Zero-generic initial ideals

TL;DR

The paper introduces zero-generic initial ideals, a new class of monomial ideals that mimic generic initial ideals in any characteristic, sharing key algebraic properties and satisfying conjectures related to local cohomology.

Contribution

It defines zero-generic initial ideals and proves they replicate many properties of generic initial ideals in characteristic zero, including stability, Hilbert series, and cohomological invariants.

Findings

Zero-generic initial ideals are strongly stable in any characteristic.

They share the same Hilbert series, regularity, and Betti numbers as the original ideal.

They satisfy Green's Crystallization Principle even in positive characteristic.

Abstract

Given a homogeneous ideal I of a polynomial ring A=K[X_1,...,X_n] and a monomial order, we construct a new monomial ideal of A associated with I. We call it the zero-generic initial ideal of I with respect to the order and denote it with gin_0(I), or with Gin_0(I) whenever the order is the reverse-lexicographic order. When the characteristic of K is zero, a zero-generic initial ideal is the usual generic initial ideal. We show that Gin_0(I) is endowed with many interesting properties, some of which are easily seen, e.g., it is a strongly stable monomial ideal in any characteristic, and has the same Hilbert series as I; some other properties are less obvious: it shares with I the same Castelnuovo-Mumford regularity, projective dimension and extremal Betti numbers. Quite surprisingly, gin_0(I) also satisfies Green's Crystallization Principle, which is known to fail in positive characteristic. Thus, zero-generic initial ideals can be used as formal analogues of generic initial ideals computed in characteristic 0. We also prove the analogue for local cohomology of Pardue's Conjecture for Betti numbers: we show that the Hilbert functions of local cohomology modules of a quotient ring of a polynomial ring modulo a weakly stable ideal are independent of the characteristic of the coefficients field.