Related papers: Edge ideals and DG algebra resolutions

For a graph $G$, the $k$-colour Ramsey number $R_k(G)$ is the least integer $N$ such that every $k$-colouring of the edges of the complete graph $K_N$ contains a monochromatic copy of $G$. Let $C_n$ denote the cycle on $n$ vertices. We show…

Let $R$ be a local commutative noetherian ring and $HKR$ the homology ring of the corresponding Koszul complex. We study the homological properties of $HKR$ in particular the so-called Avramov spectral sequence. When the embedding dimension…

We have tried to translate some graph properties of AG(R) and Gamma(R) to the topological properties of Zariski topology. We prove that Rad(Gamma(R)) and Rad(AG(R)) are equal and they are equal to 3, if and only if the zero ideal of R is an…

Our focus in this paper is in effective computation of the core core(I) of an ideal I which is defined to be the intersection of all minimal reductions of I. The first main result is a closed formula for the graded core(m) of the maximal…

Conditions on the Koszul complex of a noetherian local ring $R$ guarantee that $\mathrm{Tor}^{R}_{i}(M,N)$ is non-zero for infinitely many $i$, when $M$ and $N$ are finitely generated $R$-modules of infinite projective dimension. These…

Let $\G$ be a semisimple algebraic group over a number field $K$, $\mathcal{S}$ a finite set of places of $K$, $K_\mathcal{S}$ the direct product of the completions $K_v, v \in \mathcal{S}$, and $\OO$ the ring of $\mathcal{S}$-integers of…

The number of equations needed to cut out a variety given by an ideal is called the arithmetic rank (of the ideal). It was shown in [8] that the notion of arithmetic rank is strongly related to the concept of regular sequences on the Matlis…

Let $R$ be a $d$-dimensional standard graded ring over an Artin local ring. Let $M$ be the unique maximal homogeneous ideal of $R.$ Let $h^i(R)_n$ denote the length of $H^i_M(R)_n$, i.e. the nth graded component of the ith local cohomology…

In this paper, we prove that the path ideals of both paths and cycles have minimal cellular resolutions. Specifically, these minimal free resolutions coincide with the Barile-Macchia resolutions for paths, and their generalized counterparts…

The algebra of basic covers of a graph G, denoted by \A(G), was introduced by Juergen Herzog as a suitable quotient of the vertex cover algebra. In this paper we show that if the graph is bipartite then \A(G) is a homogeneous algebra with…

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then we show that for the edge ideal $I(G)$ of a graph $G$ the induced matching number of $G$ is an upper bound for the v-number of $I(G)$ when…

We study the minimal bigraded free resolution of an ideal with three generators of the same bidegree, contained in the bihomogeneous maximal ideal $ \langle s,t\rangle \cap \langle u,v \rangle$ of the bigraded ring K[s,t;u,v]. Our analysis…

We define a simple graph as compact if it lacks even cycles and satisfies the odd-cycle condition. Our focus is on classifying all compact graphs and examining the characteristics of their edge rings. Let $G$ be a compact graph and…

Let $R$ be a standard graded commutative algebra over a field $k$, let $K$ be its Koszul complex viewed as a differential graded $k$-algebra, and let $H$ be the homology algebra of $K$. This paper studies the interplay between homological…

Oriented graph discrepancy problems focus on finding specific subgraphs within a given oriented graph $G$ that contain a significant number of edges in one direction. This concept was first introduced by Gishboliner, Krivelevich, and…

Given a commutative unital algebra $\mathcal O$, a proper ideal $\mathcal I$ in $\mathcal O$, and a positively graded differential variety over $\mathcal O/\mathcal I$, we provide a $\mathbb Z$-graded extension, whose negative part is an…

In this paper we study homological properties of the Rees ring R of the graded maximal ideal of a standard graded k-algebra A. In particular we are interested the comparison of the depth and regularity of A and R.

A differential graded algebra can be viewed as an A-infinity algebra. By a theorem of Kadeishvili, a dga over a field admits a quasi-isomorphism from a minimal A-infinity algebra. We introduce the notion of a derived A-infinity algebra and…

This thesis is a study of various ways of measuring the size and complexity of finitely generated modules over a Noetherian local ring. The classical example is the multiplicity or degree. Here we investigate several variants of the degree…

Let $R$ be a commutative ring, $Y\subseteq \mathrm{Spec}(R)$ and $ h_Y(S)=\{P\in Y:S\subseteq P \}$, for every $S\subseteq R$. An ideal $I$ is said to be an $\mathcal{H}_Y$-ideal whenever it follows from $h_Y(a)\subseteq h_Y(b)$ and $a\in…