Infinite Free Resolutions over Monomial Rings in Two Variables

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