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- W2951804216 abstract "Fix a scheme $S$ of characteristic $p$. Let $mathscr{M}$ be an $S$-algebraic stack and let $mbox{Fdiv}(mathscr{M})$ be the stack of $mbox{F}$-divided objects, that is sequences of objects $x_iinmathscr{M}$ with isomorphisms $sigma_i:x_ito mbox{F}^*x_{i+1}$. Let $mathscr{X}$ be a flat, finitely presented $S$-algebraic stack and $mathscr{X}to Pi_1(mathscr{X}/S)$ the 'etale fundamental pro-groupoid, constructed in the present text. We prove that if $mathscr{M}$ is a quasi-separated Deligne-Mumford stack and $mathscr{X}to S$ has geometrically reduced fibres, there is a bifunctorial isomorphism of stacks [mathscr{H}!om(Pi_1(mathscr{X}/S),mathscr{M}) simeq mathscr{H}!om(mathscr{X},mbox{Fdiv}(mathscr{M})).] In particular, the system of relative Frobenius morphisms $mathscr{X}to mathscr{X}^{p/S}to mathscr{X}^{p^2/S}todots$ allows to recover the space of connected components $pi_0(mathscr{X}/S)$ and the relative 'etale fundamental gerbe. In order to obtain these results, we study the existence and properties of relative perfection for algebras in characteristic $p$." @default.
- W2951804216 created "2019-06-27" @default.
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- W2951804216 date "2019-09-26" @default.
- W2951804216 modified "2023-09-30" @default.
- W2951804216 title "Unramified F-divided objects and the 'etale fundamental pro-groupoid in positive characteristic" @default.
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