Projective Space in Synthetic Algebraic Geometry
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909864254504960 |
|---|---|
| author | Cherubini, Felix Coquand, Thierry Ritter, Matthias Wärn, David |
| author_facet | Cherubini, Felix Coquand, Thierry Ritter, Matthias Wärn, David |
| contents | Synthetic algebraic geometry is a new approach to algebraic geometry. It consists in using homotopy type theory extended with three axioms, together with the interpretation of these in a higher version of the Zariski topos, in order to do algebraic geometry internally to this topos. In this article, we will show basic properties of projective n-space $\mathbb{P}^n$ in synthetic algebraic geometry. In particular, we show that the automorphism group of $\mathbb{P}^n$ is $\mathrm{PGL}_{n+1}(R)$ and that the picard group is $\mathbb{Z}$. We will provide different proofs of the latter statement, where the most synthetic approach naturally leads to the refined statement that the type of line bundles on $\mathbb{P}^n$ is the higher type $\mathbb{Z}\times K(R^\times,1)$, where $K(R^\times,1)$ is a delooping of the group of units of the internal base ring $R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13916 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projective Space in Synthetic Algebraic Geometry Cherubini, Felix Coquand, Thierry Ritter, Matthias Wärn, David Algebraic Geometry Logic 14A99 (Primary), 03B38, 18N99 (Secondary) Synthetic algebraic geometry is a new approach to algebraic geometry. It consists in using homotopy type theory extended with three axioms, together with the interpretation of these in a higher version of the Zariski topos, in order to do algebraic geometry internally to this topos. In this article, we will show basic properties of projective n-space $\mathbb{P}^n$ in synthetic algebraic geometry. In particular, we show that the automorphism group of $\mathbb{P}^n$ is $\mathrm{PGL}_{n+1}(R)$ and that the picard group is $\mathbb{Z}$. We will provide different proofs of the latter statement, where the most synthetic approach naturally leads to the refined statement that the type of line bundles on $\mathbb{P}^n$ is the higher type $\mathbb{Z}\times K(R^\times,1)$, where $K(R^\times,1)$ is a delooping of the group of units of the internal base ring $R$. |
| title | Projective Space in Synthetic Algebraic Geometry |
| topic | Algebraic Geometry Logic 14A99 (Primary), 03B38, 18N99 (Secondary) |
| url | https://arxiv.org/abs/2405.13916 |