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On Nori’s fundamental group scheme. (English) Zbl 1137.14035
Kapranov, Mikhail (ed.) et al., Geometry and dynamics of groups and spaces. In memory of Alexander Reznikov. Partly based on the international conference on geometry and dynamics of groups and spaces in memory of Alexander Reznikov, Bonn, Germany, September 22–29, 2006. Basel: Birkhäuser (ISBN 978-3-7643-8607-8/hbk). Progress in Mathematics 265, 377-398 (2008).
Let $$X$$ be a connected proper reduced scheme over a perfect field $$k$$ which has a $$k$$-rational point. Let $$\mathcal{S}\left( X\right)$$ be the category of semi-stable degree 0 vector bundles on $$X$$; this is a full subcategory of the category of quasi-coherent sheaves on $$X$$. A vector bundle $$V$$ on $$X$$ is finite if there are distinct polynomials $$f,g$$ with non-negative integer coefficients such that $$f\left( V\right) \cong g\left( V\right) .$$ The category $$\mathcal{C}^{N}\left( X\right)$$ consisting of vector bundles which are subquotients in $$\mathcal{S}\left( X\right)$$ of a direct sum of finite bundles is a $$k$$-linear abelian rigid tensor category whose objects are called Nori finite bundles. This $$\mathcal{C}^{N}\left( X\right)$$ is a Tannaka category, and we let $$\pi^{N}\left( X,x\right)$$ be the corresponding Tannaka group scheme over $$k$$. There is a well-known equivalence between $$\mathcal{C}^{N}\left( X\right)$$ and the representation category of $$\pi^{N}\left( X,x\right) .$$ The category $$\mathcal{C}^{N}\left( X\right)$$ contains the subcategories $$\mathcal{C}^{\text{ét}}\left( X\right)$$, the bundles for which the corresponding representation of $$\pi^{N}\left( X,x\right)$$ factors through an étale group scheme; and $$\mathcal{C} ^{F}\left( X\right) ,$$ the bundles for which the corresponding representation factors through a finite local group scheme. The intersection of these two subcategories consists of the trivializable bundles. Generally, for a subcategory $$S$$ of $$\mathcal{C}^{N}\left( X\right)$$ we let $$\pi\left( X,S,x\right)$$ be the Tannaka group corresponding to $$S$$.
The categories $$\mathcal{C}^{\text{ét}}\left( X\right)$$ and $$\mathcal{C}^{F}\left( X\right)$$ are used to provide insight into the structure of $$\mathcal{C}^{N}\left( X\right) .$$ Let $$r^{\text{ét}}$$ (resp. $$r^{F}$$) be the natural map $$\pi^{N}\left( X,x\right) \rightarrow \pi^{\text{ét}}\left( X,x\right)$$ (resp. $$\pi^{N}\left( X,x\right) \rightarrow\pi^{F}\left( X,x\right)$$) induced by the inclusion of categories. For $$S$$ and $$T$$ subcategories of $$\mathcal{C}^{N}\left( X\right)$$ such that $$\pi\left( X,S,x\right)$$ étale and $$\pi\left( X,T,x\right)$$ local the authors construct a category $$\mathcal{E}\left( X_{S\cup T}\right)$$ consisting of bundles $$V$$ whose push down on $$X_{S}$$ is $$F$$-finite. Then a $$k$$-linear abelian rigid tensor category $$\mathcal{E}$$ is constructed, whose objects are pairs $$\left( X_{S\cup T},V\right)$$ where $$V$$ is an object in $$\mathcal{E}\left( X_{S\cup T}\right) .$$
The main result is as follows: the functor $$\mathcal{C}^{N}\left( X\right) \rightarrow\mathcal{E}$$ which assigns $$\left( X_{S},\pi_{S}^{\ast}\left( V\right) \right)$$ to $$V$$, where $$S$$ is the maximal étale subcategory of the subcategory $$\left\langle V\right\rangle$$ spanned by $$V\in$$Obj$$\left( C^{N}\left( X\right) \right) ,$$ identifies the representation category of $$Ker\left( r^{\text{ét}},r^{F}\right)$$ with $$\mathcal{E}.$$
For the entire collection see [Zbl 1130.00014].

##### MSC:
 14L17 Affine algebraic groups, hyperalgebra constructions 14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
##### Keywords:
fundamental group scheme; Tannaka duality; finite bundles
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