arXiv subjects

Rachel N. Diethorn

Publications and source records attributed to Rachel N. Diethorn.

5 recordsLinked to original sources

arXiv2023

Resolutions of differential operators of low order for an isolated hypersurface singularity

In this paper we develop a new approach for studying differential operators of an isolated singularity graded hypersurface ring $R$ defining a surface in affine three-space over a field of characteristic zero. With this method, we construct an explicit minimal generating set for the modules of differential operators of order two and three, as well as their minimal free resolutions; this expands results of Bernstein, Gel'fand, and Gel'fand and of Vigué. Our construction relies, in part, on a description of these modules that we derive in the singularity category of $R$. Namely, we build explicit matrix factorizations starting from that of the residue field.

math.AC↗

arXiv2022

A self-dual complete resolution

We construct a self-dual complete resolution of a module defined by a pair of embedded complete intersection ideals in a local ring. Our construction is based on a gluing construction of Herzog and Martsinkovsky and exploits the structure of Koszul homology in the embedded complete intersection case. As a consequence of our construction, we produce an isomorphism between certain stable homology and cohomology modules.

math.AC↗

arXiv2021

A survey on the Koszul homology algebra

The Koszul homology algebra of a commutative local (or graded) ring $R$ tends to reflect important information about the ring $R$ and its properties. In fact, certain classes of rings are characterized by the algebra structure on their Koszul homologies. In this paper we survey some classical results on the Koszul homology algebras of such rings and highlight some applications. We report on recent progress on the Koszul homology algebras of Koszul algebras and examine some open questions on the topic.

math.AC↗

arXiv2019

Koszul homology of quotients by edge ideals

We show that the Koszul homology algebra of a quotient by the edge ideal of a forest is generated by the lowest linear strand. This provides a large class of Koszul algebras whose Koszul homology algebras satisfy this property. We obtain this result by constructing the minimal graded free resolution of a quotient by such an edge ideal via the so called iterated mapping cone construction and using the explicit bases of Koszul homology given by Herzog and Maleki. Using these methods we also recover a result of Roth and Van Tuyl on the graded Betti numbers of quotients of edge ideals of trees.

math.AC↗

arXiv2019

Generators of Koszul homology with coefficients in a $\underline{g}$-weak complete intersection module

We discuss a class of modules, which we call $\underline{g}$-weak complete intersection modules, inspired by the weak complete intersection ideals studied by Rahmati, Striuli, and Yang and we present explicit formulas for the generators of Koszul homology with coefficients in a $\underline{g}$-weak complete intersection module. This generalizes work of Herzog and of Corso, Goto, Huneke, Polini, and Ulrich. We use these explicit formulas to study connections between $\underline{g}$-weak complete intersection ideals and weak complete intersection ideals.

math.AC↗