Connected components of qcqs schemes and projective spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915939661905920 |
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| author | Tarizadeh, Abolfazl |
| author_facet | Tarizadeh, Abolfazl |
| contents | In this article, we first prove a general result in topology which states that every quasi-component of a quasi-spectral space is connected. \\ As an application, the structure of the connected components of every quasi-compact quasi-separated (qcqs) scheme $X$ is fully characterized. They are exactly of the form $f^{-1}(C)$ where $f:X\rightarrow\Spec(R)$ is the canonical morphism, $C$ is a connected component of $\Spec(R)$ and $R=\mathscr{O}_{X}(X)$ is the ring of global sections of $X$. \\ Next, we make new advances in understanding the structure of the connected components of projective spaces. In general, for an $\mathbb{N}$-graded ring $R=\bigoplus\limits_{n\geqslant0}R_{n}$, the structure of the connected components of scheme $\Proj(R)$ is still unknown. However, we show that for any scheme $S$ the connected components of the projective space $\mathbb{P}^{n}_{S}= \mathbb{P}^{n}_{\mathbb{Z}}\times_{\Spec(\mathbb{Z})}S$ are exactly of the form $\mathbb{P}^{n}_{C}$ where $C$ is a connected component of $S$ which is equipped with a closed subscheme structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_05185 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Connected components of qcqs schemes and projective spaces Tarizadeh, Abolfazl Commutative Algebra Algebraic Geometry General Topology Rings and Algebras 14A15, 14A25, 14A05, 13A02, 13A15, 16U40 In this article, we first prove a general result in topology which states that every quasi-component of a quasi-spectral space is connected. \\ As an application, the structure of the connected components of every quasi-compact quasi-separated (qcqs) scheme $X$ is fully characterized. They are exactly of the form $f^{-1}(C)$ where $f:X\rightarrow\Spec(R)$ is the canonical morphism, $C$ is a connected component of $\Spec(R)$ and $R=\mathscr{O}_{X}(X)$ is the ring of global sections of $X$. \\ Next, we make new advances in understanding the structure of the connected components of projective spaces. In general, for an $\mathbb{N}$-graded ring $R=\bigoplus\limits_{n\geqslant0}R_{n}$, the structure of the connected components of scheme $\Proj(R)$ is still unknown. However, we show that for any scheme $S$ the connected components of the projective space $\mathbb{P}^{n}_{S}= \mathbb{P}^{n}_{\mathbb{Z}}\times_{\Spec(\mathbb{Z})}S$ are exactly of the form $\mathbb{P}^{n}_{C}$ where $C$ is a connected component of $S$ which is equipped with a closed subscheme structure. |
| title | Connected components of qcqs schemes and projective spaces |
| topic | Commutative Algebra Algebraic Geometry General Topology Rings and Algebras 14A15, 14A25, 14A05, 13A02, 13A15, 16U40 |
| url | https://arxiv.org/abs/2401.05185 |