Searcharxiv⌕ Search

arXiv subjects

Liana M. Şega

Publications and source records attributed to Liana M. Şega.

13 recordsLinked to original sources
arXiv2026

Realizing resolutions of powers of extremal ideals

Extremal ideals are a class of square-free monomial ideals which dominate and determine many algebraic invariants of powers of all square-free monomial ideals. For example, the $r^{th}$ power ${\mathcal{E}_q}^r$ of the extremal ideal on $q$ generators has the maximum Betti numbers among the $r^{th}$ power of any square-free monomial ideal with $q$ generators. In this paper we study the combinatorial and geometric structure of the (minimal) free resolutions of powers of square-free monomial ideals via the resolutions of powers of extremal ideals. Although the end results are algebraic, this problem has a natural interpretation in terms of polytopes and discrete geometry. Our guiding conjecture is that all powers ${\mathcal{E}_q}^r$ of extremal ideals have resolutions supported on their Scarf simplicial complexes, and thus their resolutions are as small as possible. This conjecture is known to hold for $r \leq 2$ or $q \leq 4$. In this paper we prove the conjecture holds for $r=3$ and any $q\geq 1$ by giving a complete description of the Scarf complex of ${\mathcal{E}_q}^3$. This effectively gives us a sharp bound on the betti numbers and projective dimension of the third power of any square-free momomial ideal. For large $i$ and $q$, our bounds on the $i^{th}$ betti numbers are an exponential improvement over previously known bounds. We also describe a large number of faces of the Scarf complex of ${\mathcal{E}_q}^r$ for any $r,q \geq 1$.

math.AC↗
arXiv2026

A study of maximal shifts in the minimal graded free resolution of the residue field

We study two invariants, rate and slant, that describe the extremal behavior of the maximal shifts in the minimal free resolution of the residue field over a standard graded algebra. We provide bounds and computations of these invariants for a variety of rings, including compressed level algebras, rings defined by general forms and rings defined by monomials. We also prove an asymptotic additivity property of the maximal shifts for certain classes of rings, including Golod rings, which allows to interpret slant as a limit.

math.AC↗
arXiv2025

Divisibility Relations and $\mathcal{D}$-Extremal ideals

A divisibility relation between the generators of a square-free monomial ideal formally encodes the situation when one generator divides the least common multiple of some other generators. The divisibility relations contribute to the deletion of some parts of the Taylor resolution of the ideal, and therefore lead to finding a resolution closer to the minimal one. Motivated by this observation, for a given set $\mathcal{D}$ of divisibility relations, we study all square-free monomials satisfying the relations in $\mathcal{D}$. We define a class of square-free monomial ideals called $\mathcal{D}$-extremal ideals $\mathcal{E}_\mathcal{D}$ , and show it is optimal in the sense that it is an ideal satisfying exactly those divisibility relations coming from $\mathcal{D}$, and no others. We then show that $\mathcal{E}_\mathcal{D}$ is extremal in the sense that the resolution and betti numbers of the powers of any square-free monomial ideal satisfying the relations in $\mathcal{D}$ are bounded by those of the same powers of $\mathcal{E}_\mathcal{D}$.

math.AC↗
arXiv2024

The mapping cone of an Eisenbud operator and applications to exact zero divisors

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 operator, on the complex $U \otimes_Q Q/(f,g)$ and use the mapping cone of the operator to study two exact sequences that relate homology over $Q$ to homology over $Q/(f)$. Several applications are given.

math.AC↗
arXiv2023

DG module structures and minimal free resolutions modulo an exact zero-divisor

Let $Q$ be a local ring with maximal ideal $\mathfrak{n}$ and let $f,g\in \mathfrak{n}\smallsetminus\mathfrak{n}^2$ with $fg=0$. When $M$ is a finite $Q$-module with $fM=0$, we show that a minimal free resolution of $M$ over $Q$ has a differential graded module structure over the differential graded algebra $Q\langle y,t\mid \partial(y)=f, \partial(t)=gy\rangle$. When $(f,g)$ is a pair of exact zero divisors, we use this structure to describe a minimal free resolution of $M$ over $Q/(f)$.

math.AC↗
arXiv2022

Asymptotic behavior of Ext for pairs of modules of large complexity over graded complete intersections

Let $M$ and $N$ be finitely generated graded modules over a graded complete intersection $R$ such that $\operatorname{Ext}_R^i(M,N)$ has finite length for all $i\gg 0$. We show that the even and odd Hilbert polynomials, which give the lengths of $\operatorname{Ext}^i_R(M,N)$ for all large even $i$ and all large odd $i$, have the same degree and leading coefficient whenever the highest degree of these polynomials is at least the dimension of $M$ or $N$. Refinements of this result are given when $R$ is regular in small codimensions.

math.AC↗
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↗
arXiv2017

Detecting Koszulness and related homological properties from the algebra structure of Koszul homology

Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $\operatorname{H}^R$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the multiplicative structure of $\operatorname{H}^R$ and the property that $R$ is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincaré series. As an application, we show that the Poincaré series of all finitely generated modules over a stretched Cohen-Macaulay local ring are rational, sharing a common denominator.

math.AC↗
arXiv2016

Conditions for the Yoneda algebra of a local ring to be generated in low degrees

The powers ${\mathfrak m}^n$ of the maximal ideal $\mathfrak m$ of a local Noetherian ring $R$ are known to satisfy certain homological properties for large values of $n$. For example, the homomorphism $R\to R/{\mathfrak m}^n$ is Golod for $n\gg 0$. We study when such properties hold for small values of $n$, and we make connections with the structure of the Yoneda Ext algebra, and more precisely with the property that the Yoneda algebra of $R$ is generated in degrees $1$ and $2$. A complete treatment of these properties is pursued in the case of compressed Gorenstein local rings.

math.AC↗
arXiv2015

Minimal quasi-complete intersection ideals

A quasi-complete intersection (q.c.i.) ideal of a local ring is an ideal with "free exterior Koszul homology"; the definition can also be understood in terms of vanishing of André-Quillen homology functors. Principal q.c.i. ideals are well understood, but few constructions are known to produce q.c.i. ideals of grade zero that are not principal. This paper examines the structure of q.c.i. ideals. We exhibit conditions on a ring $R$ which guarantee that every q.c.i. ideal of $R$ is principal. On the other hand, we give an example of a minimal q.c.i. deal $I$ which does not contain any principal q.c.i. ideals and is not embedded, in the sense that no faithfully flat extension of $I$ can be written as a quotient of complete intersection ideals. We also describe a generic situation in which the maximal ideal of $R$ is an embedded q.c.i. ideal that does not contain any principal q.c.i. ideals.

math.AC↗
arXiv2013

Quasi-complete intersection homomorphisms

Extending a notion defined for surjective maps by Blanco, Majadas, and Rodicio, we introduce and study a class of homomorphisms of commutative noetherian rings, which strictly contains the class of locally complete intersection homomorphisms, while sharing many of its remarkable properties.

math.AC↗
arXiv2009

Free resolutions over short Gorenstein local rings

Let R be a local ring with maximal ideal m admitting a non-zero element a\in\fm for which the ideal (0:a) is isomorphic to R/aR. We study minimal free resolutions of finitely generated R-modules M, with particular attention to the case when m^4=0. Let e denote the minimal number of generators of m. If R is Gorenstein with m^4=0 and e\ge 3, we show that \Poi MRt is rational with denominator \HH R{-t} =1-et+et^2-t^3, for each finitely generated R-module M. In particular, this conclusion applies to generic Gorenstein algebras of socle degree 3.

math.AC↗