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Oana Veliche

Publications and source records attributed to Oana Veliche.

At least 19 recordsLinked to original sources
arXiv2026

Homological properties of rings defined by $n+1$ general quadrics in $n$ variables

We study the almost complete intersection ring $R$ defined by $n+1$ general quadrics in a polynomial ring in $n$ variables over a field $\sf{k}$ and a corresponding linked Gorenstein ring $A$. The overarching theme is that, while not Koszul (except for some small values of $n$), these rings have homological properties that extend those of Koszul rings. We establish that finitely generated modules over these rings have rational Poincaré series and we give concrete formulas for the Poincaré series of $\sf{k}$ over both $A$ and $R$. We also show that $A$ has minimal rate and its Yoneda algebra $\text{Ext}_A(\sf{k},\sf{k})$ is generated by its elements of degrees $1$ and $2$. While the graded Betti numbers of $R$ and $A$ over the polynomial ring are not known when $n$ is odd, our approach provides bounds and yields values for two of these Betti numbers, showing in particular that $R$ is level.

math.AC↗
arXiv2026

A study of a quadratic almost complete intersection ideal and its linked Gorenstein ideal

We examine the ideal $I=(x_1^2, \dots, x_n^2, (x_1+\dots+x_n)^2)$ in the polynomial ring $Q=k[x_1, \dots, x_n]$, where $k$ is a field of characteristic zero or greater than $n$. We also study the Gorenstein ideal $G$ linked to $I$ via the complete intersection ideal $(x_1^2, \dots, x_n^2)$. We compute the Betti numbers of $I$ and $G$ over $Q$ when $n$ is odd and extend known computations when $n$ is even. A consequence is that the socle of $Q/I$ is generated in a single degree (thus $Q/I$ is level) and its dimension is a Catalan number. We also describe the generators and the initial ideal with respect to reverse lexicographic order for the Gorenstein ideal $G$.

math.AC↗
arXiv2026

On a minimal free resolution of the residue field over a local ring of codepth 3 of class $T$

Let $R$ be any noetherian local ring with residue field $k$, and $A$ the homology of the Koszul complex on a minimal set of generators of the maximal ideal of $R$. In this paper, we show that a minimal free resolution of $k$ over $R$ can be obtained from a graded minimal free resolution of $k$ over $A$. More precisely, this is done by the iterated mapping cone construction, introduced by the authors in a previous work, using specific choices of ingredients. As applications, using this general perspective, we exhibit a minimal free resolution of $k$ over a complete intersection ring of any codepth, and explicitly construct a minimal free resolution of $k$ over a noetherian local ring of codepth 3 of class $T$ in terms of Koszul blocks.

math.AC↗
arXiv2024

Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings

Let $(R,\mathfrak m, \mathsf k)$ be a complete intersection local ring, $K$ be the Koszul complex on a minimal set of generators of $\mathfrak m$, and $A=H(K)$ be its homology algebra. We establish exact sequences involving direct sums of the components of $A$ and express the images of the maps of these sequences as homologies of iterated mapping cones built on $K$. As an application of this iterated mapping cone construction, we recover a minimal free resolution of the residue field $\mathsf k$ over $R$, independent from the well-known resolution constructed by Tate by adjoining variables and killing cycles. Through our construction, the differential maps can be expressed explicitly as blocks of matrices, arranged in some combinatorial patterns.

math.AC↗
arXiv2023

Generic local rings on a spectrum between Golod and Gorenstein

Artinian quotients R of the local ring Q = k[[x,y,z]] are classified by multiplicative structures on A = Tor_Q^*(R,k); in particular, R is Gorenstein if and only if A is a Poincare duality algebra while R is Golod if and only if all products in A_{>0} are trivial. There is empirical evidence that generic quotient rings with small socle ranks fall on a spectrum between Golod and Gorenstein in a very precise sense: The algebra A breaks up as a direct sum of a Poincare duality algebra P and a graded vector space V, on which P_{>0} acts trivially. That is, A is a trivial extension, A = P \ltimes V, and the extremes A = (k \oplus Σk) \ltimes V and A = P correspond to R being Golod and Gorenstein, respectively. We prove that this observed behavior is, indeed, the generic behavior for graded quotients R of socle rank 2, and we show that the rank of P is controlled by the difference between the order and the degree of the socle polynomial of R.

math.AC↗
arXiv2022

Artinian Gorenstein algebras of embedding dimension four and socle degree three

We prove that in the polynomial ring $Q=\mathsf{k}[x,y,z,w]$, with $\mathsf{k}$ an algebraically closed field of characteristic zero, all Gorenstein homogeneous ideals $I$ such that $(x,y,z,w)^4\subseteq I \subseteq (x,y,z,w)^2$ can be obtained by \emph{doubling} from a grade three perfect ideal $J\subset I$ such that $Q/J$ is a locally Gorenstein ring. Moreover, a graded minimal free resolution of the $Q$-module $Q/I$ can be completely described in terms of a graded minimal free resolution of the $Q$-module $Q/J$ and a homogeneous embedding of a shift of the canonical module $ω_{Q/J}$ into $Q/J$.

math.AC↗
arXiv2021

Three takes on almost complete intersection ideals of grade 3

We are interested in the structure of almost complete intersection ideals of grade 3. We give three constructions of these ideals and their free resolutions: one from the commutative algebra point of view, an equivariant construction giving a nice canonical form, andfinally an interpretation in terms of open sets in certain Schubert varieties.

math.AC↗
arXiv2021

A truncated minimal free resolution of the residue field

In a paper in 1962, Golod proved that the Betti sequence of the residue field of a local ring attains an upper bound given by Serre if and only if the homology algebra of the Koszul complex of the ring has trivial multiplications and trivial Massey operations. This is the origin of the notion of Golod ring. Using the Koszul complex components he also constructed a minimal free resolution of the residue field. In this article, we extend this construction up to degree five for any local ring. We describe how the multiplicative structure and the triple Massey products of the homology of the Koszul algebra are involved in this construction. As a consequence, we provide explicit formulas for the first six terms of a sequence that measures how far the ring is from being Golod.

math.AC↗
arXiv2019

Linkage classes of grade 3 perfect ideals

While every grade 2 perfect ideal in a regular local ring is linked to a complete intersection ideal, it is known not to be the case for ideals of grade 3. We soften the blow by proving that every grade 3 perfect ideal in a regular local ring is linked to a complete intersection or a Golod ideal. Our proof is indebted to a homological classification of Cohen-Macaulay local rings of codimension 3. That debt is swiftly repaid, as we use linkage to reveal some of the finer structures of this classification.

math.AC↗
arXiv2019

Free resolutions of Dynkin format and the licci property of grade 3 perfect ideals

Recent work on generic free resolutions of length 3 attaches to every resolution a graph and suggests that resolutions whose associated graph is a Dynkin diagram are distinguished. We conjecture that in a regular local ring, every grade 3 perfect ideal whose minimal free resolution is distinguished in this way is in the linkage class of a complete intersection.

math.AC↗
arXiv2018

The Golod property of powers of the maximal ideal of a local ring

We identify minimal cases in which a power $m^i\not=0$ of the maximal ideal of a local ring $R$ is not Golod, i.e.\ the quotient ring $R/m^i$ is not Golod. Complementary to a 2014 result by Rossi and Şega, we prove that for a generic artinian Gorenstein local ring with $m^4=0\not= m^3$, the quotient $R/m^3$ is not Golod. This is provided that $m$ is minimally generated by at least $3$ elements. Indeed, we show that if $m$ is $2$-generated, then every power $m^i\not= 0$ is Golod.

math.AC↗
arXiv2017

Trimming a Gorenstein ideal

Let Q be a regular local ring of dimension 3. We show how to trim a Gorenstein ideal in Q to obtain an ideal that defines a quotient ring that is close to Gorenstein in the sense that its Koszul homology algebra is a Poincare duality algebra P padded with a non-zero graded vector space on which P_{\ge 1} acts trivially. We explicitly construct an infinite family of such rings.

math.AC↗
arXiv2014

Local rings of embedding codepth 3: a classification algorithm

Let I be an ideal of a regular local ring Q with residue field k. The length of the minimal free resolution of R=Q/I is called the codepth of R. If it is at most 3, then the resolution carries a structure of a differential graded algebra, and the induced algebra structure on $Tor_Q(R,k) provides for a classification of such local rings. We describe the Macaulay2 package CodepthThree that implements an algorithm for classifying a local ring as above by computation of a few cohomological invariants.

math.AC↗
arXiv2013

The index of a numerical semigroup ring

Let $R=k[|t^a,t^b,t^c|]$ be a complete intersection numerical semigroup ring over an infinite field $k$, where $a,b,c\in\BN$. The generalized Loewy length, which is Auslander's index in this case, is computed in terms of the minimal generators of the semigroup: $a,b$ and $c$. Examples provided show that the left hand side of Ding's inequality $\mult(R)-\inde(R)-\codim(R)+1\geq 0$ can be made arbitrarily large for rings $R$ with $\edim(R)=3$ . The index of a complete intersection numerical semigroup ring with embedding dimension greater than three is also computed.

math.AC↗
arXiv2012

Local rings of embedding codepth 3. Examples

A complete local ring of embedding codepth 3 has a minimal free resolution of length 3 over a regular local ring. Such resolutions carry a differential graded algebra structure, based on which one can classify local rings of embedding codepth 3. We give examples of algebra structures that have been conjectured not to occur.

math.AC↗
arXiv2009

Comparing complexities of pairs of modules

Let $R$ be a local ring and $M,N$ be finitely generated $R$-modules. The complexity of $(M,N)$, denoted by $\cxx RMN$, measures the polynomial growth rate of the number of generators of the modules $\Ext nRMN$. In this paper we study several basic equalities and inequalities involving complexities of different pairs of modules.

math.AC↗
arXiv2009

Growth in the minimal injective resolution of a local ring

Let R be a commutative noetherian local ring with residue field k and assume that it is not Gorenstein. In the minimal injective resolution of R, the injective envelope E of the residue field appears as a summand in every degree starting from the depth of R. The number of copies of E in degree i equals the k-vector space dimension of the cohomology module Ext^i(k,R). These dimensions, known as Bass numbers, form an infinite sequence of invariants of R about which little is known. We prove that it is non-decreasing and grows exponentially if R is Golod, a non-trivial fiber product, or Teter, or if it has radical cube zero.

math.AC↗