$G_0$ of affine, simplicial toric varieties
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911115219304448 |
|---|---|
| author | Shen, Zeyu |
| author_facet | Shen, Zeyu |
| contents | Let $X$ be an affine, simplicial toric variety over a field. Let $G_0$ denote the Grothendieck group of coherent sheaves on a Noetherian scheme and let $F^1G_0$ denote the first step of the filtration on $G_0$ by codimension of support. Then $G_0(X)\cong\mathbb{Z}\oplus F^1G_0(X)$ and $F^1G_0(X)$ is a finite abelian group. In dimension 2, we show that $F^1G_0(X)$ is a finite cyclic group and determine its order. In dimension 3, $F^1G_0(X)$ is determined up to a group extension of the Chow group $A^1(X)$ by the Chow group $A^2(X)$. We determine the order of the Chow group $A^1(X)$ in this case. A conjecture on the orders of $A^1(X)$ and $A^2(X)$ is formulated for all dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05562 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $G_0$ of affine, simplicial toric varieties Shen, Zeyu K-Theory and Homology Algebraic Geometry 19A99, 14C35, 19E08 Let $X$ be an affine, simplicial toric variety over a field. Let $G_0$ denote the Grothendieck group of coherent sheaves on a Noetherian scheme and let $F^1G_0$ denote the first step of the filtration on $G_0$ by codimension of support. Then $G_0(X)\cong\mathbb{Z}\oplus F^1G_0(X)$ and $F^1G_0(X)$ is a finite abelian group. In dimension 2, we show that $F^1G_0(X)$ is a finite cyclic group and determine its order. In dimension 3, $F^1G_0(X)$ is determined up to a group extension of the Chow group $A^1(X)$ by the Chow group $A^2(X)$. We determine the order of the Chow group $A^1(X)$ in this case. A conjecture on the orders of $A^1(X)$ and $A^2(X)$ is formulated for all dimensions. |
| title | $G_0$ of affine, simplicial toric varieties |
| topic | K-Theory and Homology Algebraic Geometry 19A99, 14C35, 19E08 |
| url | https://arxiv.org/abs/2406.05562 |