2024/06/08 by Zeyu Shen, Shen, Zeyu · 1 citation
Computer Science · Mathematics · #14C35 #19A99 #19E08 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #K-Theory and Homology (math.KT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2406.05562
openalex publication_date 2024/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be an affine, simplicial toric variety over a field. Let G0 denote the Grothendieck group of coherent sheaves on a Noetherian scheme and let F1G0 denote the first step of the filtration on G0 by codimension of support. Then G0(X)≅ℤ⊕ F1G0(X) and F1G0(X) is a finite abelian group. In dimension 2, we show that F1G0(X) is a finite cyclic group and determine its order. In dimension 3, F1G0(X) is determined up to a group extension of the Chow group A1(X) by the Chow group A2(X). We determine the order of the Chow group A1(X) in this case. A conjecture on the orders of A1(X) and A2(X) is formulated for all dimensions.