2025/05/10 by Susan Cooper, Cooper, Susan M., Sara Faridi +3
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2505.06751
openalex publication_date 2025/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given any monomial ideal I minimally generated by q monomials, we define a simplicial complex \mathbbMq2 that supports a resolution of I2 . We also define a subcomplex \mathbbM2(I), which depends on the monomial generators of I and also supports the resolution of I2 . As a byproduct, we obtain bounds on the projective dimension of the second power of any monomial ideal. We also establish bounds on the Betti numbers of I2 , which are significantly tighter than those determined by the Taylor resolution of I2 . Moreover, we introduce the permutation ideal Tq which is generated by q monomials. For any monomial ideal I with q generators, we establish that β(I2) ≤ β(Tq2). We show that the simplicial complex \mathbbMq2 supports the minimal resolution of Tq2. In fact, \mathbbMq2 is the Scarf complex of Tq2.