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This paper concerns the study of a class of clutters called simplicial subclutters. Given a clutter $\mathcal{C}$ and its simplicial subclutter $\mathcal{D}$, we compare some algebraic properties and invariants of the ideals $I, J$…

We construct Coleff-Herrera products and Bochner-Martinelli type residue currents associated with a tuple $f$ of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case, as…

Given any equigenerated monomial ideal $I$ with the property that the defining ideal $J$ of the fiber cone $ F(I)$ of $I$ is generated by quadratic binomials, we introduce a matrix such that the set of its binomial $2$-minors is a…

Let $S = K[x_1, \dots, x_n]$ be the standard graded polynomial ring over a field $K$. In this paper, we address and completely solve two fundamental open questions in Commutative Algebra: (i) For which degrees $d$, does there exist a…

We introduce the concept of quasi-linearity and prove it is necessary for a monomial ideal to have a linear resolution and identify all the quasi-linear quadratic monomial ideals. We define a strongly linear monomial for a monomial ideal…

Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms…

We use the theory of poset resolutions to construct the minimal free resolution of an arbitrary stable monomial ideal in the polynomial ring whose coefficients are from a field. This resolution is recovered by utilizing a poset of…

Given a square-free monomial ideal $I$, we define a simplicial complex labeled by the generators of $I^2$ which supports a free resolution of $I^2$. As a consequence, we obtain (sharp) upper bounds on the Betti numbers of the second power…

Using Bochner-Martinelli type residual currents we prove some generalizations of Jacobi's Residue Formula, which allow proper polynomial maps to have `common zeroes at infinity', in projective or toric situations.

Lejeune-Jalabert showed that the fundamental class of a Cohen-Macaulay ideal $\mathfrak a\subset \mathcal O_0$ admits a representation as a residue, constructed from a free resolution of $\mathfrak a$, multiplied by a certain differential…

Building on coprincipal mesoprimary decomposition [Kahle and Miller, 2014], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct…

Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the…

For each squarefree monomial ideal $I\subset S = k[x_{1},\ldots, x_{n}] $, we associate a simple graph $G_I$ by using the first linear syzygies of $I$. In cases, where $G_I$ is a cycle or a tree, we show the following are equivalent: (a) $…

We prove that a certain cohomological residue associated to an ideal of pure dimension is annihilated exactly by the ideal. The cohomological residue is quite explicit and generalizes the classical local Grothendieck residue and the…

A free resolution of free partially commutative monoids is constructed and with its help the homological dimension of these monoids is calculated.

Let $R$ be a commutative Noetherian ring and let ${\bf x} :=x_1,\ldots,x_d$ be a regular $R$-sequence contained in the Jacobson radical of $R$. An ideal $I$ of $R$ is said to be a monomial ideal with respect to ${\bf x}$ if it is generated…

Kuratowski's closure-complement problem gives rise to a monoid generated by the closure and complement operations. Consideration of this monoid yielded an interesting classification of topological spaces, and subsequent decades saw further…

In this paper we characterize, in algebraic and geometric terms, when a graded vanishing ideal is generated by binomials over any field K.

In this paper we consider monomial localizations of monomial ideals and conjecture that a monomial ideal is polymatroidal if and only if all its monomial localizations have a linear resolution. The conjecture is proved for squarefree…

In this paper, we provide a combinatorial criteria for equigenerated monomial ideals in three variables to have linear resolutions. As a consequence, we prove that in three variables, equigenerated monomial ideals with linear resolutions…