Monomial ideals with large projective dimension

TL;DR

This paper provides explicit formulas for Betti numbers of monomial ideals in polynomial rings and characterizes those with maximal projective dimension, advancing understanding of their algebraic properties.

Contribution

It offers explicit Betti number formulas and characterizes monomial ideals with projective dimension equal to the number of variables.

Findings

Explicit formulas for total, graded, and multigraded Betti numbers.

Characterization of monomial ideals with projective dimension n.

Abstract

Let S be a polynomial ring in n variables, over an arbitrary field. We give the total, graded, and multigraded Betti numbers of S/M, for every monomial ideal M in S. We also give an explicit characterization of all monomial ideals M in S for which the quotient S/M has projective dimension n.