Linearity defects of modules over commutative rings

Linearity Defects of Face Rings

Let $S = K[x_1, ..., x_n ]$ be a polynomial ring over a field $K$, and $E = K < y_1, ..., y_n >$ an exterior algebra. The "linearity defect" $ld_E(N)$ of a finitely generated graded $E$-module $N$ measures how far $N$ departs from "componentwise linear". It is known that $ld_E(N) < \infty$ for all $N$. But the value can be arbitrary large, while the similar invariant $ld_S(M)$ for an $S$-module $M$ is alway at most $n$. We show that if $I_Δ$ (resp. $J_Δ$) is the squarefree monomial ideal of $S$ (resp. $E$) corresponding to a simplicial complex $Δ$ on ${1, >..., n}$, then $ld_E(E/J_Δ) = ld_S(S/…

On the asymptotic linearity of Castelnuovo-Mumford regularity

Let R be a standard graded ring over a commutative Noetherian ring with unity and I a graded ideal of R. Let M be a finitely generated graded R-module. We prove that there exist integers e and ρ_M(I) such that for all large n, reg(I^nM)= ρ_M(I)n+e.

Linearity Defect and Regularity over a Koszul Algebra

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 is the Koszul dual S^! of a polynomial ring S. Eisenbud et al. showed that $ld_E(M) < \infty$ for all $M \in mod E$. Improving their result, we show the following (and many other facts): (*) If A is a Koszul complete intersection, then $reg_{A^!} (M) < \infty$ and $ld_{A^!} (M) < \infty$ for all $M \in mod A^!$. (**) There is a uniform bound of $ld(J)$, wher…

On the Linearity of ω-primality in Numerical Monoids

In an atomic, cancellative, commutative monoid, the ω-value measures how far an element is from being prime. In numerical monoids, we show that this invariant exhibits eventual quasilinearity (i.e., periodic linearity). We apply this result to describe the asymptotic behavior of the ω-function for a general numerical monoid and give an explicit formula when the monoid has embedding dimension 2.

On the asymptotic behavior of the linearity defect

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 minimal free resolution of $M$ is linear. In the paper, it is proved that for any ideal $I$ in a regular local ring $R$ and for any finitely generated $R$-module $M$, each of the sequences $(\text{ld}_R (I^nM))_n$ and $(\text{ld}_R (M/I^nM))_n$ is eventually constant. The first statement follows from a more general result about the eventual constancy of the…

The Gorenstein defect category

We consider the homotopy category of complexes of projective modules over a Noetherian ring. Truncation at degree zero induces a fully faithful triangle functor from the totally acyclic complexes to the stable derived category. We show that if the ring is either Artin or commutative Noetherian local, then the functor is dense if and only if the ring is Gorenstein. Motivated by this, we define the Gorenstein defect category of the ring, a category which in some sense measures how far the ring is from being Gorenstein.

通过通用选票矩阵得到仿射线性的紧界

A tight bound for affine-linearity, via universal ballot matrices

研究将陶的仿射线性拼接结果推广到 $\mathbb{F}^n$，得到所需额外直线数的紧界，证明通用0-1矩阵族的性质及假设条件的必要性，赋予仿射线性新数值不变量。

从函数在仿射线上的限制恢复其仿射线性

Recovering affine-linearity of functions from their restrictions to affine lines

本文研究从函数在仿射线上的限制恢复其全局仿射线性的条件，提出基于一般线和弱乘法B_h-集的定量界，并推广von Staudt结果至非双射情形。

Homological criteria for minimal multiplicity

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 paper gives characterizations of rings of minimal multiplicity in terms of the Ext-algebra of the ring; in particular, we show that minimal multiplicity can be detected via an Ext-algebra which is Gorenstein or Koszul AS-regular. The second part of this paper characterizes rings of minimal multiplicity via a numerical homological invariant introduced by J. Herzo…

高余维情形下的同余模与Wiles-Lenstra-Diamond数值判据

Congruence modules and the Wiles-Lenstra-Diamond numerical criterion in higher codimensions

本文通过定义同余模并建立高余维数值判据，推广了Wiles-Lenstra-Diamond的经典结果，并将其应用于正亏数情形的积分R=T定理、虚二次域上的模性提升以及权一上同调的挠Jacquet-Langlands对应。