Related papers: Homological criteria for minimal multiplicity

Let R be a local Cohen-Macaulay ring with canonical module \omega_R. We investigate the following question of Huneke: If the sequence of Betti numbers \{\beta_i^R(\omega_R)\} has polynomial growth, must R be Gorenstein? This question is…

A recent continuous family of multiplicity functions on local rings was introduced by Taylor interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicities. The obvious goal is to use this as a tool for deforming results from one to…

The notion of multiplicity of a module first arose as consequence of Hilbert's work on commutative algebra, relating the dimension of rings with the degree of certain polynomials. For noncommutative rings, the notion of multiplicity first…

In this paper we give new lower bounds on the Hilbert-Kunz multiplicity of unmixed non-regular local rings, bounding them uniformly away from one. Our results improve previous work of Aberbach and Enescu.

The linearity defect, introduced by Herzog and Iyengar, is a numerical measure for the complexity of minimal free resolutions. Employing a characterization of the linearity defect due to \c{S}ega, we study the behavior of linearity defect…

Ring epimorphisms often induce silting modules and cosilting modules, termed minimal silting or minimal cosilting. The aim of this paper is twofold. Firstly, we determine the minimal tilting and minimal cotilting modules over a tame…

In this paper we study homological dimensions of finitely generated modules over commutative Noetherian local rings, called reducing homological dimensions. We obtain new characterizations of Gorenstein and complete intersection local rings…

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…

Let $(R,\m)$ be a formally unmixed local ring of positive prime characteristic and dimension $d$. We examine the implications of having small Hilbert-Kunz multiplicity (i.e., close to 1). In particular, we show that if $R$ is not regular,…

Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild…

We present results on the Watanabe-Yoshida conjecture for the Hilbert-Kunz multiplicity of a local ring of positive characteristic. By improving on a "volume estimate" giving a lower bound for Hilbert-Kunz multiplicity, we obtain the…

Let $(R, \mathfrak{m})$ be a Noetherian local ring. This paper concerns several extremal invariants arising from the study of the relation between colength and (Hilbert--Samuel or Hilbert--Kunz) multiplicity of an $\mathfrak{m}$-primary…

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…

For a simplicial complex $\Delta$ we study the effect of barycentric subdivision on ring theoretic invariants of its Stanley-Reisner ring. In particular, for Stanley-Reisner rings of barycentric subdivisions we verify a conjecture by Huneke…

In this paper, we ask the following question: what is the minimal value of the difference $e_{HK}(I) - e_{HK}(I')$ for ideals $I' \supseteq I$ with $l_A(I'/I) =1$? In order to answer to this question, we define the notion of minimal…

We establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field, for codimension two algebras and for Gorenstein algebras of codimension three. In fact, we…

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…

Let S be a finitely generated standard multi-graded algebra over a Noetherian local ring A. This paper first expresses mixed multiplicities of S in term of Hilbert-Samuel multiplicity that explained the mixed multiplicities S as the…

The paper investigates the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities in families of ideals. It is shown that Hilbert-Samuel multiplicity is upper semicontinuous almost generally and that Hilbert-Kunz multiplicity is upper…

The notion of epsilon multiplicity was originally defined by Ulrich and Validashti as a limsup and they used it to detect integral dependence of modules. It is important to know if it can be realized as a limit. In this article we show that…