The Künneth Formula of Fundamental Group Schemes
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908835418996736 |
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| author | Li, Lingguang Tian, Niantao |
| author_facet | Li, Lingguang Tian, Niantao |
| contents | Let $k$ be a field, $f:X\rightarrow S$ a proper morphism between connected schemes proper over $k$, $x\in X(k)$ lying over $s\in S(k)$, $X_s$ the fibre of $f$ over $s$, $\mathcal{C}_X$, $\mathcal{C}_{S}$, $\mathcal{C}_{X_s}$ Tannakian categories over $X,S,X_s$ respectively, $π(\mathcal{C}_X,x)$, $π(\mathcal{C}_S,s)$, $π(\mathcal{C}_{X_s},x)$ the Tannaka group schemes respectively. We give the necessary and sufficient conditions for the exactness of the homotopy sequence $π(\mathcal{C}_{X_s},x)\rightarrow π(\mathcal{C}_X,x)\rightarrow π(\mathcal{C}_S,s)\rightarrow 1$. In particular, we obtain the equivalent conditions for the Kunneth formula of fundamental group schemes for the product $X\times_k Y$ of two connected schemes $X$ and $Y$ proper over $k$. As an application, we obtain the Kunneth formula of certain fundamental group schemes over any field, such as S, Nori, EN, F, Etale, Loc, ELoc and Unipotent fundamental group schemes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14207 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Künneth Formula of Fundamental Group Schemes Li, Lingguang Tian, Niantao Algebraic Geometry 14F35, 14L15, 14J60 Let $k$ be a field, $f:X\rightarrow S$ a proper morphism between connected schemes proper over $k$, $x\in X(k)$ lying over $s\in S(k)$, $X_s$ the fibre of $f$ over $s$, $\mathcal{C}_X$, $\mathcal{C}_{S}$, $\mathcal{C}_{X_s}$ Tannakian categories over $X,S,X_s$ respectively, $π(\mathcal{C}_X,x)$, $π(\mathcal{C}_S,s)$, $π(\mathcal{C}_{X_s},x)$ the Tannaka group schemes respectively. We give the necessary and sufficient conditions for the exactness of the homotopy sequence $π(\mathcal{C}_{X_s},x)\rightarrow π(\mathcal{C}_X,x)\rightarrow π(\mathcal{C}_S,s)\rightarrow 1$. In particular, we obtain the equivalent conditions for the Kunneth formula of fundamental group schemes for the product $X\times_k Y$ of two connected schemes $X$ and $Y$ proper over $k$. As an application, we obtain the Kunneth formula of certain fundamental group schemes over any field, such as S, Nori, EN, F, Etale, Loc, ELoc and Unipotent fundamental group schemes. |
| title | The Künneth Formula of Fundamental Group Schemes |
| topic | Algebraic Geometry 14F35, 14L15, 14J60 |
| url | https://arxiv.org/abs/2602.14207 |