arXiv subjects

Zachary Nason

Publications and source records attributed to Zachary Nason.

6 recordsLinked to original sources

arXiv2026

Intersection theorems over DG-rings revisited

In this work we generalize two recently proved intersection theorems for DG-rings. The Derived Improved New Intersection Theorem concerns the length of semi-free DG-modules over DG-rings and it was recently proved by the second author. We show that it holds under weaker hypotheses. Foxby's Intersection Theorem was generalized to DG-rings by Yang and we improve the inequality that they provided. As an application we prove a DG version of the classic result that finite length modules of finite projective dimension only exist over Cohen-Macaulay rings, generalizing another result of Yang.

math.AC↗

arXiv2026

Quasi-Gorenstein morphisms of commutative local dg-algebras

We introduce quasi-Gorenstein morphisms of commutative local dg-algebras and use a Gorenstein version of the virtually small property to characterize them, a result which is new even for homomorphisms of local rings. In a different direction, we characterize exact sequences in a noetherian local ring, in the sense of Avramov, Henriques, and \c{S}ega, in terms of quasi-Gorenstein morphisms involving Koszul complexes.

math.AC↗

arXiv2026

Finiteness of complete intersection dimensions of RHom complexes and Ext modules

In this paper, we explore the implications of the finiteness of complete intersection dimensions for RHom complexes and Ext modules. We prove various stability results and criteria for detecting finite complete intersection homological dimension of complexes and modules. In addition, we introduce and explore the concept of CI-perfect modules. We also study the vanishing of Ext when certain Hom module have finite complete intersection homological dimension. In this direction, we improve a result by Ghosh and Samanta, prove the Auslander-Reiten conjecture for finitely generated modules $M$ over a Noetherian local ring $R$ such that $\operatorname{Hom}_R(M,R)$ or $\operatorname{Hom}_R(M,M)$ has finite complete intersection injective dimension, and provide Gorenstein criteria.

math.AC↗

arXiv2025

Level Inequalities for Complexes

We prove that for all noetherian rings, the level of any homologically bounded complex $M$ with respect to the collection of projective or injective modules is bounded above by the projective dimension of $\bigoplus H(M)$ plus one or the injective dimemsion of $\bigoplus H(M)$ plus one, respectively. In addition, we also prove that if $\mathsf{C}$ is the collection of flat, Gorenstein projective, Gorenstein injective, or Gorenstein flat modules, then the level of any homologically bounded complex $M$ is bounded above by the maximum of 2 or the $\mathsf{C}$-dimension of $\bigoplus H(M)$ plus 1. These results give universal bounds for the projective, injective, and flat levels over regular local rings, and give universal bounds for the Gorenstein projective, Gorenstein injective, and Gorenstein flat levels over Gorenstein local rings. As an application of the above results, we prove a version of the Bass Formula for complexes with respect to injective level and Gorenstein injective level. We also show that the bounds achieved for each homological and Gorenstein homological level considered is optimal.

math.AC↗

arXiv2025

G-levels of perfect complexes

We prove that a commutative noetherian ring $R$ is Gorenstein of dimension at most $d$ if $d+1$ is an upper bound on the G-levels of perfect $R$-complexes. For $R$ local, we prove a formula for levels, with respect to injective or Gorenstein injective $R$-modules, of $R$-complexes with finitely generated homology; it mimics Bass' classic formula for injective dimension of finitely generated $R$-modules.

math.AC↗

arXiv2025

Maximal Cohen-Macaulay DG-complexes

Let $R$ be a commutative noetherian local differential graded (DG) ring. In this paper we propose a definition of a maximal Cohen-Macaulay DG-complex over $R$ that naturally generalizes a maximal Cohen-Macaulay complex over a noetherian local ring, as studied by Iyengar, Ma, Schwede, and Walker. Our proposed definition extends the work of Shaul on Cohen-Macaulay DG-rings and DG-modules, as any maximal Cohen-Macaulay DG-module is a maximal Cohen-Macaulay DG-complex. After proving necessary lemmas in derived commutative algebra, we establish the existence of a maximal Cohen-Macaulay DG-complex for every DG-ring with constant amplitude that admits a dualizing DG-module. We then use the existence of these DG-complexes to establish a derived Improved New Intersection Theorem for all DG-rings with constant amplitude.

math.AC↗