Multiplicities and Betti numbers in local algebra via lim Ulrich points

TL;DR

This paper investigates bounds on invariants like Euler characteristic and Dutta multiplicity for certain finite free complexes over local rings, using Ulrich modules and sheaf constructions to derive new algebraic inequalities.

Contribution

It introduces a novel approach using lim Ulrich points and sheaf sequences to establish bounds on multiplicities and Betti numbers in local algebra.

Findings

Lower bounds on Euler characteristic for complexes over strict complete intersections

Bounds on Dutta multiplicity in localized graded algebras

Construction of sequences of Ulrich modules with good convergence properties

Abstract

This work concerns finite free complexes with finite length homology over a commutative noetherian local ring $R$. The focus is on complexes that have length $\mathrm{dim}\, R$, which is the smallest possible value, and in particular on free resolutions of modules of finite length and finite projective dimension. Lower bounds are obtained on the Euler characteristic of such short complexes when $R$ is a strict complete intersection, and also on the Dutta multiplicity, when $R$ is the localization at its maximal ideal of a standard graded algebra over a field of positive prime characteristic. The key idea in the proof is the construction of a suitable Ulrich module, or, in the latter case, a sequence of modules that have the Ulrich property asymptotically, and with good convergence properties in the rational Grothendieck group of $R$. Such a sequence is obtained by constructing an appropriate sequence of sheaves on the associated projective variety.