The fundamental group-scheme
For a connected reduced scheme X over an arbitrary field k the fundamental group scheme is constructed and studied (which is a group scheme over k) and its connection with principal G-fiberings, where G is a finite (or nilpotent) group scheme, and finite linear fiberings on X is investigated. For X and its fixed k-point \(\chi_ 0:\quad Spec k\to X\) a triple \((P,\pi (X,\chi_ 0),*)\) is constructed (where \(\pi (X,\chi_ 0)\) is the group k-scheme which is the projective limit of finite group schemes, P is the principal \(\pi (X,\chi_ 0)\) fibering over X with marked point * lying over \(\chi_ 0)\) which has the universality property with respect to principal G-fiberings on X, where G is a finite group k-scheme with marked point lying over \(\chi_ 0\). The group k-scheme \(\pi (X,\chi_ 0)\) is called the fundamental group scheme of the scheme X and possesses many natural properties. The case of a complete scheme X over a perfect field k was considered by the author earlier and is connected with finite and essentially finite vector bundles over X which are multidimensional generalizations of one- dimensional bundles corresponding to the finite order points on the Jacobian of a curve [the author, Compos. Math. 33, 29-41 (1976; Zbl 0337.14016)]. Owing to the consideration of non-complete schemes the author carries over these results to linear, parabolic (defined by Seshadri) essentially finite bundles over a smooth projective curve X with a finite set S of points deleted. Such bundles are constructed by k- linear representations of the group \(\pi (X-S,\chi_ 0)\), where \(\chi_ 0\not\in S\), which are passed through representations of finite group schemes over k. The last chapter is devoted to the study of a nilpotent fundamental group-scheme \(U(X,\chi_ 0)\) (it is constructed if \(\Gamma\) (X,\({\mathcal O}_ X)=k)\). If char k\(>0\) and \(\dim H^ 1(X,{\mathcal O}_ X)<\infty\) then \(U(X,\chi_ 0)\) is a factor of \(\pi (X,\chi_ 0)\). The connection of \(U(X,\chi_ 0)\) with Pic X is studied. In the case of complete reduced curves and \(p=char k>0\), the computation of \(U(X,\chi_ 0)\) leads to non-commutative formal groups which are computed by the author for rational curves with the simplest singularities. As a corollary, an old result of I. R. Shafarevich is reproved: For a complete curve X with \(\Gamma\) (X,\({\mathcal O}_ X)=k\), the maximal p-factor of the étale fundamental group is a free pro-p-group in characteristic p. The main means used by the author are the equivalence between the Tannaka category and the category of finite-dimensional representations of a certain affine group scheme [see \textit{N. R. Saavedra}, ''Catégories tannakiennes'', Lect. Notes Math. 265 (1972; Zbl 0241.14008)].
finite-dimensional representations of affine group scheme
scientific article; zbMATH DE number 3939489
scientific article