arXiv subjects

Dorin Popescu

Publications and source records attributed to Dorin Popescu.

53 records · Page 3Linked to original sources

arXiv2012

Depth of factors of square free monomial ideals

Let $I$ be an ideal of a polynomial algebra over a field, generated by $r$ square free monomials of degree $d$. If $r$ is bigger (or equal, if $I$ is not principal) than the number of square free monomials of $I$ of degree $d+1$, then $\depth_SI= d$. Let $J\subsetneq I$, $J\not =0$ be generated by square free monomials of degree $\geq d+1$. If $r$ is bigger than the number of square free monomials of $I\setminus J$ of degree $d+1$, or more generally the Stanley depth of $I/J$ is $d$, then $\depth_SI/J= d$. In particular, Stanley's Conjecture holds in theses cases.

math.AC↗

arXiv2012

Stanley conjecture on intersections of four monomial prime ideals

We show that the Stanley's Conjecture holds for an intersection of four monomial prime ideals of a polynomial algebra $S$ over a field and for an arbitrary intersection of monomial prime ideals $(P_i)_{i\in [s]}$ of $S$ such that $P_i\not\subset Σ_{1=j\not =i}^s P_j$ for all $i\in [s]$.

math.AC↗

arXiv2011

Depth and minimal number of generators of square free monomial ideals

Let $I$ be an ideal of a polynomial algebra $S$ over a field generated by square free monomials of degree $\geq d$. If $I$ contains more monomials of degree $d$ than $(n-d)/(n-d+1)$ of the total number of square free monomials of $S$ of degree $d+1$ then $\depth_SI\leq d$, in particular the Stanley's Conjecture holds in this case.

math.AC↗

arXiv2011

Graph and depth of a monomial squarefree ideal

Let $I$ be a monomial squarefree ideal of a polynomial ring $S$ over a field $K$ such that the sum of every three different of its minimal prime ideals is the maximal ideal of $S$, or more general a constant ideal. We associate to $I$ a graph on $[s]$, $s=|\Min S/I|$ on which we may {\em read} the depth of $I$. In particular, $\depth_S\ I$ does not depend of char $K$. Also we show that $I$ satisfies the Stanley's Conjecture.

math.AC↗

arXiv2010

Stanley depth and size of a monomial ideal

Lyubeznik introduced the concept of size of a monomial ideal and showed that the size of a monomial ideal increased by $1$ is a lower bound for its depth. We show that the size is also a lower bound for its Stanley depth. Applying Alexander duality we obtain upper bounds for the regularity and Stanley regularity of squarefree monomial ideals.

math.AC↗

arXiv2010

An inequality between depth and Stanley depth

We show that Stanley's Conjecture holds for square free monomial ideals in five variables, that is the Stanley depth of a square free monomial ideal in five variables is greater or equal with its depth.

math.AC↗

arXiv2009

Computing the Stanley depth

Let $Q$ and $Q'$ be two monomial primary ideals of a polynomial algebra $S$ over a field. We give an upper bound for the Stanley depth of $S/(Q\cap Q')$ which is reached if $Q$,$Q'$ are irreducible. Also we show that Stanley's Conjecture holds for $Q_1\cap Q_2$, $S/(Q_1\cap Q_2\cap Q_3)$, $(Q_i)_i$ being some irreducible monomial ideals of $S$.

math.AC↗

arXiv2008

Stanley Depth of Multigraded Modules

The Stanley's Conjecture on Cohen-Macaulay multigraded modules is studied especially in dimension 2. In codimension 2 similar results were obtained by Herzog, Soleyman-Jahan and Yassemi. As a consequence of our results Stanley's Conjecture holds in 5 variables.

math.AC↗

arXiv2007

Stanley Conjecture in small embedding dimension

We show that Stanley's conjecture holds for a polynomial ring over a field in four variables. In the case of polynomial ring in five variables, we prove that the monomial ideals with all associated primes of height two, are Stanley ideals.

math.AC↗

arXiv2007

Sequentially Cohen-Macaulay monomial ideals of embedding dimension four

Let $I$ be a monomial ideal of the polynomial ring $S=K[x_1,...,x_4]$ over a field $K$. Then $S/I$ is sequentially Cohen-Macaulay if and only if $S/I$ is pretty clean. In particular, if $S/I$ is sequentially Cohen-Macaulay then $I$ is a Stanley ideal.

math.AC↗

arXiv2005

The strong Lefschetz property and simple extensions

Stanley showed that monomial complete intersections have the strong Lefschetz property. Extending this result we show that a simple extension of an Artinian Gorenstein algebra with the strong Lefschetz property has again the strong Lefschetz property.

math.AC↗

arXiv2005

Criterions for Shellable Multicomplexes

After [J.Herzog, D.Popescu, Finite filtrations of modules and shellable multicomplexes, Preprint IMAR no 4/2005, Bucharest, 2005], the shellability of multicomplexes $Γ$ is given in terms of some special faces of $Γ$ called facets. Here we give a criterion for the shellability in terms of maximal facets. Multigraded pretty clean filtration is the algebraic counterpart of a shellable multicomplex. We give also a criterion for the existence of a multigraded pretty clean filtration.

math.AC↗

arXiv2005

Monomial Cycle Basis on Koszul Homology Modules

It gives a class of $p$-Borel principal ideals of a polynomial algebra over a field $K$ for which the graded Betti numbers do not depend on the characteristic of $K$ and the Koszul homology modules have monomial cyclic basis. Also it shows that all principal $p$-Borel ideals have binomial cycle basis on Koszul homology modules.

math.AC↗

arXiv2005

Finite Filtrations of Modules and Shellable Multicomplexes

We introduce pretty clean modules, extending the notion of clean modules by Dress, and show that pretty clean modules are sequentially Cohen-Macaulay. We also extend a theorem of Dress on shellable simplicial complexes to multicomplexes.

math.AC↗

arXiv2004

Rank two Cohen-Macaulay modules over singularities of type $x_1^3+x_2^3+x_3^3+x_4^3$

We describe, by matrix factorizations, all the rank two maximal Cohen-Macaulay modules over singularities of type $x_1^3+x_2^3+x_3^3+x_4^3$.

math.AC↗

arXiv2003

Rank one Maximal Cohen-Macaulay modules over singularities of type Y_1^3+Y_2^3+Y_3^3+Y_4^3

We describe, by matrix factorizations, the rank one graded maximal Cohen-Macaulay modules over the hypersurface Y_1^3+Y_2^3+Y_3^3+Y_4^3.

math.AC↗

arXiv2001

Groebner bases and regularity of Rees algebras

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.

math.AC↗