On the growth of deviations

TL;DR

This paper investigates the behavior of deviations in graded algebras, showing they are maximized by Lex-segment ideals and grow exponentially in Golod rings and specific quadratic monomial algebras.

Contribution

It establishes that deviations do not decrease under initial ideals and identifies conditions for their exponential growth, extending understanding of algebraic deviations.

Findings

Deviations are maximized by Lex-segment ideals.

Deviations grow exponentially in Golod rings.

Deviations do not decrease when passing to initial ideals.

Abstract

The deviations of a graded algebra are a sequence of integers that determine the Poincare series of its residue field and arise as the number of generators of certain DG algebras. In a sense, deviations measure how far a ring is from being a complete intersection. In this paper we study extremal deviations among those of algebras with a fixed Hilbert series. In this setting, we prove that, like the Betti numbers, deviations do not decrease when passing to an initial ideal and are maximized by the Lex-segment ideal. We also prove that deviations grow exponentially for Golod rings and for certain quadratic monomial algebras.