交换代数

In this paper, we give a new criterion for the Cohen-Macaulayness of vertex splittable ideals, a family of monomial ideals recently introduced by Moradi and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and provides an…

Let $G$ be a finite simple graph and let $I(G)$ be its edge ideal. In this article, we deeply investigate the squarefree powers of $I(G)$ by means of Betti splittings. When $G$ is a forest, it is shown that the normalized depth function of…

In this article we show that for every prime number $p$, any irreducible homogeneous locally nilpotent derivations of rank $2$ and degree $p-2$ are triangularizable. Further, we describe the structure of irreducible non-triangularizable…

Let $k$ be a field of characteristic zero and $R$ a $k$-algebra. In this paper we study homogeneous $R$-lnds $D$ on $R[X,Y,Z]$ with respect to the standard weights $(1,1,1)$. We show that when $R$ is a PID, $rank(D)$ can be at most $2$ if…

The exact value of the separating Noether number of an arbitrary finite abelian group of rank two is determined. This is done by a detailed study of the monoid of zero-sum sequences over the group.

Bernstein's inequality is a central result in the theory of $D$-modules on smooth varieties. While Bernstein's inequality fails for rings of differential operators on general singularities, recent work of \`{A}lvarez Montaner, Hern\'andez,…

An atomic monoid $M$ is called length-factorial if for every non-invertible element $x \in M$, no two distinct factorizations of $x$ into irreducibles have the same length (i.e., number of irreducible factors, counting repetitions). The…

Let $f$ be a diagonal hypersurface in $A_p=\mathbb{F}_p[[x_1,\dots,x_n]]$. We study the behavior of the function $\phi_{f,p}({a}/{p^e})=p^{-ne}\dim_{\mathbb{F}_p}\big(A_p/(x_1^{p^e},\dots,x_n^{p^e},f^a)\big)$ which encodes information about…

In 1993, just about a century after the epoch of Classical Invariant Theory and almost 30 years after Mumford's seminal book on Geometric Invariant Theory, Bernd Sturmfels approached the subject from a new, algorithmic perspective in his…

Let k^[6] denote a polynomial ring in 6 variables over an algebraically closed field k of characteristic zero and consider the action of SL2(k) on k^[6] induced by the irreducible representation of SL2 of degree 5 (the binary quintic…

Let $(Q,\mathfrak{m},{\sf k})$ be a local ring that admits an exact pair of zero divisors $(f,g)$, $M$ a $Q$-module with $fM = 0$ and $U$ a free resolution of $M$ over $Q$. We construct a degree $-2$ chain map, which we call an Einsenbud…

We compute the minimum distance of the parameterized code of order 1 over an even cycle.

We classify all unicycle graphs whose edge-binomials form a $d$-sequence, particularly linear type binomial edge ideals. We also classify unicycle graphs whose parity edge-binomials form a $d$-sequence. We study the regularity of powers of…

In this article, we study the generalized Poincare problem from the opposite perspective, by establishing lower bounds on the degree of the vector field in terms of invariants of the variety.

Given a monomial ideal $I$, we study two functions that quantify ways to measure the difference between symbolic powers and usual powers of $I$. In many cases we determine the asymptotic growth rate of these two functions. We also perform…

Consider a complex affine variety $\tilde V$ and a real analytic Zariski-dense submanifold V of $\tilde V$. We compare modules over the ring $O (\tilde V)$ of regular functions on $\tilde V$ with modules over the ring $C^\infty (V)$ of…

Let $R_n=K[x_1,\dots,x_n]$ be the $n$-variable polynomial ring over a field $K$. Let $S_n$ denote the set of monomials in $R_n$. A monomial $u \in S_n$ is a \textit{Gotzmann monomial} if the Borel-stable monomial ideal $\langle u \rangle$…

Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d \geq 1$. Let $\mathcal{D}^2_{fg}(A)$ denote the derived category of $2$-periodic complexes with finitely generated cohomology modules. Let $\mathcal{K}^2(\proj A) $ denote the…

For $K$ a field, consider a finite subgroup $G$ of $\operatorname{GL}_n(K)$ with its natural action on the polynomial ring $R:=K[x_1,\dots,x_n]$. Let $\mathfrak{n}$ denote the homogeneous maximal ideal of the ring of invariants $R^G$. We…

We introduce a new class of commutative noetherian DG-rings which generalizes the class of regular local rings. These are defined to be local DG-rings $(A,\bar{\mathfrak{m}})$ such that the maximal ideal $\bar{\mathfrak{m}} \subseteq…