Some new invariants of Noetherian local rings related to square of the maximal ideal

2022/04/27 by Dylan C. Beck, Souvik Dey, Beck, Dylan C. +1
Mathematics · #Commutative Algebra and Its Applications #Algebraic Geometry and Number Theory #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2205.01658

Abstract

We introduce two new invariants of a Noetherian (standard graded) local ring (R, \mathfrak m) that measure the number of generators of certain kinds of reductions of \mathfrak m, and we study their properties. Explicitly, we consider the minimum among the number of generators of ideals I such that either I² = \mathfrak m² or I ⊇ \mathfrak m² holds. We investigate subsequently the case that R is the quotient of a polynomial ring k[x₁, …, x_n] by an ideal I generated by homogeneous quadratic forms, and we compute these invariants. We devote specific attention to the case that R is the quotient of a polynomial ring k[x₁, …, x_n] by the edge ideal of a finite simple graph G.

Related

• Generators of reductions of ideals in a local Noetherian ring with\n finite residue field
• The Golod property of powers of the maximal ideal of a local ring
• The Proj of the Rees algebra of a graded family of ideals
• Structure theorems for certain Gorenstein ideals
• The multiplicity and the number of generators of an integrally closed ideal