We consider classes of ideals which generalize the mixed product ideals introduced by Restuccia and Villarreal, and also generalize the expansion construction by Bayati and the first author cite{BH}. We compute the minimal graded free resolution of g
eneralized mixed product ideals and show that the regularity of a generalized mixed product ideal coincides with regularity of the monomial ideal by which it is induced.
We show that the graded maximal ideal of a graded $K$-algebra $R$ has linear quotients for a suitable choice and order of its generators if the defining ideal of $R$ has a quadratic Grobner basis with respect to the reverse lexicographic order, and s
how that this linear quotient property for algebras defined by binomial edge ideals characterizes closed graphs. Furthermore, for algebras defined by binomial edge ideals attached to a closed graph and for join-meet rings attached to a finite distributive lattice we present explicit Koszul filtrations.
The purpose of this note is to introduce a multiplication on the set of homogeneous polynomials of fixed degree d, in a way to provide a duality theory between monomial ideals of K[x_1,ldots,x_d] generated in degrees leq n and block stable ideals (a
class of ideals containing the Borel fixed ones) of K[x_1,ldots,x_n] generated in degree d. As a byproduct we give a new proof of the characterization of Betti tables of ideals with linear resolution given by Murai.
Let $S=K[x_1,ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $Isubset S$ a squarefree monomial ideal. In the present paper we are interested in the monomials $u in S$ belonging to the socle $Soc(S/I^{k})$ of $S/I^{k}$, i.e.,
$u otin I^{k}$ and $ux_{i} in I^{k}$ for $1 leq i leq n$. We prove that if a monomial $x_1^{a_1}cdots x_n^{a_n}$ belongs to $Soc(S/I^{k})$, then $a_ileq k-1$ for all $1 leq i leq n$. We then discuss squarefree monomial ideals $I subset S$ for which $x_{[n]}^{k-1} in Soc(S/I^{k})$, where $x_{[n]} = x_{1}x_{2}cdots x_{n}$. Furthermore, we give a combinatorial characterization of finite graphs $G$ on $[n] = {1, ldots, n}$ for which $depth S/(I_{G})^{2}=0$, where $I_{G}$ is the edge ideal of $G$.
We present some partial results regarding subadditivity of maximal shifts in finite graded free resolutions.
