2013/08/01 by Gwyneth Whieldon, Whieldon, Gwyneth R.
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1308.0179
openalex publication_date 2013/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M in k[x,y] be a monomial ideal M=(m1,m2,...,mr), where the mi 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 βiS(k) are given by β0S(k)=1, β1S(k)=2, and βiS(k)=βi-1(k)+(r-1)βi-2S(k), where r is the number of minimal generators of M. This specializes to the classic example S=k[x,y]/(x2,xy), which has βiS(k)=Fi+1, where Fi+1 is the (i+1)st Fibonacci number. Macaulay2 code producing these resolutions is available at: http://cs.hood.edu/~whieldon/pages/research.html