Related papers: Notes on the linearity defect and applications

The linearity defect is a measure for the non-linearity of minimal free resolutions of modules over noetherian local rings. A tantalizing open question due to Herzog and Iyengar asks whether a noetherian local ring $(R,\mathfrak{m})$ is…

This article concerns linear parts of minimal resolutions of finitely generated modules over commutative local, or graded rings. The focus is on the linearity defect of a module, which marks the point after which the linear part of its…

This work concerns the linearity defect of a module $M$ over a noetherian local ring $R$, introduced by Herzog and Iyengar in 2005, and denoted by $\text{ld}_R M$. Roughly speaking, $\text{ld}_R M$ is the homological degree beyond which the…

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…

Let A be a Koszul algebra, and $mod A$ the category of finitely generated graded left A-modules. The "linearity defect" ld_A(M) of $M \in mod A$ is an invariant defined by Herzog and Iyengar. An exterior algebra E is a Koszul algebra which…

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…

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 $M$.…

Lower bounds on Hilbert-Samuel multiplicity are known for several types of commutative noetherian local rings, and rings with multiplicities which achieve these lower bounds are said to have minimal multiplicity. The first part of this…

We prove that truncations of nonstandard graded polynomial rings are (nonstandard) Koszul modules in the sense of Herzog and Iyengar. This provides an analogue of the fact that such truncations have linear resolutions in the standard graded…

This article is concerned with graded modules M with linear resolutions over a standard graded algebra R. It is proved that if such an M has Hilbert series $H_M(s)$ of the form $ps^d+qs^{d+1}$, then the algebra R is Koszul; if, in addition,…

Given a Serre class $\mathcal{S}$ of modules, we compare the containment of the Koszul homology, Ext modules, Tor modules, local homology, and local cohomology in $\mathcal{S}$ up to a given bound $s \geq 0$. As some applications, we give a…

Given a commutative Noetherian local ring $R$, the linearity defect of a finitely generated $R$-module $M$, denoted $\ld_R(M)$, is an invariant that measures how far $M$ and its syzygies are from having a linear resolution. Motivated by a…

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 $\varphi \colon P\to…

We define a local homomorphism $(Q,k)\to (R,\ell)$ to be Koszul if its derived fiber $R \otimes^{\mathsf{L}}_Q k$ is formal, and if $\operatorname{Tor}^Q(R,k)$ is Koszul in the classical sense. This recovers the classical definition when…

Numerical invariants of a minimal free resolution of a module $M$ over a regular local ring $(R,\n)$ can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable $\n$-stable filtrations ${\mathbb…

Let $M$ be a finite module over a noetherian ring $R$ with a free resolution of length 1. We consider the generalized Koszul complexes $\mathcal{C}_{\bar\lambda}(t)$ associated with a map $\bar\lambda:M\to\mathcal{H}$ into a finite free…

The Koszul homology of modules of the polynomial ring $R$ is a central object in commutative algebra.It is strongly related with the minimal free resolution of these modules, and thus with regularity, Hilbert functions, etc. Here we…

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…

We define generalized Koszul modules and rings and develop a generalized Koszul theory for $\mathbb{N}$-graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings…

Let $k$ be an infinite field of characteristic $p > 0$ and let $R = k[Y_1,\ldots, Y_d]$ (or $R = k[[Y_1,\ldots, Y_d]]$). Let $F \colon \text{Mod}(R) \rightarrow \text{Mod}(R)$ be the Frobenius functor and let $\mathcal{M}$ be a $F_R$-finite…