arXiv subjects

Rasoul Ahangari Maleki

Publications and source records attributed to Rasoul Ahangari Maleki.

8 recordsLinked to original sources

arXiv2020

The absolutely Koszul and Backelin-Roos properties for spaces of quadrics of small codimension

Let $\kk$ be a field, $R$ a standard graded quadratic $\kk$-algebra with $\dim_{\kk}R_2\le 3$, and let $\ov\kk$ denote an algebraic closure of $\kk$. We construct a graded surjective Golod homomorphism $φ\colon P\to R\otimes_{\kk}\ov{\kk}$ such that $P$ is a complete intersection of codimension at most $3$. Furthermore, we show that $R$ is absolutely Koszul (that is, every finitely generated $R$-module has finite linearity defect) if and only if $R$ is Koszul if and only if $R$ is not a trivial fiber extension of a standard graded $\kk$-algebra with Hilbert series $(1+2t-2t^3)(1-t)^{-1}$. In particular, we recover earlier results on the Koszul property of Backelin, Conca and D'Alì.

math.AC↗

arXiv2018

The Golod property for powers of ideals and Koszul ideals

Let $S$ be a regular local ring or a polynomial ring over a field and $I$ be an ideal of $S$. Motivated by a recent result of Herzog and Huneke, we study the natural question of whether $I^m$ is a Golod ideal for all $m\geq 2$. We observe that the Golod property of an ideal can be detected through the vanishing of certain maps induced in homology. This observation leads us to generalize some known results from the graded case to local rings and obtain new classes of Golod ideals.

math.AC↗

arXiv2017

Koszul cycles and Golod rings

Let $S$ be the power series ring or the polynomial ring over a field $K$ in the variables $x_1,\ldots,x_n$, and let $R=S/I$, where $I$ is proper ideal which we assume to be graded if $S$ is the polynomial ring. We give an explicit description of the cycles of the Koszul complex whose homology classes generate the Koszul homology of $R=S/I$ with respect to $x_1,\ldots,x_n$. The description is given in terms of the data of the free $S$-resolution of $R$. The result is used to determine classes of Golod ideals, among them proper ordinary powers and proper symbolic powers of monomial ideals. Our theory is also applied to stretched local rings.

math.AC↗

arXiv2017

On the rate of graded modules

Let $K$ be a field, $R$ a standard graded $K$-algebra and $M$ be a finitely generated graded $R$-module. The rate of $M$, $rate_R(M)$, is a measure of the growth of the shifts in the minimal graded free resolution of $M$. In this paper, we find upper bounds for this invariant. More precisely, let $(A,\mathfrak{n})$ be a regular local ring and $I\subseteq \mathfrak{n} ^t$ be an ideal of $A$, where $t\geq 2$. We prove that if $(B=A/I, \mathfrak{m} =\mathfrak{n} /I)$ is a Cohen-Macaulay local ring with multiplicity $e(B)= \binom{h+t-1}{h}$, where $h=embdim(B)-dim B$, then $rat(gr_{\mathfrak{m}}(B))=t-1$ and for every $B$-module $N$, which annihilated by a minimal reduction of $\mathfrak{m}$, $rate_{gr_{\mathfrak{m}}(B)}(gr_{\mathfrak{m}}(N))\leq t-1$.

math.AC↗

arXiv2016

Linearity defect of the residue field of short local rings

Let $(R,\m,k)$ be a Noetherian local ring with maximal ideal $\m$ and residue field $k$. The linearity defect of a finitely generated $R$-module $M$, which is denoted $\ld_R(M)$, is a numerical measure of how far $M$ is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether $\ld_R(k)<\infty$ implies $\ld_R(k)=0$, in the case when $\m^4=0$.

math.AC↗

arXiv2014

Rate and syzigies of modules over Veronese subrings

Let $K$ be a field, $R$ be a standard graded $K$-algebra and $M$ be a finitely generated graded $R$-module. The rate of $M$, $\rate_R(M)$, is a measure of the growth of the shifts in the minimal graded free resolution of $M$. In this paper, we study the rate of Veronese modules of $M$. More precisely, it is shown that $\rate_{R^{(c)}}(M)\leq \lceil \max\{\rate_{R}(M),\rat(R)\}/c\rceil+\max\{0,\lceil t^{R}_{0}(M)/c\rceil\},$ for all $c\geq 1$. This extends a result of Herzog et al. As a consequence of this, if $M$ is generated in degree zero, then $\reg_{R^{(c)}}(M)=0$, for all $c\geq \max\{\rate_{R}(M), \rat(R)\}$. Also, for powers of the homogeneous maximal ideal $\m$ of $R$, it is shown that $\rate_{R^{(c)}}(\m^{s}(s))\leq \lceil \rat(R)/c\rceil$, for all $c\geq 1$. In particular case, we give a simple proof to a theorem of Backelin.

math.AC↗

arXiv2013

Regularity and linearity defect of modules over local rings

Given a finitely generated module $M$ over a commutative local ring (or a standard graded $k$-algebra) $(R,\m,k) $ we detect its complexity in terms of numerical invariants coming from suitable $\m$-stable filtrations $\mathbb{M}$ on $M$. We study the Castelnuovo-Mumford regularity of $gr_{\mathbb{M}}(M) $ and the linearity defect of $M,$ denoted $\ld_R(M),$ through a deep investigation based on the theory of standard bases. If $M$ is a graded $R$-module, then $\reg_R(gr_{\mathbb{M}}(M)) <\infty$ implies $\reg_R(M)<\infty$ and the converse holds provided $M$ is of homogenous type. An analogous result can be proved in the local case in terms of the linearity defect. Motivated by a positive answer in the graded case, we present for local rings a partial answer to a question raised by Herzog and Iyengar of whether $\ld_R(k)<\infty$ implies $R$ is Koszul.

math.AC↗

arXiv2013

On the regularity and Koszulness of modules over local rings

Koszul modules over Noetherian local rings $R$ were introduced by Herzog and Iyengar and they possess good homological properties, for instance their Poincare' series is rational. It is an interesting problem to characterize classes of Koszul modules. Following the idea traced by Avramov, Iyengar and Sega, we take advantage of the existence of special filtration on $R$ for proving that large classes of $R$-modules over Koszul rings are Koszul modules. By using this tool we reprove and extend some results obtained by Fitzgerald.

math.AC↗