arXiv subjects

Nawaj KC

Publications and source records attributed to Nawaj KC.

5 recordsLinked to original sources

arXiv2026

Regularity is bounded on a quasi-excellent Noetherian scheme

A point of a scheme has an associated tangent cone, the spectrum of a standard graded algebra encoding the local singularity. Its homological complexity can be measured by its graded Betti table: a matrix that records a part of the structure of its graded, minimal free resolution over a polynomial ring. A natural question is whether the homological complexity of the tangent cones varies arbitrarily across a scheme. In this paper, we show that this is not the case for a quasi-excellent Noetherian scheme; over such schemes, only finitely many graded Betti tables can occur. More generally, we show that a coherent sheaf over a quasi-excellent Noetherian scheme admits finitely many graded Betti tables, and that the constancy loci for the graded Betti table are constructible. As an immediate consequence, regularity is bounded on a quasi-excellent Noetherian scheme.

math.AC↗

arXiv2026

Lifting systems for finite length modules

This paper is concerned with lifting modules along a surjective map of noetherian local rings, say $\varphi \colon R \twoheadrightarrow S$. A finitely generated $R$-module $L$ is a naive lift of an $S$-module $M$ if $L \otimes_R S \cong M$. We are concerned with the maximum depth and dimension among all naive lifts of $M$, which we call the liftable depth and liftable dimension, respectively, of $M$ along $\varphi$. We approach this via a notion of lifting systems that we introduce in this paper. We then provide a necessary and sufficient condition for a module of finite length to lift and Serre lift to a regular local ring in terms of lifting systems.

math.AC↗

arXiv2024

Lower bounds on Loewy lengths of modules of finite projective dimension

This article is concerned with nonzero modules of finite length and finite projective dimension over a local ring. We show the Loewy length of such a module is larger than the regularity of the ring whenever the ring is strict Cohen-Macaulay, establishing a conjecture of Corso--Huneke--Polini--Ulrich for such rings. In fact, we show the stronger result that the Loewy length of a nonzero module of finite flat dimension is at least the regularity for strict Cohen-Macaulay rings, which significantly strengthens a theorem of Avramov--Buchweitz--Iyengar--Miller. As an application we simultaneously verify a Lech-like conjecture, comparing generalized Loewy length along flat local extensions, and a conjecture of Hanes for strict Cohen-Macaulay rings. Finally, we also give notable improvements to known lower bounds for Loewy lengths without the strict Cohen-Macaulay assumption. The strongest general bounds we achieve are over complete intersection rings.

math.AC↗

arXiv2024

On liftings of modules of finite projective dimension

We introduce and study a notion of Serre liftable modules; these are modules that are liftable to modules of the maximal possible dimension over a regular local ring. We establish new cases of Serre's positivity conjecture over ramified regular local rings by proving it for Serre liftable modules. Furthermore, we show that the length of a nonzero Serre liftable module is bounded below by the Hilbert-Samuel multiplicity of the local ring. This establishes special cases of the Length Conjecture of Iyengar-Ma-Walker.

math.AC↗

arXiv2024

Singular loci of algebras over ramified discrete valuation rings

When $k$ is a field, the classical Jacobian criterion computes the singular locus of an equidimensional, finitely generated $k$-algebra as the closed subset of an ideal generated by appropriate minors of the so-called Jacobian matrix. Recently, Hochster-Jeffries and Saito have extended this result for algebras over any unramified discrete valuation ring of mixed characteristic via the use of $p$-derivations. Motivated by these results, in this paper, we state and prove an analogous Jacobian criterion for algebras over a ramified discrete valuation ring of mixed characteristic.

math.AC↗