainly interested in two notions related to syzygies: one is the property $\mathbf{N}_{d,p}~(d\ge 2, ~p\geq 1)$, which means that $X$ is $d$-regular up to $p$-th…