Infinite Free Resolutions over Monomial Rings in Two Variables

TL;DR

This paper constructs explicit free resolutions and recursive formulas for Betti numbers of monomial ideals in two variables, providing a comprehensive method for understanding their algebraic structure.

Contribution

It introduces a new explicit construction of free resolutions for monomial ideals in two variables and derives recursive formulas for their Betti numbers.

Findings

Explicit free resolutions for all monomial ideals in two variables.

Recursive formulas for Betti numbers based on the number of generators.

Special case recovering Fibonacci sequence for a specific ideal.

Abstract

Let M in k[x,y] be a monomial ideal M=(m_1,m_2,...,m_r), where the m_i are a minimal generating set of M. We construct an explicit free resolution of k over S=k[x,y]/M for all monomial ideals M, and provide recursive formulas for the Betti numbers. In particular, if M is any monomial ideal (excepting five degenerate cases,) the total Betti numbers \beta_i^S(k) are given by \beta_0^S(k)=1, \beta_1^S(k)=2, and \beta_i^S(k)=\beta_{i-1}(k)+(r-1)\beta_{i-2}^S(k), where r is the number of minimal generators of M. This specializes to the classic example S=k[x,y]/(x^2,xy), which has \beta_i^S(k)=F_{i+1}, where F_{i+1} is the (i+1)st Fibonacci number. Macaulay2 code producing these resolutions is available at: http://cs.hood.edu/~whieldon/pages/research.html