Related papers: On simple A-multigraded minimal resolutions

To any lattice $L \subset \mathbb{Z}^{m}$ one can associate the lattice ideal $I_{L} \subset K[x_{1},...,x_{m}]$. This paper concerns the study of the relation between the binomial arithmetical rank and the minimal number of generators of…

For an ideal I in a regular local ring (R,m)$ with residue class field K = R/m or a standard graded K-algebra R we show that for k >> 0 --> the Artin--Rees number of the syzygy modules of I^k as submodules of the free modules from a free…

Let $A$ be a commutative noetherian ring and $I$ an ideal in $A$. We characterize algebraically when all the minimal primes of the associated graded ring $G_I A$ contract to minimal primes of $A/I$. This, applied to intersection theory,…

We study the almost complete intersection ring $R$ defined by $n+1$ general quadrics in a polynomial ring in $n$ variables over a field $\sf{k}$ and a corresponding linked Gorenstein ring $A$. The overarching theme is that, while not Koszul…

A minimal monomial ideal is the combinatorially simplest monomial ideal whose lcm-lattice equals a given finite atomic lattice $\hat{L}$. The minimal ideal inherits many nice properties of any ideal $I$ whose lcm-lattice also equals…

Among other results, we prove that if $I$ is a monomial ideal of $S=K[x_1,\ldots,x_n]$, where $K$ is a field, and $a\geq b-1\geq0$ are integers such that $a+b\leq\mathrm{proj~dim}(S/I)$, then $$t_{a+b}\leq…

Let $R=\mathbb{K}[x_1,\dots,x_n]$, a graded algebra $S=R/I$ satisfies $N_{k,p}$ if $I$ is generated in degree $k$, and the graded minimal resolution is linear the first $p$ steps, and the $k$-index of $S$ is the largest $p$ such that $S$…

Consider the polynomial ring $R_n = k[x_1,...,x_n]$, where $k$ is a field. Let $m = (x_1,...,x_n)$ and $I$ be an $m$-primary monomial ideal in $R$. We consider the problem of determining whether such ideals are in the Gorenstein liasion…

Let $R = S/I$ be a quotient of a standard graded polynomial ring $S$ by an ideal $I$ generated by quadrics. If $R$ is Koszul, a question of Avramov, Conca, and Iyengar asks whether the Betti numbers of $R$ over $S$ can be bounded above by…

We construct a (shellable) polyhedral cell complex that supports a minimal free resolution of a Borel fixed ideal, which is minimally generated (in the Borel sense) by just one monomial in S=k[x_1,x_2,...,x_n]; this includes the case of…

Let $A$ be a regular domain containing a field $K$ of characteristic zero, $G$ be a finite subgroup of the group of automorphisms of $A$ and $B=A^G$ be the ring of invariants of $G$. Let $S= A[X_1,\ldots, X_m]$ and $R= B[X_1, \ldots, X_m]$…

Let $R_0$ be any domain, let $R=R_0[U_1, ..., U_s]/I$, where $U_1, ..., U_s$ are indeterminates of some positive degrees, and $I\subset R_0[U_1, ..., U_s]$ is a homogeneous ideal. The main theorem in this paper is states that all the…

A restricted $d$th power of an ideal $I$ is obtained by restricting the exponent vectors allowed to appear on the "natural" generating set of $I^d$, for some integer $d$. In this paper, we study homological properties of restricted powers…

We prove the subadditivity property for the maximal degrees of the syzygies of facet ideals simplicial forests. For such an ideal $I$, if the $i$-th Betti number is nonzero and $i=a+b$, we show that there are monomials in the lcm lattice of…

Let $\mathbb{K}$ be a field and $A$ a Noetherian $\mathbb{K}$-algebra. In a paper of 2020, M. Albert, C. Bertone, M. Roggero and W. M. Seiler proved that, given a quasi-stable module $U \subset R^m$ with $R=\mathbb{K}[x_0,\dots,x_n]$, any…

Guo and the second author have shown that the closure $[I]$ in the Drury-Arveson space of a homogeneous principal ideal $I$ in $\mathbb{C}[z_1,...,z_n]$ is essentially normal. In this note, the authors extend this result to the closure of…

Let $G$ be a linear semisimple Lie group without compact factors. We show that uniform approximate lattices $\Lambda$ arising as regular model sets in $G$ determine the ambient group $G$ in a strong sense. Specifically, for every…

For an ideal $I_{m,n}$ generated by all square-free monomials of degree $m$ in a polynomial ring $R$ with $n$ variables, we obtain a specific embedding of a canonical module of $R/I_{m,n}$ to $R/I_{m,n}$ itself. The construction of this…

In this paper we investigate the Rees algebras of squarefree monomial ideals $I \subset S=K[x_1,\dots,x_n]$ generated in degree $n-2$, where $K$ is a field. Every such ideal arises as the complementary edge ideal $I_c(G)$ of a finite simple…

We consider a nodal curve $C$ in the complex projective plane whose irreducible components $C_i$ are smooth. A minimal set of generators $G$ for the first and second syzygy modules of the Jacobian ideal of $C$ are described, using recent…