Characteristic-free bounds for the Castelnuovo–Mumford regularity

Abstract

We study bounds for the Castelnuovo-Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular, our aim is to give a positive answer to a question posed by Bayer and Mumford in What can be computed in algebraic geometry? (Computational algebraic geometry and commutative algebra, Symposia Mathematica, vol. XXXIV (1993), 1-48) by showing that the known upper bound in characteristic zero holds true also in positive characteristic. We first analyse Giusti's proof, which provides the result in characteristic zero, giving some insight into the combinatorial properties needed in that context. For the general case, we provide a new argument which employs the Bayer-Stillman criterion for detecting regularity.

Bounding Projective Dimension

“…This bound can be deduced from work of Galligo [22] and Giusti [23] in characteristic zero, as was observed by Bayer and Mumford [2,Theorem 3.7]. It was later proved in all characteristics by Caviglia and Sbarra [10].…”

Some Remarks on Borel Type Ideals

“…We mention that this concept appears also in [3,Definition 1.3] as the so called weakly stable ideal. Herzog, Popescu and Vladoiu proved in [8] that I is of Borel type, if and only if for any monomial u ∈ I and for any 1 ≤ j < i ≤ n with x i |u, there exists an integer t > 0 such that x t j u/x ν i (u) i ∈ I, where ν i (u) is the exponent of x i in u.…”

A Stable Property of Borel Type Ideals

“…We mention that this concept appears also in [3,Definition 1.3] as the so called weakly stable ideal. Herzog, Popescu and Vladoiu proved in [7] that I is of Borel type, if and only if for any monomial u ∈ I and for any 1 ≤ j < i ≤ n and q > 0 with x q i |u, there exists an integer t > 0 such that x t j u/x q i ∈ I.…”