The homotopy sequence of Nori's fundamental group (Q2438385)

\textit{A. Grothendieck} [Séminaire de géométrie algébrique du Bois Marie 1960/61 (SGA 1). New York: Springer-Verlag (1971; Zbl 0234.14002)] showed that for a proper separable morphism \(X\to S\) of connected schemes with geometrically connected fibres there is an exact sequence of étale fundamental groups \[ \pi_1(X_{\bar s}, \bar x)\to \pi_1(X, \bar x)\to \pi_1(S, \bar s)\to 1 \] where \(\bar x\) is a geometric point of \(X\) with image \(\bar s\) in \(S\), and \(X_{\bar s}\) is the geometric fibre. In a preprint, \textit{H. Esnault, P.-H. Hai} and \textit{E. Viehweg} [``On the homotopy exact sequence for Nori's fundamental group'', \url{arXiv:0908.0498}] gave a counterexample showing that such an exact sequence does not necessarily exist for \textit{M. V. Nori}'s fundamental group scheme [Proc. Indian Acad. Sci., Math. Sci. 91, 73--122 (1982; Zbl 0586.14006)], even for \(X\) and \(S\) projective and smooth over an algebraically closed field of positive characteristic. In the present paper, the author gives necessarily and sufficient conditions (a bit complicated to check in practice) for the corresponding sequence of Nori fundamental group schemes to be exact. As applications, he gives partly new proofs of Grothendieck's sequence above and of a result of \textit{V. B. Mehta} and \textit{S. Subramanian} [Invent. Math. 148, No. 1, 143--150 (2002; Zbl 1020.14006)] according to which Nori's fundamental group scheme is compatible with products of proper connected schemes. The paper also contains a counterexample to product compatibility with one of the schemes non-proper: a product of the affine line with a supersingular elliptic curve in characteristic 2.