Unramified F-divided objects and the \'etale fundamental pro-groupoid in positive characteristic

TL;DR

This paper establishes a deep connection between the étale fundamental pro-groupoid of a scheme in positive characteristic and the stack of F-divided objects, enabling recovery of connected components and fundamental gerbes.

Contribution

It introduces a bifunctorial isomorphism linking the étale fundamental pro-groupoid and F-divided objects, and studies relative perfection in characteristic p.

Findings

Isomorphism between hom-stacks and F-divided objects stacks

Recovery of connected components via Frobenius system

Analysis of relative perfection in characteristic p

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 $x_i\in\mathscr{M}$ with isomorphisms $\sigma_i:x_i\to \mbox{F}^*x_{i+1}$. 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 $\mathscr{X}\to \mathscr{X}^{p/S}\to \mathscr{X}^{p^2/S}\to\dots$ 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 $p$.

Equations

$$\mathcal{H}om({\Pi }_1(\mathcal{X}/S),\mathcal{M})\stackrel{\sim }{\to }\mathcal{H}om(\mathcal{X},\mathrm{Fdiv}(\mathcal{M}))$$

$$F_i:{\mathcal{X}}^{p^i/S}⟶{\mathcal{X}}^{p^{i+1}/S}$$

$$X$$

$$\mathcal{X}$$

$$F_i:A^{p^{i+1}/R}\to A^{p^i/R}$$

$$A^{p^{\infty}/R}=\lim\,\big{(}\!\!\leavevmode\hbox to127.78pt{\vbox to18.35pt{\pgfpicture\makeatletter\hbox{\hskip 63.89162pt\lower-9.22621pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-63.89162pt}{-9.12637pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 20.06194pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.7564pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\cdots\ A^{p^{2}/R}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 20.06194pt\hfil&\hfil\hskip 35.77414pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.70639pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A^{p/R}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 13.01193pt\hfil&\hfil\hskip 30.81775pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 8.05554pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-23.56775pt}{-6.62637pt}\pgfsys@lineto{-1.6055pt}{-6.62637pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.40552pt}{-6.62637pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.67137pt}{-2.98473pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\operatorname{F}_{1}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{25.21832pt}{-6.62637pt}\pgfsys@lineto{47.18057pt}{-6.62637pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{47.38055pt}{-6.62637pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ 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$$\mathcal{O}({\pi }_0(A/R))\stackrel{\sim }{\to }{A}^{p^{\infty }/R}.$$

$$X⟶{\pi }_0(X/S)⟶{\pi }^a(X/S).$$

$$A^{\acute{\mathrm{e}}\mathrm{t}/R}\stackrel{\sim }{\to }\mathcal{O}(\pi (A/R))\stackrel{\sim }{\to }{A}^{p^{\infty }/R}.$$

$$A^{\operatorname{pf}}=\lim\big{(}\!\!\leavevmode\hbox to100.67pt{\vbox to16.42pt{\pgfpicture\makeatletter\hbox{\hskip 50.33356pt\lower-8.25955pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-50.33356pt}{-8.15971pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 14.30554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\cdots\ A}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 14.30554pt\hfil&\hfil\hskip 27.97247pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ 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{}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-21.52248pt}{-5.65971pt}\pgfsys@lineto{-2.4055pt}{-5.65971pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.20552pt}{-5.65971pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-14.88208pt}{-1.94028pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\operatorname{F}_{\!A}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{14.50554pt}{-5.65971pt}\pgfsys@lineto{33.62251pt}{-5.65971pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{33.8225pt}{-5.65971pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.14594pt}{-1.94028pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\operatorname{F}_{\!A}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\!\!\big{)};\qquad A^{\operatorname{copf}}=\operatorname{colim}\big{(}\!\!\leavevmode\hbox to100.67pt{\vbox to16.42pt{\pgfpicture\makeatletter\hbox{\hskip 50.33356pt\lower-8.25955pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ 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}\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\!\!\big{)}.$$

$$X$$

$$f_{!}(X^{\mathrm{copf}})\stackrel{\sim }{\to }(f_{!}X{)}^{\mathrm{copf}}.$$

$${\mathrm{Hom}}_{{\mathrm{Sp}}_S}(f_{!}X,i_{S}Y)$$

$$={\mathrm{Hom}}_{{\mathrm{Sp}}_T}(X,i_T{f}^{\ast }Y)$$

$$={\mathrm{Hom}}_{{\mathrm{Perf}}_T}(X^{\mathrm{copf}},f^{\ast }Y)$$

$$={\mathrm{Hom}}_{{\mathrm{Perf}}_S}(f_{!}{X}^{\mathrm{copf}},Y).$$

$$\mathscr{X}^{\operatorname{copf}/S}=\operatorname{colim}\big{(}\!\!\leavevmode\hbox to121.71pt{\vbox to18.35pt{\pgfpicture\makeatletter\hbox{\hskip 60.853pt\lower-9.22621pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-60.853pt}{-9.12637pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 8.05554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\mathscr{X}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 8.05554pt\hfil&\hfil\hskip 35.50484pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.43709pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ 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$$G_i:F_{S,\ast }^{i+1}\mathcal{M}\to F_{S,\ast }^i\mathcal{M}$$

$$G_i(x)\cdot =F_{T^{p^i/S}/S}^{\ast }x$$

$$\mathscr{M}^{\operatorname{pf}/S}=\lim\big{(}\!\!\leavevmode\hbox to166.39pt{\vbox to19.06pt{\pgfpicture\makeatletter\hbox{\hskip 83.1944pt\lower-9.57953pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-83.1944pt}{-9.47969pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 8.05554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\dots}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 8.05554pt\hfil&\hfil\hskip 38.81554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.74779pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\operatorname{F}_{S,*}^{2}\mathscr{M}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 16.05333pt\hfil&\hfil\hskip 38.81554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.74779pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\operatorname{F}_{S,*}\mathscr{M}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 16.05333pt\hfil&\hfil\hskip 31.6511pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ 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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.5852pt}{-3.33804pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{G_{1}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{42.85442pt}{-6.97969pt}\pgfsys@lineto{64.81667pt}{-6.97969pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{65.01665pt}{-6.97969pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{50.28366pt}{-3.33804pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{G_{0}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\!\!\big{)},$$

$$\mathcal{H}om(\mathcal{X},{\mathcal{M}}^{\mathrm{pf}})=\mathcal{H}om({\mathcal{X}}^{\mathrm{copf}},{\mathcal{M}}^{\mathrm{pf}})=\mathcal{H}om({\mathcal{X}}^{\mathrm{copf}},\mathcal{M}).$$

$${\mathcal{M}}^{\mathrm{pf}}(T)$$

$$=\lim \mathcal{H}om(T,F_{S,\ast }^i\mathcal{M})=\mathcal{H}om(T,\lim F_{S,\ast }^i\mathcal{M})=(\lim F_{S,\ast }^i\mathcal{M})(T).$$

$$\underline{\mathrm{Hom}}\left(x,y\right)\stackrel{\sim }{\to }{\mathrm{Fdiv}}_S(\underline{\mathrm{Hom}}\left(x_0,y_0\right))$$

$$\underline{\mathrm{Hom}}(x,y)⟶\underline{\mathrm{Hom}}(x_0,y_0)$$

$$\underline{\mathrm{Hom}}\left(x,y\right)$$

$$\underline{\mathrm{Hom}}\left(x_0,y_0\right)$$

$${\text{EtAff}}_S$$

$$E_1$$

$$♯\in \{\varnothing ,\mathrm{sep},\mathrm{aff},\mathrm{surj},\mathrm{dom}\}.$$

$${\mathcal{O}}_{S,s}$$

Full text

1. 1 Introduction

2. 2 Perfection and coperfection

3. 2.1 Categorical definitions

4. 2.2 Base restriction

5. 2.3 The case of stacks; $\operatorname{F}$-divided objects

6. 2.4 Perfect algebraic stacks

7. 3 Étale hulls and connected components

8. 3.1 Étale affine hulls and largest étale subalgebras

9. 3.2 Affine hull of the space of components

10. 3.3 Computing the space of components

11. 4 Perfection of algebras

12. 4.1 Base change in preperfection

13. 4.2 Perfection over artinian rings

14. 4.3 Preperfection over noetherian rings

15. 4.4 Perfection over regular or unibranch one-dimensional rings

16. 4.5 Examples

17. 5 Unramified $\operatorname{F}$-divided objects and the étale fundamental pro-groupoid

18. 5.1 The case of algebraic spaces

19. 5.2 The étale fundamental pro-groupoid

1 Introduction

** 1.1**

Motivation.

Using Cartier’s Theorem on the descent of vector bundles under Frobenius, Gieseker and Katz were able to give another viewpoint on stratified vector bundles on a smooth variety of characteristic $p$. Namely, they showed that these objects are equivalent to $\operatorname{F}$-divided vector bundles, that is, sequences $\{E_{i}\}_{i\geqslant 0}$ of vector bundles and isomorphisms $E_{i}\simeq\operatorname{F}^{*}E_{i+1}$ where $\operatorname{F}$ is the Frobenius endomorphism of the variety, see [Gi75]. Since then, these have occupied an important place in the research on vector bundles in characteristic $p$. Looking only at the recent literature, we can mention the works of dos Santos [DS07], [DS11], Esnault and Mehta [EM10], Berthelot [Be12], Tonini and Zhang [TZ17].

More generally, one can expect that in the study of curves, or morphisms, or torsors (etc) in characteristic $p$, the $\operatorname{F}$-divided curves, morphisms, torsors (etc) are natural objects which are likely to play an important role. In the present article, for any algebraic stack $\mathscr{M}$ we introduce the stack $\operatorname{Fdiv}(\mathscr{M})$ of $\operatorname{F}$-divided objects of $\mathscr{M}$ and we seek to understand it (see Remark 2.3.5 for a warning on notation). Note that $\operatorname{F}$-divided vector bundles correspond to the case where $\mathscr{M}$ is the classifying stack $B\operatorname{GL}_{n}$, a typical example of Artin stack with affine positive-dimensional inertia. In this article, we study the somehow opposite case where $\mathscr{M}$ is a Deligne-Mumford stack. In this case we call the objects of $\operatorname{Fdiv}(\mathscr{M})$ unramified $\operatorname{F}$-divided objects. Roughly speaking, our main result says that unramified $\operatorname{F}$-divided objects defined over geometrically reduced bases are quasi-isotrivial. In order to achieve this, we establish various results on the perfection of algebras, and on the coperfection of algebraic spaces and stacks, which have independent interest. Let us now give more precise statements.

** 1.2**

Quasi-isotriviality of unramified $\operatorname{F}$-divided objects.

Let $S$ be an algebraic space and $\mathscr{X}\to S$ an algebraic stack. Throughout the paper, we say that $\mathscr{X}\to S$ is separable if it has geometrically reduced fibres. When $\mathscr{X}\to S$ is flat, finitely presented and separable, we construct its étale fundamental pro-groupoid $\mathscr{X}\to\Pi_{1}(\mathscr{X}/S)$. This is a 2-pro-object of the 2-category of étale algebraic stacks, with coarse moduli space the space of connected components $\pi_{0}(\mathscr{X}/S)$, see [Rom11], seen as a constant 2-pro-object. When $S$ is the spectrum of a field $k$ and $\mathscr{X}$ is geometrically connected, the étale fundamental pro-groupoid $\Pi_{1}(\mathscr{X}/S)$ is representable in the 2-category of stacks by the étale fundamental gerbe $\Pi^{\operatorname{\acute{e}t}}_{\mathscr{X}/k}$ of Borne and Vistoli [BV15, § 8]. In characteristic $p$, étale morphisms are perfect and it follows that the natural map $\operatorname{Fdiv}(\Pi_{1}(\mathscr{X}/S))\to\Pi_{1}(\mathscr{X}/S)$ is an isomorphism.

Theorem A. Let $S$ be a noetherian algebraic space of characteristic $p$. Let $\mathscr{X}\to S$ be a flat, finitely presented, separable algebraic stack. Let $\mathscr{M}\to S$ be a quasi-separated Deligne-Mumford stack. Then by applying $\operatorname{Fdiv}$ and precomposing with $\mathscr{X}\to\Pi_{1}(\mathscr{X}/S)$, we obtain an isomorphism

[TABLE]

between the stacks of morphisms of pro-Deligne-Mumford stacks (with $\mathscr{M}$ seen as a constant 2-pro-object) on the source, and morphisms of stacks on the target. This isomorphism is functorial in $\mathscr{X}$ and $\mathscr{M}$.

See LABEL:theo:coperfection_stacks. Intuitively, this means that any $\operatorname{F}$-divided object of $\mathscr{M}$ over the base $\mathscr{X}$ becomes constant after étale surjective base changes on $S$ and on $\mathscr{X}$, i.e. is quasi-isotrivial in a suitable sense. Here is a simple illustration. Let us assume that $X$ is a connected, simply connected variety over a separably closed field $k$. Then Theorem A implies that all $\operatorname{F}$-divided families $C\to X$ of stable $n$-pointed curves of genus $g$ with $2g-2+n>0$ are constant. The same assertion with vector bundles replacing curves is the (almost exact) analogue of Gieseker’s conjecture, proved by Esnault and Mehta [EM10]. However, Esnault and Mehta’s situation and ours are different in nature. In fact, in loc. cit. as well as in our work, the approach has two comparable steps. First one uses the fact that objects are described by a morphism from a suitable fundamental group(oid) scheme $\Pi$ (the étale fundamental pro-groupoid for us, and the stratified fundamental group scheme in [EM10]). Second one proves that under the given assumptions, the group scheme $\Pi$ vanishes. The crucial difference is that in our setting, the first step is the difficult part of the argument and the second step is almost trivial, while for Esnault and Mehta the first step is easy and the second step is where all the effort lies.

If contemplated with a focus on $\mathscr{X}$, Theorem A gives information on its coperfection. The viewpoint being substantially different, it is worth giving the corresponding version of the statement. For this we denote by $\mathscr{X}^{p^{i}/S}$ the $i$-th Frobenius twist of $\mathscr{X}/S$ and

[TABLE]

the relative Frobenius morphism.

Theorem A’. Let $S$ be a noetherian algebraic space of characteristic $p$.

• *  *(1)

Let $X\to S$ be a flat, finitely presented, separable morphism of algebraic spaces. The inductive system of relative Frobenii

[TABLE]

admits a colimit in the category of algebraic spaces over $S$. This colimit is the algebraic space of connected components $\pi_{0}(X/S)$; it is a coperfection of $X\to S$.

• *  *(2)

Let $\mathscr{X}\to S$ be a flat, finitely presented, separable algebraic stack. The inductive system of relative Frobenii

[TABLE]

admits a 2-colimit in the 2-category of pro-quasi-separated Deligne-Mumford stacks over $S$. This 2-colimit is the pro-étale stack $\Pi_{1}(\mathscr{X}/S)$; it is a 2-coperfection of $\mathscr{X}/S$ in the 2-category of pro-Deligne-Mumford stacks.

See Remarks 5.1.2 and LABEL:rem:on_coperfection_2. Statement (2) is equivalent to Theorem A as explained in Remark 2.3.3. Note that (2) includes (1) as a special case, because $\Pi_{1}(\mathscr{X}/S)$ has coarse moduli space $\pi_{0}(\mathscr{X}/S)$. We include (1) for emphasis and also because the proof actually proceeds by deducing (2) from (1).

Theorem A’ seems to suggest that taking coperfection in the higher category of pro-Deligne-Mumford $n$-stacks would eventually recover the whole relative étale homotopy type of $X\to S$. We plan to investigate this eventuality in a future article.

** 1.3**

Perfection of algebras; largest étale subalgebras.

Within the category of algebras, the situation is somehow more subtle. Given a characteristic $p$ ring $R$ and an algebra $R\to A$, denote by

[TABLE]

the relative Frobenius of $A^{p^{i}/R}$, the $i$-th Frobenius twist of $A$. Define the preperfection of $A/R$:

[TABLE]

The name is explained by a surprising fact: the algebra $A^{p^{\infty}/R}$ is not perfect in general, even if $R\to A$ is flat, finitely presented and separable. We give an example of this with $R$ equal to the local ring of a nodal curve singularity (see 4.5.2). In our example the double preperfection is perfect but we do not know if iterated preperfections should converge to a perfect algebra in general. In the affine case $S=\operatorname{Spec}(R)$ and $X=\operatorname{Spec}(A)$, we write $\pi_{0}(A/R)$ instead of $\pi_{0}(X/S)$. What Theorem A’ implies in this case is that there is an isomorphism of $R$-algebras:

[TABLE]

Here ${\cal O}(-)$ is the functor of global functions. Given the bad properties of the rings under consideration, this could not really be anticipated: indeed, in general ${\cal O}(\pi_{0}(A/R))$ is not étale and $A^{p^{\infty}/R}$ is not perfect. Although we present the above isomorphism of $R$-algebras as a corollary to Theorem A’, the structure of the proof is actually to first establish this isomorphism of algebras (see 4.3.2) and then deduce the geometric statement for spaces and stacks (Theorem A’).

This begs for a further study of perfection of algebras. Our general expectation is that for algebras of finite type, there should exist a largest étale subalgebra and this should be (at least close to) the perfection of $R\to A$. In striving to materialize this picture, we study étale hulls in more detail. We take up recent work of Ferrand [Fe19] and prove the following result which is not special to characteristic $p$.

Theorem B. Let $S$ be a noetherian, geometrically unibranch algebraic space without embedded points. Let $f:X\to S$ be a faithfully flat, finitely presented morphism of algebraic spaces.

• *  *(1)

The category of factorizations $X\to E\to S$ such that $X\to E$ is a schematically dominant morphism of algebraic spaces and $E\to S$ is étale and affine is a lattice, that is, any two objects have a supremum and an infimum (for the obvious relation of domination). Moreover it has a largest element $\pi^{a}(X/S)$.

• *  *(2)

The functor $X\mapsto\pi^{a}(X/S)$ is left adjoint to the inclusion of the category of étale, affine $S$-schemes into the category of faithfully flat, finitely presented $S$-algebraic spaces.

See Theorem 3.1.7 and Corollary 3.1.9. The largest element $\pi^{a}(X/S)$ is the relative spectrum of a sheaf of ${\cal O}_{S}$-algebras which is the largest étale subalgebra of $f_{*}{\cal O}_{X}$. It is called the étale affine hull of $X\to S$. When $S$ is artinian or $X\to S$ is separable, the functor $\pi_{0}(X/S)$ is an étale algebraic space and we have morphisms:

[TABLE]

We can take advantage of this to analyze perfection of algebras in characteristic $p$. When $S=\operatorname{Spec}(R)$ and $X=\operatorname{Spec}(A)$, the largest étale subalgebra is written $A^{\operatorname{\acute{e}t}/R}\subset A$, that is $\pi^{a}(A/R)=\operatorname{Spec}(A^{\operatorname{\acute{e}t}/R})$. We then obtain the following positive results.

Theorem C. Let $R\to A$ be a flat, finite type morphism of noetherian rings of characteristic $p$. Assume that one of the following holds:

• *  *(1)

$R$* is artinian,*

• *  *(2)

$R$* is geometrically $\mathbb{Q}$-factorial (e.g. regular) and $R\to A$ is separable,*

• *  *(2)

$R$* is one-dimensional, reduced, geometrically unibranch, and $R\to A$ is separable.*

Then the natural maps give rise to isomorphisms:

[TABLE]

See Theorem 4.2.1, Proposition 3.2.2 and Corollary 4.4.1.

** 1.4**

Overview of the paper and notations.

Each section starts with a small description of contents, where the reader will find more detail. In Section 2 we give definitions and basic facts on perfect stacks, perfection and coperfection. In Section 3 which makes no assumption on the characteristic, we give complements on the functor $\pi_{0}$. We study factorizations through an étale affine scheme, and we prove Theorem B. Finally we prove two pushout results that allow to view $\pi_{0}(X/S)$ as glued from simpler pieces (the simpler pieces being either $\pi_{0}$ of an atlas or a completion from a closed fibre), to be used in the last two sections. In Section 4 where we study the commutative algebra of perfection, proving the results summarized in Theorem C. Finally in Section 5 we prove Theorems A and A’, first for algebraic spaces and then for algebraic stacks.

All sheaves and stacks are considered for the fppf topology unless explicitly stated otherwise. We denote sets, sheaves and stacks of homomorphisms by the symbols $\operatorname{Hom}$, $\underline{\operatorname{Hom}}$ and $\mathscr{H}\!om$ respectively.

** 1.5**

Acknowledgements.

We express warm thanks to Daniel Ferrand for his detailed reading of Section 3 and his successful efforts to dissuade us from trying to prove the existence of the étale affine hull in exceeding generality. We are grateful to Fabio Tonini and Lei Zhang for enlightening discussions on theétale fundamental pro-groupoid. We also thank Niels Borne, Johann Haas and Angelo Vistoli for kind answers to our questions. The three authors are supported by the Centre Henri Lebesgue, program ANR-11-LABX-0020-01 and would like to thank the executive and administrative staff of IRMAR and of the Centre Henri Lebesgue for creating an attractive mathematical environment.

2 Perfection and coperfection

Throughout this section, we let $S$ be an algebraic space of characteristic $p$. Our purpose is to make some preliminary remarks on perfection and coperfection: definitions and formal properties (2.1 and 2.2), description in the 2-category of stacks (2.3), and structure of perfect algebraic stacks (2.4).

There is unfortunately no uniform use of the word “perfection” in the literature. Our convention is to call perfection resp. coperfection the right adjoint, resp. the left adjoint, to the inclusion of the full subcategory of perfect objects in the ambient category. This choice is prompted by the fact that in most cases of existence, the construction of perfections uses limits while the construction of coperfections uses colimits. For example, this is the way one can form the perfection $A^{\operatorname{pf}}$ and the coperfection $A^{\operatorname{copf}}$ of an $\mathbb{F}_{p}$-algebra $A$ with absolute Frobenius $\operatorname{F}_{\!A}$:

[TABLE]

We emphasize that our interest is in perfection of algebras, and coperfection of algebraic spaces and stacks. This means that our setting is relative (over a possibly imperfect base) and geometric (with schemes, spaces and stacks). Both features introduce difficulties; we do not know if perfection of algebras and coperfection of algebraic spaces and stacks exist in full generality.

2.1 Categorical definitions

** 2.1.1**

Frobenius, perfect objects. Let $f:X\to S$ be a fibred category over $S$ and let $X^{p/S}=X\times_{S,\operatorname{F}_{S}}S$ be its Frobenius twist. The absolute Frobenius is the functor $\operatorname{F}_{X}:X\to X$ defined by $\operatorname{F}_{X}(x)=\operatorname{F}_{T}^{*}x$, for all $T/S$ and $x\in X(T)$. The relative Frobenius is the functor \operatorname{F}_{X/S}\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=(\operatorname{F}_{X},f):X\to X^{p/S}. Note that $\operatorname{F}_{X}$ is not a morphism of fibred categories over $S$ while $\operatorname{F}_{X/S}$ is. We say that $X\to S$ is perfect if $\operatorname{F}_{X/S}$ is an isomorphism of fibred categories.

** 2.1.2**

Perfection and coperfection. Let $C$ be a fibred 2-category over $S$ whose objects are fibred categories of the type just discussed. We write $\mathscr{H}\!om_{C}(X,Y)$ the categories of morphisms in $C$, and $\operatorname{Hom}_{C}(X,Y)$ the object sets of the latter. The objects $X\in C$ which are perfect form a full 2-subcategory $\operatorname{\bf Perf}(C)$ whose inclusion we denote $i:\operatorname{\bf Perf}(C)\to C$. Now let $X\in C$ be any object. If the functor $\operatorname{\bf Perf}(C)\to\operatorname{\bf Cat}$, $P\mapsto\mathscr{H}\!om_{C}(iP,X)$ is 2-representable then we call the representing object the 2-perfection of $X$ and denote it $X^{\operatorname{pf}}$. If the functor $\operatorname{\bf Perf}(C)^{\circ}\to\operatorname{\bf Cat}$, $P\mapsto\mathscr{H}\!om_{C}(X,iP)$ is 2-representable then we call the representing object the 2-coperfection of $X$ and denote it $X^{\operatorname{copf}}$. We often simply that perfection and coperfection for simplicity. Hence, if all objects have perfections (resp. coperfections) then the functor $X\to X^{\operatorname{pf}}$ (resp. the functor $X\to X^{\operatorname{copf}}$) is right (resp. left) adjoint to the inclusion $i$. Note that if a given $X$ of interest may be seen as an object of different fibred 2-categories $C$ and $C^{\prime}$, then its hypothetical perfections in $C$ and $C^{\prime}$ differ in general, and similarly for its hypothetical coperfections.

** 2.1.3**

Cofibred setting. While algebraic spaces and stacks and the 2-categories that contain them fall under the scope of the “fibred” categorical setting, algebras and the categories that contain them live in the “cofibred” categorical setting. The cofibred analogues of the notions just presented exist with the obvious modifications; notably, for a cofibred category $A\to S$, the relative Frobenius is a functor $\operatorname{F}_{A/S}:A^{p/S}\to A$. In this setting, perfection (resp. coperfection) is again defined as the right (resp. left) adjoint of the inclusion of perfect objects.

** 2.1.4**

Formal properties of Frobenius.

If $f:X\to Y$ is a morphism of fibred categories over $S$, we can define X^{p/Y}\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=X\times_{Y,\operatorname{F}_{Y}}Y and relative Frobenius \operatorname{F}_{X/Y}\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=(\operatorname{F}_{X},f):X\to X^{p/Y}. We say that $f$ is (relatively) perfect if $\operatorname{F}_{X/Y}$ is an isomorphism. If $g:Y\to Z$ is another morphism of fibred categories over $S$, we have $\operatorname{F}_{X/Z}=(f^{p/Z})^{*}\operatorname{F}_{Y/Z}\circ\operatorname{F}_{X/Y}$ as one can see from the diagram with cartesian squares:

[TABLE]

Using these remarks, one checks the following facts:

• (i)

perfect morphisms are stable by base change;

• (ii)

perfect morphisms are stable by composition;

• (iii)

morphisms between fibred categories perfect over $Z$ are perfect over $Z$;

• (iv)

if $X\to Y$, $X\to Z$ are perfect and $f^{p/Z}$ descends isomorphisms (e.g. $f$ is a perfect and faithfully flat quasi-compact morphism of algebraic stacks), then $Y\to Z$ is perfect.

2.2 Base restriction

For the sake of simplicity, let us come back to algebraic spaces. Let $f:S^{\prime}\to S$ be a morphism of algebraic spaces. The base restriction along $f$ is the functor that sends an $S^{\prime}$-algebraic space $X^{\prime}$ to the $S$-algebraic space $X^{\prime}\to S^{\prime}\to S$. We denote by $f_{!}X^{\prime}$ the base restriction. The functor $f_{!}$ is left adjoint to the pullback $f^{*}$. It should not be confused with the Weil restriction functor $f_{*}$ which is right adjoint to $f^{*}$. We will need to use the fact that coperfection commutes with base restriction. This is a consequence of the simple categorical fact that if two functors commute and have left adjoints, then the left adjoints commute. Here is a precise statement in our context.

** 2.2.1**

Lemma.* Let $X,T,S$ be $\mathbb{F}_{p}$-algebraic spaces. Let $f:T\to S$ be a morphism which is relatively perfect, and $X\to T$ a morphism which admits a coperfection $X^{\operatorname{copf}}$. Then $f_{!}(X^{\operatorname{copf}})$ is a coperfection for $f_{!}X$. In a formula, we obtain an isomorphism:*

[TABLE]

Proof :

Let $\operatorname{\bf Sp}_{S}$ be the category of $S$-algebraic spaces, and $i_{S}:\operatorname{\bf Perf}_{S}\to\operatorname{\bf Sp}_{S}$ the inclusion of perfect objects. Since $f:T\to S$ is relatively perfect and relatively perfect morphisms are stable by composition, the functor $f_{!}$ maps $\operatorname{\bf Perf}_{T}$ into $\operatorname{\bf Perf}_{S}$, that is, it commutes with $i_{S}$ and $i_{T}$. Similarly $f^{*}$ maps $\operatorname{\bf Perf}_{S}$ into $\operatorname{\bf Perf}_{T}$. For each $Y\in\operatorname{\bf Perf}_{S}$ we have canonical bijections:

[TABLE]

This shows that $f_{!}X^{\operatorname{copf}}$ is the coperfection of $f_{!}X$. $\square$

The same result holds, with the same proof, for pairs of commuting adjoints in similar situations.

2.3 The case of stacks; $\operatorname{F}$-divided objects

In this section we describe concretely the perfection and coperfection of fppf stacks over $S$, and highlight some properties. As we said in the introduction, all sheaves and stacks are considered for the fppf topology so most of the time we omit the adjective.

** 2.3.1**

Coperfection of stacks.

Let $\mathscr{X}$ be a stack over $S$. We let

[TABLE]

be the colimit in the 2-category of stacks. The inductive system being filtered, the prestack colimit satisfies the stack property for coverings of affine schemes $\operatorname{Spec}(A^{\prime})\to\operatorname{Spec}(A)$, and its Zariski stackification is an fppf stack, hence is the fppf stackification. One checks the following facts:

• (i)

$\mathscr{X}^{\operatorname{copf}/S}$ is perfect and is a coperfection of $\mathscr{X}$ in the 2-category of $S$-stacks;

• (ii)

the formation of $\mathscr{X}^{\operatorname{copf}/S}$ commutes with all base changes $S^{\prime}\to S$ and is functorial in $\mathscr{X}$;

• (iii)

$\mathscr{X}^{\operatorname{copf}/S}$ is locally of finite presentation (that is, limit-preserving) if $\mathscr{X}$ is;

• (iv)

if $\mathscr{X}$ is an algebraic stack, then $\mathscr{X}^{\operatorname{copf}}$ is far from algebraic in general. For example if $\mathscr{X}$ is the affine line over $\mathbb{F}_{p}$ then for an $\mathbb{F}_{p}$-algebra $A$, the set $\mathscr{X}^{\operatorname{copf}}(A)$ is equal to $A^{\operatorname{copf}/\mathbb{F}_{p}}$, the absolute coperfection of $A$. In particular, for $A=\mathbb{F}_{p}[[t]]$ the set $\mathscr{X}^{\operatorname{copf}}(A)=\mathbb{F}_{p}[[t^{p^{-\infty}}]]$ is much bigger than $\lim\mathscr{X}^{\operatorname{copf}}(A/t^{n})=\mathbb{F}_{p}$.

** 2.3.2**

Perfection of stacks; $\operatorname{F}$-divided objects. Let $\mathscr{M}$ be a stack over $S$. For each $i\geqslant 0$ let $\operatorname{F}_{S,*}^{i}$ be the Weil restriction along the $i$-th absolute Frobenius of $S$, and

[TABLE]

the morphism which maps a $T$-valued object $x\in\mathscr{M}(T^{p^{i+1}/S})=(\operatorname{F}_{S,*}^{i+1}\mathscr{M})(T)$ to the pullback

[TABLE]

under the Frobenius $\operatorname{F}=\operatorname{F}_{T^{p^{i}/S}/S}:T^{p^{i}/S}\to T^{p^{i+1}/S}$. Then we define:

[TABLE]

the limit being taken in the 2-category of stacks. One has the following facts:

• (i)

$\mathscr{M}^{\operatorname{pf}/S}$ is perfect and is a perfection of $\mathscr{X}$ in the 2-category of $S$-stacks;

• (ii)

the formation of $\mathscr{M}^{\operatorname{pf}/S}$ commutes with all base changes $S^{\prime}\to S$ and is functorial in $\mathscr{M}$;

• (iii)

$\mathscr{M}^{\operatorname{pf}/S}$ is not locally of finite presentation in general, even if $\mathscr{M}$ is;

• (iv)

assume that $\operatorname{F}_{S}:S\to S$ is finite locally free. If $\mathscr{M}$ is a Deligne-Mumford stack, then $\mathscr{M}^{\operatorname{pf}/S}$ also. For schemes, this is proven in Kato [Ka86], Prop. 1.4. In general, one uses the fact that the diagonal being unramified, its relative Frobenius is a monomorphism, hence the transitions $G_{i}:\operatorname{F}_{S,*}^{i+1}\mathscr{M}\to\operatorname{F}_{S,*}^{i}\mathscr{M}$ are representable affine morphisms. For Artin stacks, the same argument proves that the diagonal of $\mathscr{M}^{\operatorname{pf}/S}$ is representable by algebraic spaces, but in general it is not locally of finite type and $\mathscr{M}^{\operatorname{pf}/S}$ is not algebraic. For instance, in the case of $\mathscr{M}=B\mathbb{G}_{m}$ over $S=\operatorname{Spec}(\mathbb{F}_{p})$ the diagonal is a torsor under $\mu_{p^{\infty}}=\lim\mu_{p^{i}}$. Finally if $\operatorname{F}_{S}$ is not finite locally free, then already the diagonal may fail to be representable;

• (v)

If $\mathscr{M}^{\prime}\to\mathscr{M}$ is perfect, the natural morphism $\mathscr{M}^{\prime\operatorname{pf}/S}\to\mathscr{M}^{\operatorname{pf}/S}\times_{\mathscr{M}}\mathscr{M}^{\prime}$ is an isomorphism of stacks.

** 2.3.3**

Remark. For arbitrary $S$-stacks $\mathscr{X}$ and $\mathscr{M}$, we have canonical isomorphisms:

[TABLE]

This equality is what explains the dual interpretation of our result embodied by Theorems A and A’ in the introduction. Indeed, assume we have a satisfactory understanding of the above object as a bifunctor in $\mathscr{X}$ and $\mathscr{M}$. Then letting $\mathscr{X}$ vary we obtain a description of the perfection of $\mathscr{M}$, while letting $\mathscr{M}$ vary we obtain a description of the coperfection of $\mathscr{X}$. Going still further, since $T^{\operatorname{copf}}=\operatorname{colim}T^{p^{i}/S}$ we have

[TABLE]

This shows that once we know coperfection in the 2-category of stacks, the construction of the perfection is forced upon us.

The points of the stack $\mathscr{M}^{\operatorname{pf}/S}$ are exactly the $\operatorname{F}$-divided objects of $\mathscr{M}$. We want to give the latter an existence of their own, independent of the adjointness property.

** 2.3.4**

Definition. We denote by $\operatorname{Fdiv}_{S}(\mathscr{M})$ the stack described as follows.

• (1)

An $\operatorname{F}$-divided object of $\mathscr{M}$ over an $S$-scheme $T$ is a collection of pairs $(x_{i},\sigma_{i})_{i\geqslant 0}$ where $x_{i}\in\mathscr{M}(T^{p^{i}/S})$ and $\sigma_{i}:x_{i}\to\operatorname{F}^{*}x_{i+1}$ is an isomorphism; here $\operatorname{F}=\operatorname{F}_{T^{p^{i}/S}/S}:T^{p^{i}/S}\to T^{p^{i+1}/S}$ is Frobenius.

• (2)

A morphism between $(x_{i},\sigma_{i})_{i\geqslant 0}$ and $(y_{i},\tau_{i})_{i\geqslant 0}$ is a collection of morphisms $u_{i}:x_{i}\to y_{i}$ such that $\tau_{i}\circ u_{i}=\operatorname{F}^{*}u_{i+1}\circ\sigma_{i}$ for all $i\geqslant 0$.

To make things clear: $\operatorname{Fdiv}_{S}(\mathscr{M})$ and $\mathscr{M}^{\operatorname{pf}/S}$ are really two names for the same object.

** 2.3.5**

Remark. In most of the existing literature, e.g. [DS07], [TZ17], the notation $\operatorname{Fdiv}({\cal Z})$ is used for the category of $\operatorname{F}$-divided vector bundles on ${\cal Z}$. In Tonini and Zhang [TZ17], Def. 6.20, the notation is extended to the effect that $\operatorname{Fdiv}({\cal Z},{\cal Y})$ denotes the category of $\operatorname{F}$-divided objects of a stack ${\cal Y}$ over the base ${\cal Z}$. In the present paper, our emphasis is on the stack where divided objects take their values rather than the base that supports them. We are therefore led to drop ${\cal Z}$ from the notation, so that our $\operatorname{Fdiv}(\mathscr{M})$ is Tonini and Zhang’s $\operatorname{Fdiv}(-,\mathscr{M})$. We warn the reader that as a result, the notation $\operatorname{Fdiv}(\mathscr{M})$ does not have the same meaning in both works. Writing $\operatorname{\bf Vect}$ for the stack of vector bundles, the following table gives a summary of the correspondence of notations.

[TABLE]

We end this subsection with a lemma which is a consequence of the fact that the diagonal of the perfection is the perfection of the diagonal. This will be useful later in Section 5.

** 2.3.6**

Lemma.* Let $S$ be an algebraic space of characteristic $p$ and $\mathscr{Y}\to S$ a perfect stack. Let $\mathscr{M}$ be a stack and $f\colon\operatorname{Fdiv}_{S}(\mathscr{M})\to\mathscr{M}$ the perfection morphism. Let $x,y\colon\mathscr{Y}\to\operatorname{Fdiv}_{S}(\mathscr{M})$ be two morphisms, and write $x_{0},y_{0}\colon\mathscr{Y}\to\mathscr{M}$ for the compositions $fx$, $fy$. Then there is an isomorphism of sheaves on $S$*

[TABLE]

identifying the morphism

[TABLE]

with the $S$-perfection morphism.

Proof :

As $\mathscr{M}^{\operatorname{pf}/S}=\operatorname{Fdiv}_{S}(\mathscr{M})$ is defined as a limit, the formation of $\operatorname{Fdiv}_{S}$ commutes with products, and the natural equivalence \operatorname{Fdiv}_{S}(\mathscr{M})\times_{S}\operatorname{Fdiv}_{S}(\mathscr{M})\xrightarrow{\,\smash{\raisebox{-2.15277pt}{\scriptstyle\sim}}\,}\operatorname{Fdiv}_{S}(\mathscr{M}\times_{S}\mathscr{M}) identifies the diagonal $\Delta_{\operatorname{Fdiv}_{S}(\mathscr{M})}$ with $\operatorname{Fdiv}_{S}(\Delta_{\mathscr{M}})$. We have a 2-cartesian diagram of stacks on $S$:

[TABLE]

Because $\mathscr{Y}\to S$ is perfect, the morphism $\operatorname{Fdiv}_{S}(\mathscr{Y})\to\mathscr{Y}$ is an isomorphism of stacks. Applying $\operatorname{Fdiv}_{S}$ to the 2-cartesian diagram

[TABLE]

we obtain the desired isomorphism \underline{\operatorname{Hom}}\left(x,y\right)\xrightarrow{\,\smash{\raisebox{-2.15277pt}{\scriptstyle\sim}}\,}\operatorname{Fdiv}_{S}(\underline{\operatorname{Hom}}\left(x_{0},y_{0}\right)). $\square$

2.4 Perfect algebraic stacks

Perfect algebraic stacks have a very simple structure.

** 2.4.1**

Lemma.* Let $\mathscr{X}$ be an algebraic stack over $S$. Consider the following conditions:*

• *  *(1)

$\mathscr{X}$* is a perfect $S$-stack.*

• *  *(2)

There exists an étale, surjective morphism $U\to\mathscr{X}$ from a perfect $S$-algebraic space.

• *  *(3)

$\mathscr{X}$* is an étale gerbe over a perfect $S$-algebraic space.*

Then we have the implications (1) $\iff$ (2) $\Longleftarrow$ (3), and if the diagonal of $\mathscr{X}\to S$ is locally of finite presentation then all three conditions are equivalent. In particular, all perfect algebraic stacks are Deligne-Mumford.

To obtain an example of a perfect algebraic stack that does not satisfy (3), take a positive-dimensional scheme $X$ over a perfect field $k$ with a non-free action of a finite group $G$, and let $\mathscr{X}=[X^{\operatorname{pf}/k}/G]$.

Proof :

We use the facts collected in 2.1.4 without explicit mention.

(1) $\Rightarrow$ (2) If $\mathscr{X}\to S$ is perfect, then so is $\mathscr{X}\times_{S}\mathscr{X}\to S$ and hence also the diagonal $\Delta:\mathscr{X}\to\mathscr{X}\times_{S}\mathscr{X}$. In particular $\Delta$ is formally unramified. Being locally of finite type ([SP19], Tag 04XS), it is unramified in the sense of [Ra70] and [SP19]. It follows that $\mathscr{X}$ is Deligne-Mumford ([SP19], Tag 06N3). Let $U\to\mathscr{X}$ be an étale surjective morphism from an algebraic space; then $U\to\mathscr{X}$ is perfect and it follows that $U\to S$ is perfect.

(2) $\Rightarrow$ (1) By 2.1.4, if $U$ is perfect and $U\to\mathscr{X}$ is étale surjective then $\mathscr{X}$ is perfect.

(3) $\Rightarrow$ (1) This is clear because an étale gerbe is perfect.

(1) $\Rightarrow$ (3) If $\Delta$ is locally of finite presentation, it is formally étale hence étale. It follows that the inertia stack $I_{\mathscr{X}}\to\mathscr{X}$ is étale and therefore there is an algebraic space $X$ and an étale gerbe morphism $\mathscr{X}\to X$, see [SP19], Tag 06QJ. $\square$

3 Étale hulls and connected components

In this section, we provide some complements on the functor $\pi_{0}$ introduced in [Rom11]. Although these results hold for algebraic stacks, we restrict most of the time to algebraic spaces because this simplifies the treatment a little and is enough for our needs. There are two viewpoints on the functor $\pi_{0}$, and we consider both.

Firstly $\pi_{0}$ is a left adjoint to the inclusion of the category of étale quasi-compact spaces in the category of flat, finitely presented, separable spaces. In the study of such “étalification” functors, Ferrand [Fe19] recently highlighted the importance of the category of factorizations $X\to E\to S$ where the second arrow is étale. He proved that when the base $S$ has finitely many irreducible components, there is a left adjoint $\pi^{s}$ to the inclusion of étale, separated spaces into all flat, finitely presented spaces. In § 3.1 we prove that the category of factorizations as well as some interesting subcategories satisfy topological invariance (in the sense of [SGA4.2], Exp. VIII, Th. 1.1). Then we prove that when $S$ is noetherian, geometrically unibranch and without embedded points, there is a left adjoint $\pi^{a}$ to the inclusion of étale, affine spaces into all flat, finitely presented spaces. In § 3.2 we compare $\pi^{a}$ with the affine hull of $\pi_{0}$.

Secondly $\pi_{0}$ is the functor of connected components of a relative space. In § 3.3 we describe ways to compute $\pi_{0}(X/S)$ by using an atlas of $X$, or completing along a closed fibre of $X\to S$.

We sometimes impose some finiteness or regularity assumptions on the base $S$, but nothing on the characteristics; it is only in later sections that we specialize to characteristic $p$.

3.1 Étale affine hulls and largest étale subalgebras

Let us briefly recall what is know on étale hulls, also called étalification functors. Consider the following diagram of fully faithful subcategories of the category of $S$-spaces (“fp” stands for finitely presented):

[TABLE]

Here are some positive facts on the existence of these adjoints:

• (i)

$\pi_{0}$ is constructed in [Rom11]. It has a moduli description in terms of connected components. When $X\to S$ is flat, finitely presented, the functor $\pi_{0}(X/S)$ is representable by an algebraic space when either $X$ is separable, or $S$ is zero-dimensional, see [Rom11], 2.1.3. Its main properties (representability, adjointness, commutation with base change) hold with no assumption on $S$. The morphism $X\to\pi_{0}(X/S)$ is surjective with connected geometric fibres.

• (ii)

$\pi^{s}$ is constructed in [Fe19] when $S$ has finitely many irreducible components, and is not known to exist otherwise. It has no known moduli description. It has functoriality and base change properties available only in restricted cases. The morphism $X\to\pi^{s}(X/S)$ is surjective but its geometric fibres are usually not connected.

• (iii)

$\pi^{a}$ is constructed is the present subsection when $S$ is noetherian, geometrically unibranch, without embedded points. It shares the same features as those just listed for $\pi^{s}$, except that $X\to\pi^{a}(X/S)$ is schematically dominant but maybe not surjective.

Here are some negative facts:

• (iv)

$\pi$ is not known to exist unless $S$ is zero-dimensional (in which case $\pi=\pi_{0}$).

• (v)

$\pi_{0}$ does extend naturally to a functor $\textbf{Flat}_{S}\to\textbf{Et}_{S}$ but this is not a left adjoint to the inclusion $i:\textbf{Et}_{S}\to\textbf{Flat}_{S}$. Indeed [Rom11], 2.1.3 implies that for all flat, finitely presented $X\to S$ the functor $\pi_{0}(X/S)$ defined as an étale sheaf is constructible, hence an étale quasi-compact algebraic space. Moreover, for each étale $E\to S$ there is a map $\operatorname{Hom}(X,E)\to\operatorname{Hom}(\pi_{0}(X/S),E)$. However, in general there is no map in the other direction; in particular there is no morphism $X\to\pi_{0}(X/S)$ and this prevents $\pi_{0}$ from being an adjoint of $i$. For instance, let $S$ be the spectrum of a discrete valuation ring $R$ with fraction field $K$ and let $X=\operatorname{Spec}(R[x]/(x^{2}-\pi x))$. Then $\pi_{0}(X/S)\simeq\operatorname{Spec}(K)\sqcup\operatorname{Spec}(K)$ and the map $\pi_{0}(X/S)\to S$ is not even surjective.

We now start our investigations on $\pi^{a}$. To start with, we recall the definition of the category of factorizations from [Fe19]. In order to make Theorem 3.1.7 possible, we modify the definition slightly by relaxing the assumption of surjectivity.

** 3.1.1**

Definition. Let $X\to S$ be a morphism of algebraic spaces. The category of factorizations is the category ${\sf E}(X/S)$ whose objects are the factorizations $X\to E\to S$ such that $E\to S$ is étale, and whose morphisms are the commutative diagrams:

[TABLE]

The category ${\sf E}^{\operatorname{surj}}(X/S)$, resp. ${\sf E}^{\operatorname{dom}}(X/S)$ is the full subcategory of factorizations such that $X\to E$ is surjective, resp. schematically dominant. The category ${\sf E}^{\operatorname{sep}}(X/S)$, resp. ${\sf E}^{\operatorname{aff}}(X/S)$ is the full subcategory of factorizations such that $E\to S$ is separated, resp. affine. We write ${\sf E}^{\operatorname{aff},\operatorname{dom}}(X/S)={\sf E}^{\operatorname{aff}}(X/S)\cap{\sf E}^{\operatorname{dom}}(X/S)$ and similarly for other intersections.

We will often denote a factorization $X\to E\to S$ simply by using the letter $E$. We draw the attention of the reader to the fact that for the subcategories ${\sf E}^{\sharp}(X/S)$ defined above, the property “$\sharp$” applies either to $E\to S$ or to $X\to S$, depending on the case.

** 3.1.2**

Lemma.* Let $X\to S$ be a morphism of algebraic spaces $X\to S$. Let $f:S^{\prime}\to S$ be a morphism of spaces which is integral, radicial and surjective. Let $X^{\prime}=X\times_{S}S^{\prime}$.*

• *  *(1)

The pullback functor $f^{*}:{\sf E}(X/S)\to{\sf E}(X^{\prime}/S^{\prime})$ is an equivalence which preserves the subcategories ${\sf E}^{\operatorname{sep}}$, ${\sf E}^{\operatorname{aff}}$ and ${\sf E}^{\operatorname{surj}}$.

• *  *(2)

If moreover $S,S^{\prime}$ are locally noetherian, $f$ induces a bijection $\operatorname{Emb}(S^{\prime})\to\operatorname{Emb}(S)$ of embedded points, and $X\to S$ is faithfully flat, then $f^{*}$ preserves also the subcategory ${\sf E}^{\operatorname{dom}}$.

Proof :

First, we recall basic facts on the topological invariance of the étale site. Let $f:S^{\prime}\to S$ be a morphism of algebraic spaces which is integral, radicial and surjective. Then the pullback functor $f^{*}$ induces an equivalence between the category of étale $S$-spaces and the category of étale $S^{\prime}$-spaces: see [SGA4.2], Exp. VIII for schemes and [SP19], Tag 05ZG for spaces. This equivalence preserves affine objects, see [SP19], Tag 07VW.

(1) We prove that $f^{*}$ is essentially surjective. Let $X^{\prime}\to E^{\prime}\to S^{\prime}$ be a factorization. By topological invariance of the étale site, there exists an essentially unique $E\to S$ such that $E^{\prime}\simeq E\times_{S}S^{\prime}$. In order to descend $u^{\prime}:X^{\prime}\to E^{\prime}$ to a morphism $u:X\to E$, by descent of morphisms to an étale scheme along universal submersions ([SGA1], Exp. IX, prop. 3.2) it is enough to prove that $\operatorname{pr}_{1}^{*}u^{\prime}=\operatorname{pr}_{2}^{*}u^{\prime}$ where $\operatorname{pr}_{1},\operatorname{pr}_{2}:S^{\prime}\times_{S}S^{\prime}\to S^{\prime}$ are the projections. By [SGA1], Exp. IX, prop. 3.1 it is enough to find a surjective morphism $g:S^{\prime\prime\prime}\to S^{\prime}\times_{S}S^{\prime}$ such that the two maps agree after base change along $g$. We can take $S^{\prime\prime\prime}=S^{\prime}$ and $g$ the diagonal map. This proves essential surjectivity; we leave full faithfulness to the reader. We now prove that $f^{*}$ preserves the indicated subcategories. Since the diagonal of $E\to S$ is a closed immersion if and only if the diagonal of $E^{\prime}\to S^{\prime}$ is a closed immersion, we see that $f^{*}$ preserves ${\sf E}^{\operatorname{sep}}(X/S)$. The fact that $f^{*}$ preserves ${\sf E}^{\operatorname{aff}}$ was recalled above. Finally $f^{*}$ preserves ${\sf E}^{\operatorname{surj}}$ because $f$ is a universal homeomorphism.

(2) Here the morphisms $X\to E$ in the factorizations are automatically flat. Thus such a morphism is schematically dominant if and only if its image contains the set of associated points $\operatorname{Ass}(E)$. Since $\operatorname{Ass}(E)=\cup_{s\in\operatorname{Ass}(S)}E_{s}$ by [EGA] IV.3.3.1, we see that $X\to E$ is schematically dominant if and only if the image of $X\to E$ contains all fibres $E_{s}$ with $s\in\operatorname{Ass}(S)$. But $f$ induces a bijection of the non-embedded associated points since it is a homeomorphism, and a bijection on embedded points by assumption. Hence it is equivalent to say that the image of $X^{\prime}\to E^{\prime}$ contains all fibres $E^{\prime}_{s^{\prime}}$ with $s^{\prime}\in\operatorname{Ass}(S^{\prime})$. $\square$

** 3.1.3**

Suprema and infima. We say that $E_{1}$ and $E_{2}$ have a supremum if the category of factorizations $E$ mapping to $E_{1}$ and $E_{2}$ has a terminal element. We say that $E_{1}$ and $E_{2}$ have a infimum if the category of factorizations $E$ receiving maps from $E_{1}$ and $E_{2}$ has an initial element. In pictures:

[TABLE]

Note that in the three categories ${\sf E}^{\operatorname{surj}}(X/S)$, ${\sf E}^{\operatorname{sep},\operatorname{dom}}(X/S)$ and ${\sf E}^{\operatorname{aff},\operatorname{dom}}(X/S)$, if there is a morphism between $E_{1}$ and $E_{2}$ then it is unique. In other words, these categories really are posets.

** 3.1.4**

Corollary.* Let ${\sf E}^{\sharp}(X/S)\subset{\sf E}(X/S)$ be any subcategory with*

[TABLE]

Let $f:S^{\prime}\to S$ be a morphism of spaces which is integral, radicial and surjective. In case $\sharp=\operatorname{dom}$ assume moreover that $f$ and $X$ satisfy the assumptions of 3.1.2(2). Then the following hold.

• *  *(1)

${\sf E}^{\sharp}(X/S)$* has an initial element if and only if ${\sf E}^{\sharp}(X^{\prime}/S^{\prime})$ has one.*

• *  *(2)

Let $E_{1},E_{2}$ be factorizations in ${\sf E}^{\sharp}(X/S)$ and $E^{\prime}_{1},E^{\prime}_{2}$ their images in ${\sf E}^{\sharp}(X^{\prime}/S^{\prime})$. Then $E_{1},E_{2}$ have a supremum, resp. an infimum, if and only if $E^{\prime}_{1},E^{\prime}_{2}$ have a supremum, resp. an infimum.

Proof :

Suprema and infima are defined in terms of morphisms and are therefore preserved by the equivalences $f^{*}:{\sf E}^{\sharp}(X/S)\to{\sf E}^{\sharp}(X^{\prime}/S^{\prime})$. $\square$

We arrive at the main existence result of this subsection. We prepare the proof with two lemmas. The first is classical; the proof given here was suggested to us by Daniel Ferrand.

** 3.1.5**

Lemma.* Let $E\to S$ be an étale, quasi-compact, separated morphism of schemes. Then after an étale surjective base change $S^{\prime}\to S$, the $S$-scheme $E$ is a disjoint union of a finite number of open subschemes of $S$. If moreover $E\to S$ is surjective and birational, it is an isomorphism.*

Proof :

Since $E\to S$ is of finite presentation, we can assume that $S$ is affine noetherian. Let $m(E/S)$ be the maximum of the number of geometric connected components of the fibres of $E\to S$; this is finite by [EGA], IV3.9.7.8 and noetherian induction. The base change S^{\prime}_{1}\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=E\to S produces an open and closed section whose complement has $m$-number strictly less. By induction on $m$, we obtain a splitting of $E$ as a disjoint union of finitely many opens, as asserted. The second claim follows because assuming birationality, the number of opens has to be one. $\square$

** 3.1.6**

Lemma.* Let $S$ be a separated noetherian scheme, and $U\subset S$ a nonempty dense open. Then the set of opens $V$ containing $U$ and such that $V\to S$ is affine is finite and has a smallest element for inclusion.*

Proof :

If $V$ is such an open, the complement $S\setminus V$ is included in $S\setminus U$ and has pure codimension 1 in $S$ by [EGA] IV.21.12.7. This proves that $S\setminus V$ is a union of one-codimensional irreducible components of $S\setminus U$. Since these are finite in number, we see the set of interest is finite. Since $S$ is separated, the intersection of all its elements is again $S$-affine and is the smallest element. $\square$

** 3.1.7**

Theorem.* Let $f:X\to S$ be a faithfully flat, finitely presented morphism of algebraic spaces. Assume that $S$ is noetherian, geometrically unibranch, without embedded points. Then the category ${\sf E}^{\operatorname{aff},\operatorname{dom}}(X/S)$ is a lattice, that is, any two objects have a supremum and an infimum. Moreover ${\sf E}^{\operatorname{aff},\operatorname{dom}}(X/S)$ has a largest element.*

A similar statement holds in the category ${\sf E}^{\operatorname{surj},\operatorname{sep}}(X/S)$ where existence of suprema and maximum are due to Ferrand [Fe19].

Proof :

Throughout the proof we write ${\sf E}={\sf E}^{\operatorname{aff},\operatorname{dom}}(X/S)$. Note that for each factorization $X\to E\to S$, the morphism $X\to E$ is flat and finitely presented.

We start with the proof that any two factorizations $E_{1},E_{2}\in{\sf E}$ have a supremum. By topological invariance of the étale site, we can assume that $S$ is reduced. Let $E$ be the schematic image of the morphism $X\to E_{1}\times_{S}E_{2}$. As a closed subscheme of $E_{1}\times_{S}E_{2}$, it is affine and unramified over $S$. By the theorem on unramified morphisms over unibranch schemes ([EGA], IV.18.10.1), it is enough to prove that for each $e\in E$ with image $s\in S$, the map of local rings ${\cal O}_{S,s}\to{\cal O}_{E,e}$ is injective. Let $\eta_{1},\dots,\eta_{n}$ be the associated points of $S$ and let ${\cal O}_{E,\eta_{i}}$ be the semi-local rings of the fibres of $E\to S$ at $\eta_{i}$. Like in the proof of 3.1.2, we have $\operatorname{Ass}(E)=E_{\eta_{1}}\cup\dots\cup E_{\eta_{n}}$. We have a commutative diagram:

[TABLE]

The left and right maps are injective. The bottom map is injective also because $E_{\eta_{i}}$ is in the image of $X\to E$ and $X\to S$ is faithfully flat. Therefore ${\cal O}_{S,s}\to{\cal O}_{E,e}$ is injective and this concludes the argument.

Now we prove that there is a largest element. For each $E\in{\sf E}$, the image of $X\to E$ is an open subscheme $U\subset E$, étale, separated, quasi-compact over $S$, which we call the “image” of the factorization $E$. It is determined by the scheme R\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=X\times_{E}X=X\times_{U}X which is the graph in $X\times_{S}X$ of an open and closed equivalence relation: indeed, we recover $U$ as the quotient algebraic space $X/R$. Because $S$ is noetherian, there are finitely many open and closed equivalence relations ([Fe19], 3.2.1, 3.2.2) hence finitely many “images” $U$. By the existence of suprema in ${\sf E}$, the poset of “images” forms a directed finite set, hence it has a largest element.

We fix $E\in{\sf E}$ whose “image” $U$ is largest. It is now enough to prove that the directed set of maps $u:F\to E$ in ${\sf E}$ has a largest element $u^{max}\colon E^{max}\to E$. Since ${\sf E}$ is a directed set, $E^{max}$ will automatically be a largest element for it, concluding the proof.

Given a map $u\colon E^{\prime}\to E$, we observe that there is an induced isomorphism $U^{\prime}\simeq U$ between the “images”. Moreover $U\subset E$ and $U^{\prime}\subset E^{\prime}$ are schematically dense in $E$. It follows that the induced étale surjective separated morphism from $E^{\prime}$ onto its image $u(E^{\prime})\subset E$ is birational, hence an isomorphism by Lemma 3.1.5. Since $E^{\prime}$ is affine over $S$, then so is $u(E^{\prime})$; hence Lemma 3.1.6 applied to the open $U\subset E$ implies that the directed set of maps $F\to E$ stabilizes, so eventually an $E^{max}$ is achieved.

Finally, we construct an infimum for $E_{1}$ and $E_{2}$. Let $E_{0}$ be the pushout of the diagram $E_{1}\leftarrow X\to E_{2}$, that is, the quotient of $E_{1}\sqcup E_{2}$ by the étale equivalence relation that identifies the image of $X\to E_{1}$ and the image of $X\to E_{2}$. Let $E$ be the largest element of the category ${\sf E}^{\operatorname{aff},\operatorname{dom}}(E_{0}/S)$. This is the infimum of $E_{1}$ and $E_{2}$. $\square$

** 3.1.8**

Definition. With the notations and assumptions of Theorem 3.1.7, the largest element of the poset ${\sf E}^{\operatorname{aff},\operatorname{dom}}(X/S)$ is called the étale affine hull of $X/S$ and denoted $\pi^{a}(X/S)$. Its ${\cal O}_{S}$-sheaf of functions is called the largest (quasi-coherent) étale ${\cal O}_{S}$-subalgebra of $f_{*}{\cal O}_{X}$.

Giving an existence proof which is more constructive than the one given above is not easy because of the limited formal properties of the étale affine hull (compatibility with base change, with the formation of products, etc). Such properties are of course essential in most situations where the etale affine hull is useful. A sample of base change results for the étale separated hull is given in [Fe19], § 7. Similar results can be proven for the étale affine hull.

** 3.1.9**

Corollary.* Let $S$ be a noetherian geometrically unibranch scheme without embedded points. Let $u:X\to Y$ be a morphism between faithfully flat, finitely presented $S$-algebraic spaces.*

• *  *(1)

There is an induced morphism of étale affine hulls $\pi^{a}(X/S)\to\pi^{a}(Y/S)$.

• *  *(2)

The functor $\pi^{a}$ is left adjoint to the inclusion of the category of étale, affine $S$-schemes into the category of faithfully flat, finitely presented $S$-algebraic spaces.

Proof :

(1) By topological invariance of the étale site (Lemma 3.1.2), we can assume that $S$ is reduced. Let $E$ be the schematic image of $X\to Y\to\pi^{a}(Y/S)$. It follows from the theorem on unramified morphisms over unibranch schemes ([EGA], IV.18.10.1) that $E\to S$ is étale. By the definition of $\pi^{a}(X/S)$ we obtain a morphism $\pi^{a}(X/S)\to\pi^{a}(Y/S)$.

(2) Let $u:X\to E$ be an $S$-morphism from a faithfully flat, finitely presented space to an étale, affine scheme. By (1) there is an induced morphism $\pi^{a}(X/S)\to\pi^{a}(E/S)$. Since $E\to\pi^{a}(E/S)$ is an isomorphism, we obtain a morphism $\pi^{a}(X/S)\to E$. $\square$

3.2 Affine hull of the space of components

Let $S$ be a noetherian scheme and $X\to S$ a flat separable morphism of finite type. A priori, there is no reason to expect that $\pi_{0}(X/S)^{\operatorname{aff}}\to S$, the affine hull of $\pi_{0}(X/S)\to S$, be étale. There are two reasons for this: the first, is that a priori $\pi_{0}(X/S)^{\operatorname{aff}}$ may not be of finite type. The second reason is that, even when it is of finite type, it may well be ramified. This may happen already over a dimension $1$ base with a nodal singularity, as Example 4.5.2 illustrates.

Here we describe a case where $\pi_{0}(X/S)^{\operatorname{aff}}$ is étale, for some geometrically unibranch reduced base schemes $S$. More precisely, in this situation the étale affine hull $\pi^{a}(X/S)\to S$ exists, and there is a natural map $\pi_{0}(X/S)^{\operatorname{aff}}\to\pi^{a}(X/S)$. We will prove that under some local factoriality-type conditions, this is an isomorphism.

** 3.2.1**

Definition. A noetherian local ring $R$ is called geometrically set-theoretically factorial if its strict henselization is integral, and each pure one-codimensional closed subscheme of $\operatorname{Spec}(R)$ has the same support as a principal closed subscheme.

Although a little ill-looking, this definition includes many examples of interest such as regular rings, $\mathbb{Q}$-factorial rings like the quadratic cone singularity $xy=z^{2}$, and all reduced unibranch curves. We note moreover that these examples are also $S_{2}$ and hence satisfy all the assumptions of the following statement.

** 3.2.2**

Proposition.* Let $X\to S$ be a morphism of algebraic spaces which is flat, separable, and finitely presented. Assume that $S$ is locally noetherian, $S_{2}$, with geometrically set-theoretically factorial local rings. Then the natural map $\pi_{0}(X/S)^{\operatorname{aff}}\to\pi^{a}(X/S)$ is an isomorphism.*

Proof :

It is enough to prove that $\pi_{0}(X/S)^{\operatorname{aff}}\to S$ is étale. We prove more generally that for all étale, quasi-compact algebraic spaces $E\to S$ the map $E^{\operatorname{aff}}\to S$ is étale. For this, we can work étale-locally on $S$. First let us see that we can reduce to the case where $E\to S$ is separated. By Ferrand [Fe19], Th. 3.2.1 there is an étale separated hull $\pi^{s}(E/S)\to S$. By [Fe19], Prop. 8.1.2 the map $E\to\pi^{s}(E/S)$ is initial among maps to separated schemes; note that Ferrand assumes normality of $S$ but really uses only the unibranch hypothesis (in loc. cit., this is said explicitly before Lemma 6.1.1 which is the key to Lemma 8.1.1). Since $E^{\operatorname{aff}}\to S$ is separated, we obtain a factorization $E\to\pi^{s}(E/S)\to E^{\operatorname{aff}}$. Taking global sections, the map ${\cal O}(E^{\operatorname{aff}})\to{\cal O}(\pi^{s}(E/S))\to{\cal O}(E)={\cal O}(E^{\operatorname{aff}})$ is the identity; since $E\to\pi^{s}(E/S)$ is dominant we see that $E$ has the same affine hull as $\pi^{s}(E/S)$. Hence replacing $E$ by $\pi^{s}(E/S)$ if necessary, we can assume that it is separated. By Lemma 3.1.5, working étale-locally around a fixed point $s\in S$ we can reduce to the case where $S$ is affine and $E$ is an open of $S$. Let us write the closed complement as Z\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=S\setminus E=Z_{1}\cup Z^{\prime} where $Z_{1}$ has pure codimension 1 in $S$ and $Z^{\prime}$ has codimension at least 2. By the assumption that the strictly local ring of $s$ is geometrically set-theoretically factorial, the 1-cycle $Z_{1}$ is set-theoretically principal on a small enough étale neighbourhood of $s$ in $S$. We replace $S$ by such a neighbourhood and let $f$ be a local equation for $Z_{1}$. Then the morphism ${\cal O}(S)\to{\cal O}(S\setminus Z_{1})$ is the localization-by-$f$ map which is étale. Since moreover $S$ has the $S_{2}$ property, the restriction ${\cal O}(S\setminus Z_{1})\to{\cal O}(S\setminus Z)={\cal O}(E)$ is an isomorphism. The result follows. $\square$

In the one-dimensional case, removing the unibranch condition in 3.2.2 yields a weaker result:

** 3.2.3**

Proposition.* Let $S$ be a reduced noetherian excellent scheme of dimension $\leqslant 1$. Let $X\to S$ be a flat separable morphism of finite presentation. Then $\pi_{0}(X/S)^{\operatorname{aff}}$ is quasi-finite.*

Proof :

Quasi-finiteness of $\pi_{0}(X/S)^{\operatorname{aff}}$ may be checked étale locally on $S$. So we let $s$ be a geometric point of $S$ and $(S^{\prime},s^{\prime})\to(S,s)$ an étale neighbourhood such that:

• i)

the irreducible components $S_{1},\ldots,S_{n}$ of $S^{\prime}$ are geometrically unibranch;

• ii)

for every $i\neq j$, $S_{i}\cap S_{j}=\{s^{\prime}\}$;

• iii)

the fibre $\pi_{0}(X/S)_{s^{\prime}}$ is a disjoint union of copies of $\operatorname{Spec}k(s^{\prime})$;

• iv)

$S^{\prime}=\operatorname{Spec}R^{\prime}$ is affine.

The reason why an étale neighbourhood satisfying condition i) exists, is that the regular locus of $S$ is open dense by excellence, hence so is the geometrically unibranch locus. So we may replace $S$ by $S^{\prime}$ and assume that $S=\operatorname{Spec}R$ satisfies the properties above.

Write $\pi=\pi_{0}(X/S)$. Then $\pi=\pi^{\prime}\sqcup\pi^{*}$, where $\pi^{*}$ is the union of those connected components that do not meet the fibre $\pi_{s^{\prime}}$. Then $\pi^{*}$ lives over $S^{\prime}\setminus\{s^{\prime}\}$ which by condition ii) is geometrically unibranch. By Proposition 3.2.2, the map $\pi^{\operatorname{aff}}\to S$ is étale, and in particular quasi-finite. It remains to check that $\pi^{\prime\operatorname{aff}}$ is quasi-finite.

Up to restricting $S$ by a further étale neighbourhood of $s$, we may assume that the isomorphism $\bigsqcup_{i=1}^{n}\operatorname{Spec}k(s)\to\pi^{\prime}_{s}$ extends to an open immersion $\alpha\colon\bigsqcup_{i=1}^{n}S\to\pi^{\prime}$. We claim that $\alpha$ has dense image. Indeed, let $Z$ be an irreducible component of $\pi^{\prime}$. Then $Z$ maps to some irreducible component $S_{i}$ of $S$. By assumption, $S_{i}$ is geometrically unibranch, so by [EGA], th. 18.10.1, $Z\to S_{i}$ is étale. In particular $Z\to\pi^{\prime}_{S_{i}}$ is an étale, closed immersion, that is, $Z$ is a connected component of $\pi^{\prime}_{S_{i}}$. Thanks to condition ii), $Z$ is also a connected component of $\pi^{\prime}$, and therefore meets the closed fibre. In particular it meets the image of $\alpha$. This proves the claim.

The morphism $\alpha$ is dominant and induces an injective $R$-algebra morphism ${\cal O}(\pi^{\prime})\hookrightarrow R^{n}.$ It follows that ${\cal O}(\pi^{\prime})$ is finite as an $R$-module. In particular $\pi^{\prime\operatorname{aff}}\to S$ is finite. $\square$

3.3 Computing the space of components

In this subsection, we collect some ways to compute the space of connected components $\pi_{0}(X/S)$ in various situations: when we apply changes of the base, when we use an atlas of $X$, and when we complete along a closed fibre of $X\to S$.

The first two results deal with arbitrary base change and étale base restriction. Both results hold whether $\pi_{0}(X/S)$ is representable or not.

** 3.3.1**

Lemma.* Let $\mathscr{X}/S$ be an $S$-algebraic stack and let $S^{\prime}\to S$ be a base change. Then we have a canonical isomorphism of $S^{\prime}$-functors \pi_{0}(\mathscr{X}\times_{S}S^{\prime}/S^{\prime})\xrightarrow{\,\smash{\raisebox{-2.15277pt}{\scriptstyle\sim}}\,}\pi_{0}(\mathscr{X}/S)\times_{S}S^{\prime}.*

Proof :

This follows from the definition of $\pi_{0}$ because both sides of the map in the statement parametrize relative connected components of $\mathscr{X}\times_{S}T^{\prime}$ for variable $S^{\prime}$-schemes $T^{\prime}$. $\square$

The following statement is related to factorizations in the sense of Definition 3.1.1.

** 3.3.2**

Lemma.* Let $\mathscr{X}\stackrel{{\scriptstyle h}}{{\longrightarrow}}\mathscr{E}\stackrel{{\scriptstyle f}}{{\longrightarrow}}S$ be morphisms of algebraic stacks.*

• *  *(1)

If $\mathscr{E}\to S$ is an étale algebraic space, there is a morphism of $S$-functors

[TABLE]

which is an isomorphism when $\mathscr{X}\to\mathscr{E}$ is universally open.

• *  *(2)

If $\mathscr{X},\mathscr{E}$ are finitely presented over $S$ and $\mathscr{X}\to\mathscr{E}$ is a universal submersion with connected geometric fibres, there is an isomorphism

[TABLE]

Proof :

Since (2) is easy to prove and not used in the paper, we only prove (1). Note that if $\mathscr{X}\to\mathscr{E}$ is flat, finitely presented and separable, this follows from Lemma 2.2.1. However, here we assume much less. The morphism in the statement is constructed as follows. For each $S$-scheme $T$, a point of $f_{!}\pi_{0}(\mathscr{X}/\mathscr{E})$ with values in $T$ is a pair composed of an $S$-morphism $u:T\to\mathscr{E}$ and a $T$-relative connected component $\mathscr{C}^{\prime}\subset\mathscr{X}\times_{\mathscr{E}}T$. Since $\mathscr{E}\to S$ is étale, the map $\mathscr{X}\times_{\mathscr{E}}T\to\mathscr{X}\times_{S}T$ is an open immersion globally and a closed immersion in the fibres, showing that \mathscr{C}\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=\mathscr{C}^{\prime} is a $T$-relative connected component of $\mathscr{X}\times_{S}T$ i.e. a $T$-valued point of $\pi_{0}(f_{!}\mathscr{X}/S)$. Let us describe the inverse morphism, assuming $\mathscr{X}\to\mathscr{E}$ universally open. Let $\mathscr{C}\subset\mathscr{X}\times_{S}T$ be a $T$-relative connected component. By the assumption on $\mathscr{X}\to\mathscr{E}$, the image $\mathscr{D}$ of $\mathscr{C}$ in $\mathscr{E}\times_{S}T$ is open, hence étale over $T$ with nonempty geometrically connected $T$-fibres. It follows that $\mathscr{D}\to T$ is an isomorphism. Using its inverse, we obtain a morphism $T\to\mathscr{D}\to\mathscr{E}$ and the pair $(T\to\mathscr{E},\mathscr{C})$ is a $T$-point of $f_{!}\pi_{0}(\mathscr{X}/\mathscr{E})$. These constructions are inverse to each other. $\square$

We continue with a description of $\pi_{0}(X/S)$ in terms of an atlas. This takes the form of a pushout property which is a consequence of the right exactness of the functor $\pi_{0}$, and will have an important refinement in the context of stacks in the later Lemma LABEL:lemma:stacky-pushout.

** 3.3.3**

Lemma.* Let $X\to S$ be a flat, finitely presented morphism of separable algebraic spaces and let $U\to X$ be an fppf, separable, surjective morphism.*

• *  *(1)

Let $R\subset U\times U$ be the fppf equivalence relation defined by $U\to X$, so that $X$ is identified with the coequalizer $\operatorname{coeq}(R\rightrightarrows U)$. Then we have $\pi_{0}(X/S)=\operatorname{coeq}(\pi_{0}(R/S)\rightrightarrows\pi_{0}(U/S))$.

• *  *(2)

The diagram

[TABLE]

is a pushout in the category of sheaves.

We warn the reader that $\pi_{0}(R/S)\to\pi_{0}(U/S)\times_{S}\pi_{0}(U/S)$ may fail to be injective; e.g. $R$ may be disconnected in a connected $U$.

Proof :

Throughout, we write $\pi_{0}(X)$ instead of $\pi_{0}(X/S)$ and we omit $S$ from fibred products.

(1) Let $\tilde{\pi}_{0}(R)$ denote the equivalence relation generated by the image of $\pi_{0}(R)\to\pi_{0}(U)\times\pi_{0}(U)$. Let us prove that the formation of $\tilde{\pi}_{0}(R)$ commutes with fppf surjective refinements $f:U^{\prime}\to U$. That is, if $f^{*}R\subset U^{\prime}\times U^{\prime}$ is the preimage of the equivalence relation $R$ under $U^{\prime}\times U^{\prime}\to U\times U$ and $f^{*}(\tilde{\pi}_{0}(R))$ is the preimage of the relation $\tilde{\pi}_{0}(R)$ under $\pi_{0}(U^{\prime})\times\pi_{0}(U^{\prime})\to\pi_{0}(U)\times\pi_{0}(U)$ then we want to prove that the natural map

[TABLE]

is an isomorphism. For this it is enough to prove that $\pi_{0}(f^{*}R)\to f^{*}\pi_{0}(R)$ is surjective. Since the spaces are étale, we may assume that $S$ is the spectrum of an algebraically closed field, and we can represent each connected component by a point lying on it. Since $f:U^{\prime}\to U$ is surjective, a point of

[TABLE]

can be represented by a triple $(r,u_{1},u_{2})\in R(k)\times U^{\prime}(k)\times U^{\prime}(k)$, which is what we wanted to prove. Since any two atlases for $X$ have a common refinement, it follows that the quotient space $\pi_{0}(U)/\tilde{\pi}_{0}(R)$ does not depend on the choice of $U$ up to a canonical isomorphism, and taking $U=X$ and $U^{\prime}=U$ we see that

[TABLE]

(2) Let $Y$ be a sheaf and let $a:X\to Y$, $b:\pi_{0}(U)\to Y$ be maps that coincide on $U$. Denote by $u:U\to X$ the chosen atlas and $s,t:R\to U$ the projections. Let $\sigma,\tau$ be the maps $\pi_{0}(s),\pi_{0}(t):\pi_{0}(U)\to\pi_{0}(X)$. Using that $R\to\pi_{0}(R)$ is an epimorphism of sheaves, from $aus=aut$ we deduce $b\sigma=b\tau$. Then (1) implies that $b$ factors through a map $\pi_{0}(X)\to Y$. $\square$

** 3.3.4**

Completion. We finish this subsection with a description of $\pi_{0}(X/S)$ over a complete local base which will be crucial for the proof of Theorem 4.3.2. Let $S$ be the spectrum of a complete noetherian local ring $R$ with maximal ideal $\mathfrak{m}$. For each $n\geqslant 0$ let $S_{n}=\operatorname{Spec}R/\mathfrak{m}^{n+1}$. By [EGA] IV.18.5.15, restriction to $S_{0}$ yields an equivalence $\operatorname{\bf F\acute{E}t}/S\simeq\operatorname{\bf F\acute{E}t}/S_{0}$ between the categories of finite étale algebras. In particular, given $X\to S$ flat of finite type and separable, there exists a unique finite étale scheme $\widehat{\pi}/S$ restricting to $\pi_{0}(X\times_{S}{S_{n}}/S_{n})$ over each $S_{n}$. Alternatively, one can see $\widehat{\pi}$ as the algebraization of the formal completion of $\pi_{0}(X/S)$, which explains the choice of notation $\widehat{\pi}$. As $\widehat{\pi}$ is finite over $S$, it is a product of complete local rings. By [SP19], Tag 0AQH there is a natural morphism of $S$-algebraic spaces

[TABLE]

which restricts to an isomorphism over each $S_{n}$.

** 3.3.5**

Proposition.* Let $R$ be a complete noetherian ring, $A$ a flat separable $R$-algebra of finite type. Write $X=\operatorname{Spec}A$, $S=\operatorname{Spec}R$, $s$ for the closed point of $S$, and let $V=S\setminus\{s\}$. The commutative diagram of $S$-algebraic spaces*

{\widehat{\pi}_{V}}$${\widehat{\pi}}$${\pi_{V}}$${\pi_{0}(X/S)}$$\scriptstyle{\psi_{V}}$$\scriptstyle{\psi}

is a pushout in the category of fppf sheaves over $S$.

Proof :

In the proof we write \pi\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=\pi_{0}(X/S). In order to prove the claim, it suffices to show that any diagram of solid arrows

{\widehat{\pi}_{V}}$${\widehat{\pi}}$${\pi_{V}}$${\pi}$${Z}$$\scriptstyle{\psi_{V}}$$\scriptstyle{\psi}$$\scriptstyle{a}$$\scriptstyle{b}

where $Z$ is an $S$-sheaf, admits a unique dashed arrow making the diagram commute.

First of all, notice that $\psi\colon\widehat{\pi}\to\pi$ is étale; writing $U=\pi_{V}\sqcup\widehat{\pi}$, it follows that $U\to\pi$ is faithfully flat of finite presentation, hence it is a coequalizer for $U\times_{\pi}U\to U$. Therefore, in order to obtain a unique dashed arrow, it suffices to check that $a\circ p_{1}=a\circ p_{2}$, where $p_{1},p_{2}$ are the projections $\widehat{\pi}\times_{\pi}\widehat{\pi}\to\widehat{\pi}$.

The $S$-scheme $\widehat{\pi}$ is finite étale, hence the map $\psi\colon\widehat{\pi}\to\pi$ is separated and quasi-finite, and so is also the base change $p_{1}\colon\widehat{\pi}\times_{\pi}\widehat{\pi}\to\widehat{\pi}$. Moreover, we know that $\widehat{\pi}$ is a finite disjoint union of spectra of completed local rings; by the classification of separated quasi-finite schemes over henselian local rings, $\widehat{\pi}\times_{\pi}\widehat{\pi}$ decomposes into a disjoint union $P^{f}\sqcup P^{\prime}$ such that $p_{1}\colon P^{f}\to\widehat{\pi}$ is finite (and étale), and $P^{\prime}=P^{\prime}_{V}$ has empty closed fibre. One obtains a similar decomposition for the map $p_{2}$, let us say $\widehat{\pi}\times_{\pi}\widehat{\pi}=Q^{f}\sqcup Q^{\prime}$. However, the compositions $\widehat{\pi}\times_{\pi}\widehat{\pi}\xrightarrow{p_{i}}\widehat{\pi}\to\pi\to S$ are the same map for $i=1,2$, and are both quasi-finite, separated; so both $P^{f}$ and $Q^{f}$ are equal to the finite part of the composition, and we find $P^{f}=Q^{f}$.

The restriction of $\psi$ to the closed fibre, $\psi_{s}\colon\widehat{\pi}_{s}\to\pi_{s}$, is an isomorphism by construction of $\widehat{\pi}$, and therefore so is $P^{f}_{s}=(\widehat{\pi}\times_{\pi}\widehat{\pi})_{s}\xrightarrow{p_{1}}\widehat{\pi}_{s}.$ The isomorphism extends uniquely to an isomorphism $P^{f}\to\widehat{\pi}$.

Consider the diagram of solid arrows

{\widehat{\pi}\sqcup P^{\prime}}$${\widehat{\pi}}$${\widehat{\pi}}$${Z}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{a}$$\scriptstyle{a}

where we have identified $P^{f}$ with $\widehat{\pi}$. We want to show that it is commutative.

For $i=1,2$, the morphism $p_{i}$ is the identity on $\widehat{\pi}$, so we really only need to show that $a\circ p_{1}$ agrees with $a\circ p_{2}$ on $P^{\prime}$. As $P^{\prime}$ is contained in $(\widehat{\pi}\times_{\pi}\widehat{\pi})_{V}$, we have $a\circ(p_{1})_{V}=b\circ\psi_{V}\circ(p_{1})_{V}=b\circ\psi_{V}\circ(p_{2})_{V}=a\circ(p_{2})_{V}$ and the proof is complete. $\square$

4 Perfection of algebras

The commutative algebra developed in this section has independent interest but is also fruitfully introduced with an eye towards the geometric applications of the next section. Let $X\to S$ be a flat, finitely presented morphism of algebraic spaces of characteristic $p$. In order to study the coperfection of $X$ in the category of $S$-algebraic spaces, we will use the étale algebraic spaces $\pi_{0}(X/S)$ and $\pi^{a}(X/S)$ (assuming they exist). Since étale implies relatively perfect, the morphism $X\to\pi_{0}(X/S)$ extends to the direct Frobenius system and we have a diagram:

[TABLE]

The present section is devoted to the case where $S=\operatorname{Spec}(R)$ and $X=\operatorname{Spec}(A)$. The main question is whether there exists a perfection functor, right adjoint to the inclusion of perfect $R$-algebras into all $R$-algebras. In such generality we do not know if such perfection exists. At least an obvious approximation should be the preperfection:

[TABLE]

The above diagram of spaces provides a diagram of algebras

[TABLE]

where $A^{\operatorname{\acute{e}t}/R}={\cal O}(\pi^{a}(A/R))$ is the largest étale subalgebra of $A$, see Definition 3.1.8. Our goal is roughly to find as many situations as possible where both maps above are isomorphisms.

We start in § 4.1 with preliminary material on base change in the formation of the preperfection. Then we prove that both maps above are indeed isomorphisms when $R$ is artinian and $R\to A$ is of finite type, see § 4.2, or $R$ is regular and $R\to A$ is of finite type and separable, see § 4.4. Over a general noetherian ring, only the map ${\cal O}(\pi_{0}(A/R))\to A^{p^{\infty}/R}$ is an isomorphism, see § 4.3. This is already remarkable, given the poor properties of both algebras: in general ${\cal O}(\pi_{0}(A/R))$ is not étale and $A^{p^{\infty}/R}$ is not perfect, even when $R\to A$ is flat, of finite type and separable. One may expect that after iterating the preperfection functor $(-)^{p^{\infty}/R}$ a finite (sufficiently high) number of times, one reaches a perfect $R$-algebra. With the hope that this might be true, we establish in § 4.4 some finiteness properties of $A^{p^{\infty}/R}$. We conclude the section with counterexamples.

4.1 Base change in preperfection

For each morphism of $\mathbb{F}_{p}$-algebras $R\to A$ and each base change morphism $R\to R^{\prime}$ we have a natural base change map for preperfection:

[TABLE]

It is important to understand this map for at least two reasons. The first is that the study of $A^{p^{\infty}/R}$ with the usual tools (localization, completion on $R$…) involves many base changes. The second is that the base change map along Frobenius $\operatorname{F}:R\to R$ controls the success or failure of $A^{p^{\infty}/R}$ to be perfect; we elaborate on this in Remark 4.1.4. Before stating the first lemma devoted to properties of $\phi$, we recall a result of T. Dumitrescu.

** 4.1.1**

Theorem.*

Let $R\to A$ be a morphism of noetherian commutative rings. Let $\operatorname{F}_{A/R}:A^{p/R}\to A$ be the relative Frobenius morphism. Then the following are equivalent:*

• (i)

$R\to A$* is flat and separable,*

• (ii)

$\operatorname{F}_{A/R}$* is injective and its cokernel is a flat $R$-module.*

Proof :

See [Du95], Theorem 3. $\square$

** 4.1.2**

Remark. If we do not assume that $R$ and $A$ are noetherian but $R\to A$ is of finite presentation, then (i) $\Rightarrow$ (ii) is true. Indeed $R\to A$ is the base change of a map $R_{0}\to A_{0}$ along a morphism $R_{0}\to R$ with $R_{0}$ noetherian and we may choose $R_{0}\to A_{0}$ flat and separable, see [EGA] IV3, 11.2.7 and 12.1.1(vii). Then by the noetherian case, it follows that $\operatorname{F}_{A_{0}/R_{0}}$ is injective with $R_{0}$-flat cokernel. By base change $\operatorname{F}_{A/R}$ is injective with $R$-flat cokernel.

** 4.1.3**

Lemma.* The base change map $\phi_{R,R^{\prime},A}:A^{p^{\infty}/R}\otimes_{R}R^{\prime}\longrightarrow(A\otimes_{R}R^{\prime})^{p^{\infty}/R^{\prime}}$ is:*

• *  *(1)

an isomorphism if $R\to R^{\prime}$ is finite locally free.

• *  *(2)

injective in each of the following cases:

• (i)

$R\to R^{\prime}$* is projective.*

• (ii)

$R\to R^{\prime}$* is flat and $R\to A$ is flat, finitely presented, with reduced geometric fibres.*

• (iii)

$R^{\prime}=\operatorname{colim}R$* is the absolute coperfection of a ring $R$ such that $\operatorname{F}:R\to R$ is projective. *

Proof :

Note that since $(A\otimes_{R}R^{\prime})\otimes_{R^{\prime},\operatorname{F}^{i}}R^{\prime}=A^{p^{i}/R}\otimes_{R}R^{\prime}$, the map $\phi_{R,R^{\prime},A}$ is just a special case for the $R$-module M\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=R^{\prime} of the map $\phi_{R,M,A}$ that appears as the upper horizontal row in the following commutative diagram:

{(\lim A^{p^{i}/R})\otimes_{R}M}$${\lim(A^{p^{i}/R}\otimes_{R}M)}$${\left(\prod_{i\geqslant 0}A^{p^{i}/R}\right)\otimes_{R}M}$${\prod_{i\geqslant 0}(A^{p^{i}/R}\otimes_{R}M).}$$\scriptstyle{\phi_{R,M,A}}$$\scriptstyle{\psi_{R,M,A}}

In the sequel we assume that $M$ is flat, so the left-hand vertical map is injective. If $M$ is free, resp. free of finite rank, then $\psi_{R,M,A}$ is injective, resp. an isomorphism. It follows that also $\phi_{R,M,A}$ is injective, resp. an isomorphism. If $M$ is projective, one reaches the same conclusions by embedding it in a free module, resp. a free module of finite rank, and using the facts that $\phi_{R,M,A}$ and $\psi_{R,M,A}$ are additive in $M$. This settles cases (1) and (2.i).

In case (2.ii), by Dumitrescu’s theorem 4.1.1 all the maps $A^{p^{i}/R}\to A^{p^{i+1}/R}$ are injective; it follows that $\lim A^{p^{i}/R}\to A^{p^{j}/R}$ is injective for each fixed $j$. By flatness of $R\to R^{\prime}$ the tensored map $(\lim A^{p^{i}/R})\otimes_{R}R^{\prime}\to A^{p^{j}/R}\otimes_{R}R^{\prime}$ is injective. Therefore $\phi_{R,R^{\prime},A}$ is also injective.

In case (2.iii) we can write the coperfection as $R^{\prime}=\operatorname{colim}R^{p^{-j}}$. Since the absolute Frobenius of $R$ is projective, it is in fact faithfully flat. It follows that the maps $R^{p^{-j}}\to R^{p^{-(j+1)}}$ are faithfully flat, hence universally injective. Thus for each $i,j$ the map

[TABLE]

is injective. Then for each $i$

[TABLE]

is injective. Taking limits

[TABLE]

is injective, which implies that

[TABLE]

is injective. Since also by (2.i) the map

[TABLE]

is injective, by composition we obtain the result. $\square$

** 4.1.4**

Remarks. (1) Let $R\to A$ be a map of rings of characteristic $p>0$. When inquiring whether the preperfection $A^{p^{\infty}/R}$ is perfect, we are led to ask if the Frobenius of the preperfection (“Frobenius of the limit”) is an isomorphism. In general it is not; an example is given in in 4.5.2. In contrast, the morphism obtained as the limit of the Frobenius maps of the individual rings of the system (“the limit of Frobenius”) is an isomorphism: it is essentially a shift by one in the indices, which is invisible in the infinite system. In fact, “Frobenius of the limit” and “the limit of Frobenius” are the two edges of a commutative triangle whose third edge, the base change map in preperfection, serves to compare them:

[TABLE]

Since $\lim\operatorname{F}$ is an isomorphism, we see that $A^{p^{\infty}/R}$ is a perfect $R$-algebra if and only if the base change map $\phi_{R,R,A}$ is an isomorphism. According to Lemma 4.1.3(1), this happens when $R$ is regular and $\operatorname{F}$-finite, for then absolute Frobenius is finite locally free (see [Ku69]).

(2) In case (2.ii), it will be a consequence of Theorem 5.1.1 that the base change map is in fact an isomorphism.

(3) Here is an example where the base change map is not surjective. Let $k$ be a field of characteristic $p$ and $k^{\prime}$ an infinite-dimensional field extension. Let

[TABLE]

and $A^{\prime}=A\otimes_{k}k^{\prime}$. Let $t_{0},t_{1},\dots$ be an infinite family of elements of $k^{\prime}$ that is $k$-linearly independent. Let $x^{\prime}_{i}=\epsilon_{0}t_{i}+\epsilon_{1}t_{i-1}+\dots+\epsilon_{i}t_{0}\in(A^{\prime})^{p^{i}/k^{\prime}}$. Then $\operatorname{F}_{A^{\prime}/k^{\prime}}(x^{\prime}_{i+1})=x^{\prime}_{i}$ so $x^{\prime}=(x^{\prime}_{i})$ is an element of $(A^{\prime})^{p^{\infty}/k^{\prime}}$ which obviously does not come from $A^{p^{\infty}/k}\otimes k^{\prime}$.

We end this section with a case where preperfection commutes with base change; since it is not used in the paper, we omit the proof.

** 4.1.5**

Lemma.* Let $A$ be an $R$-algebra, flat of finite presentation, such that the induced morphism $\operatorname{Spec}A\to\operatorname{Spec}R$ has geometrically reduced fibres. Let $f\in R$ be a non-zero divisor, with $R/fR$ reduced. The natural map $\phi\colon A^{p^{\infty}/R}\otimes_{R}R_{f}\to(A\otimes_{R}R_{f})^{p^{\infty}/R_{f}}$ is an isomorphism. $\square$*

4.2 Perfection over artinian rings

In this subsection we consider the case where $R$ is an artinian ring. For such a ring, Theorem 3.1.7 implies that any flat, finitely generated algebra $R\to A$ has a largest étale subalgebra $A^{\operatorname{\acute{e}t}}$. Below we prove that the natural map $A^{\operatorname{\acute{e}t}}\to A^{p^{\infty}}$ to the preperfection is an isomorphism. In particular, the preperfection is perfect, hence a perfection. We point out that in this special situation the separability of $R\to A$ is not needed.

** 4.2.1**

Theorem.* Let $R$ be an artinian local ring of characteristic $p$, and let $A$ be a flat $R$-algebra of finite type. Then the maps $A^{\operatorname{\acute{e}t}}\to{\cal O}(\pi_{0}(A))\to A^{p^{\infty}}$ are isomorphisms.*

Proof :

It follows from [Rom11], 2.1.3 that $\pi_{0}(A)$ is an étale quasi-compact $R$-algebraic space. Since $R$ is artinian, this space is finite. In particular it is affine and the map $A^{\operatorname{\acute{e}t}}\to{\cal O}(\pi_{0}(A))$ is an isomorphism. It remains to prove that $A^{\operatorname{\acute{e}t}}\to A^{p^{\infty}}$ is an isomorphism. The proof of this is in five steps.

Step 1. We reduce to the case where $R=k$ is a field. Let $\mathfrak{m}$ resp. $k$ be the maximal ideal resp. residue field. Let $\operatorname{F}:R\to R$ be the absolute Frobenius and $e$ an integer such that $\mathfrak{m}=\ker\operatorname{F}^{e}$. Then $\operatorname{F}^{e}$ induces a ring map $\alpha:k\to R$ which we use to view $R$ as a $k$-algebra. We compute the perfection of $A$ using the cofinal system of indices $e\mathbb{N}\subset\mathbb{N}$. For each $i\geqslant 0$ the morphism $\operatorname{F}^{ei}:R\to R$ has a factorization:

[TABLE]

Writing $A_{0}=A\otimes_{R}k$, it follows that $A^{p^{ei}/R}=A_{0}^{p^{e(i-1)}/k}\otimes_{k}R$. Passing to the limit and using 4.1.3 (1), we deduce an isomorphism:

[TABLE]

On the other hand, the $e$-fold absolute Frobenius $\operatorname{F}^{e}_{A}:A_{0}^{\operatorname{\acute{e}t}\!/k}\to A^{\operatorname{\acute{e}t}\!/R}$ extends the map $\alpha:k\to R$, providing an isomorphism:

[TABLE]

Since $\lambda$ and $\mu$ fit together in a commutative square, the reduction step follows.

Step 2. We reduce to the case where $k$ is algebraically closed. Let $k^{\prime}$ be an algebraic closure of $k$, and A^{\prime}\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=A\otimes_{k}k^{\prime}. We have injections

[TABLE]

where the first is deduced from $A^{\operatorname{\acute{e}t}/k}\hookrightarrow A^{p^{\infty}/k}$ and the second comes from case (2.i) of Lemma 4.1.3. It is classical that $A^{\operatorname{\acute{e}t}/k}\otimes_{k}k^{\prime}=(A^{\prime})^{\operatorname{\acute{e}t}/k^{\prime}}$, see [Wa79], Th. 6.5. It follows that if $(A^{\prime})^{\operatorname{\acute{e}t}/k^{\prime}}\to(A^{\prime})^{p^{\infty}/k^{\prime}}$ is an isomorphism, then $A^{\operatorname{\acute{e}t}/k}\otimes_{k}k^{\prime}\hookrightarrow A^{p^{\infty}/k}\otimes_{k}k^{\prime}$ is an isomorphism and hence $A^{\operatorname{\acute{e}t}/k}\to A^{p^{\infty}/k}$ is an isomorphism.

Step 3. We reduce to the case where $A$ is reduced. Let $A_{{\rm red}}$ be the reduced quotient. On the separable closure side, since $A^{\operatorname{\acute{e}t}/k}$ does not meet the nilradical $\operatorname{Nil}(A)$ and all separable elements of $A_{{\rm red}}$ lift to $A$, we have an isomorphism A^{\operatorname{\acute{e}t}/k}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{\scriptstyle\sim}}\,}(A_{{\rm red}})^{\operatorname{\acute{e}t}/k}. On the preperfection side, we use the isomorphisms A^{p^{i}/k}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{\scriptstyle\sim}}\,}A, $a\otimes\lambda\mapsto a\lambda^{p^{-i}}$ to obtain an isomorphism of rings A^{p^{\infty}/k}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{\scriptstyle\sim}}\,}A^{p^{\infty}/\mathbb{F}_{p}}, and similarly for $A_{{\rm red}}$. Since $\operatorname{Nil}(A)$ is finitely generated, there is $e\geqslant 0$ such that $\operatorname{Nil}(A)=\ker\operatorname{F}^{e}$ where $\operatorname{F}:A\to A$ is the absolute Frobenius. Then the computation of the perfection can be carried out along the cofinal system of indices $e\mathbb{N}\subset\mathbb{N}$, showing that the projection $A^{p^{\infty}/\mathbb{F}_{p}}\to(A_{{\rm red}})^{p^{\infty}/\mathbb{F}_{p}}$ is an isomorphism. Contemplating the commutative diagram below, we see that if $(A_{{\rm red}})^{\operatorname{\acute{e}t}/k}\to(A_{{\rm red}})^{p^{\infty}/k}$ is an isomorphism then $A^{\operatorname{\acute{e}t}/k}\to A^{p^{\infty}/k}$ also.

[TABLE]

Step 4. We reduce to the case where $A$ has connected spectrum. This is straightforward, because if $A=A_{1}\times\dots\times A_{d}$ is the decomposition of $A$ as a product of rings with connected spectrum, we have $(\prod A_{i})^{\operatorname{\acute{e}t}/k}\simeq\prod A_{i}^{\operatorname{\acute{e}t}/k}$ and $(\prod A_{i})^{p^{\infty}/k}\simeq\prod A_{i}^{p^{\infty}/k}$.

Step 5. We conclude that $A^{\operatorname{\acute{e}t}/k}\to A^{p^{\infty}/k}$ is surjective. Let $x$ be an element of the ring

[TABLE]

with $x=x_{n}^{p^{n}}$ and $x_{n}\in A$, for each $n$. By noetherianity, the increasing sequence of ideals $(x_{i})$ stabilizes at some $N$. It follows that y\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=x_{N} satisfies $(y)=(y^{p})$, in particular $(y)=(y^{2})$. Since $X=\operatorname{Spec}(A)$ is connected, we deduce that $y=0$ or $y$ is a unit; therefore $x=0$ or $x$ is a unit. Let $A_{i}$ be the quotients of $A$ by the minimal primes. Again by connectedness, the injection $A\hookrightarrow A_{1}\times\dots\times A_{n}$ induces a morphism of groups of units modulo constants $A^{\times}/k^{\times}\hookrightarrow(A_{1}^{\times}/k^{\times})\dots\times(A_{n}^{\times}/k^{\times})$ which is injective. It is a classical result of Rosenlicht ([Ros57], lemma to Prop. 3) that each $A_{i}^{\times}/k^{\times}$ is a finitely generated free abelian group; hence the same is true for $A^{\times}/k^{\times}$. In particular the class of $x$ in this group cannot be infinitely $p$-divisible, so $x\in k^{\times}$ and this proves the claim. $\square$

4.3 Preperfection over noetherian rings

The aim of this section is to generalize the statement that ${\cal O}(\pi_{0}(A))\to A^{p^{\infty}}$ is an isomorphism to the case of a general noetherian base ring $R$, in the case of separable algebras. The proof proceeds by thickening from an artinian base to a complete local base, then a Zariski-local base and then to a general base by induction on the dimension.

** 4.3.1**

Lemma.* Let $R$ be a complete noetherian local ring and $A$ a flat separable $R$-algebra of finite type. Write $\widehat{A}$ for the completion of $A$ with respect to the maximal ideal of $R$, and write $\widehat{\pi}$ for the finite étale $R$-scheme built from $\pi_{0}(A/R)$ as in the situation of § 3.3.4. Then the natural map ${\cal O}(\widehat{\pi})\to(\widehat{A})^{p^{\infty}/R}$ is an isomorphism.*

Proof :

Let $\mathfrak{m}$ be the maximal ideal of $R$. Write $B={\cal O}(\widehat{\pi})$. For every $n\geqslant 0$, let $R_{n}=R/\mathfrak{m}^{n+1}$, $A_{n}=A\otimes_{R}R_{n}$, $B_{n}=B\otimes_{R}R_{n}$. As $B_{n}={\cal O}(\pi_{0}(A_{n}/R_{n}))$, for every $n$ we have an inclusion $B_{n}\hookrightarrow A_{n}$. Taking the limit over $n$, and noticing that $B$ is finite over $R$ hence complete, we obtain an inclusion $B\hookrightarrow\widehat{A}$. As $B$ is also étale over $R$, it is in fact contained in $(\widehat{A}/R)^{p^{\infty}}$.

On the other hand, a section to the inclusion $B\hookrightarrow\widehat{A}^{p^{\infty}}$ is given by the map

[TABLE]

Here, the second-to-last equality comes from Theorem 4.2.1. To complete the proof it suffices to show that $\widehat{A}^{p^{\infty}}\to B$ is injective, or that $(\lim_{n}A_{n})^{p^{i}}\to\lim_{n}(A_{n}^{p^{i}})$ is injective. The latter is the completion morphism:

[TABLE]

As $R$ and $\widehat{A}$ are both noetherian, the hypotheses of Theorem 4.1.1 are satisfied, and we deduce that $(\widehat{A})^{p^{i}}$ is a subalgebra of $\widehat{A}$. As the latter is $\mathfrak{m}$-adically separated (that is, $\bigcap_{i=1}^{n}\mathfrak{m}^{i}\widehat{A}=0$), so is its subalgebra $(\widehat{A})^{p^{i}}$. Hence the completion morphism above is injective and we conclude. $\square$

** 4.3.2**

Theorem.* Let $R$ be a noetherian ring and $A$ a flat, separable $R$-algebra of finite type. Then the natural map*

[TABLE]

is an isomorphism.

Proof :

As a first step, we claim that we may reduce to the case of $R$ complete local. Indeed, let $R\to R^{\prime}$ be the completion of the local ring at some prime $\mathfrak{p}\subset R$. The morphism $R\to R^{\prime}$ is flat. We have a map

[TABLE]

The first equality is compatibility of global sections and flat base change, the second arrow is $\phi\otimes_{R}R^{\prime}$, while the last arrow is injective by 4.1.3. We see that if the composition is an isomorphism, then also the central arrow $\phi\otimes_{R}R^{\prime}$ is an isomorphism. As $R_{\mathfrak{p}}\to R^{\prime}$ is faithfully flat, the map $\phi\otimes_{R}R_{\mathfrak{p}}$ is also an isomorphism. Repeating the argument for all $\mathfrak{p}\subset R$, we find that $\phi$ is an isomorphism. This proves the claim.

We argue by induction on the dimension of $R$. If $R$ is of dimension zero, it is a product of finitely many artinian local rings; we reduce to $R$ local and the result follows by Theorem 4.2.1.

Now let $d$ be the dimension of $R$, and assume the result true for base rings of dimension at most $d-1$. We may assume $R$ local and complete with respect to its maximal ideal. Let $s$ be the closed point of $\operatorname{Spec}R$, and $V=S\setminus\{s\}$. Notice that $V$ is of dimension $d-1$. Cover $V$ with open affines $U_{i}=\operatorname{Spec}R_{i}$. Consider the commutative diagram of solid arrows st

{A}$${\widehat{A}}$${\prod_{i}A\otimes_{R}R_{i}}$${\prod_{i}\widehat{A}\otimes_{R}R_{i}}$${0}$${\prod_{i,j}A\otimes_{R}R_{i}\otimes_{R}R_{j}}$${\prod_{i,j}\widehat{A}\otimes_{R}R_{i}\otimes_{R}R_{j}}

Clearly, $A$ admits natural compatible maps towards the diagram, represented by dashed arrows in the diagram.

Next, we take the preperfection of the diagram. By Lemma 4.3.1 we have $\widehat{A}^{p^{\infty}}={\cal O}(\widehat{\pi})$. Moreover, for every $R$-algebra $R^{\prime}$, there is a natural map ${\cal O}(\widehat{\pi}\otimes_{R}R^{\prime})=\widehat{A}^{p^{\infty}}\otimes_{R}R^{\prime}\to(\widehat{A}\otimes_{R}R^{\prime})^{p^{\infty}}$. Finally, by the induction hypothesis $(A\otimes R_{i})^{p^{\infty}}={\cal O}(\pi(X_{U_{i}}/U_{i}))$. We get a commutative diagram

{A^{p^{\infty}}}$${{\cal O}(\widehat{\pi})}$${\prod_{i}{\cal O}(\pi_{0}(X_{U_{i}}/U_{i}))}$${\prod_{i}{\cal O}(\widehat{\pi}_{U_{i}})}$${0}$${\prod_{i,j}{\cal O}(\pi_{0}(X_{U_{ij}}/U_{ij}))}$${\prod_{i,j}{\cal O}(\widehat{\pi}_{U_{ij}})}

where the horizontal arrows are those induced by the natural morphism $\psi\colon\widehat{\pi}\to\pi_{0}(X/S)$ of § 3.3.4. The limit of the diagram of solid arrows coincides with the limit of the subdiagram of solid arrows in the commutative diagram

[TABLE]

Taking global sections in the pushout diagram of Lemma 3.3.5, we see that ${\cal O}(\pi_{0}(X/S))$ is a fibre product for the subdiagram (2) of solid arrows. Therefore we get a natural map $\chi\colon A^{p^{\infty}}\to{\cal O}(\pi_{0}(X/S))$. The maps

[TABLE]

are compatible with the natural inclusions of $A^{p^{\infty}}$ and ${\cal O}(\pi_{0}(X/S))$ into $A$. Hence $\phi$ is injective, and because $\phi\circ\chi$ is the identity, it is also surjective, as we wished to show. $\square$

With the notation of 4.3.2, the algebraic space $\pi_{0}(X/S)$ is étale; however, its $R$-algebra of global sections ${\cal O}(\pi_{0}(X/S))$ may fail to be unramified (and therefore étale and perfect); see for instance Example 4.5.2. In particular, the preperfection $A^{p^{\infty}/R}$ needs not be perfect.

4.4 Perfection over regular or unibranch one-dimensional rings

Recall from Remark 4.1.4 that if $R$ is regular and $\operatorname{F}$-finite, then for all $R\to A$ the preperfection $A^{p^{\infty}}$ is perfect. For the separable $R$-algebras that we have been studying in this section, Theorem 4.3.2 provides an explicit description of $A^{p^{\infty}}$ which allows to find more cases when preperfection is perfect.

** 4.4.1**

Corollary.* Let $R$ be a noetherian $\mathbb{F}_{p}$-algebra and $A$ a flat and separable $R$-algebra of finite type.*

• *  *(1)

If $R$ is either

• –

geometrically $\mathbb{Q}$-factorial (e.g. regular), or

• –

integral, geometrically unibranch and one-dimensional,

then we have isomorphisms:

[TABLE]

In particular $A^{p^{\infty}}$ is étale, hence perfect and of finite type.

• *  *(2)

If $R$ is reduced, excellent, of dimension $\leqslant 1$, then $A^{p^{\infty}}$ is quasi-finite, and in particular of finite type.

Proof :

(1) We proved that the map $A^{\operatorname{\acute{e}t}}\to{\cal O}(\pi_{0}(A))$ is an isomorphism in Proposition 3.2.2, and that the map ${\cal O}(\pi_{0}(A))\to A^{p^{\infty}}$ is an isomorphism in Theorem 4.3.2.

(2) This follows immediately from Proposition 3.2.3. $\square$

4.5 Examples

We shall see that the coperfection of the spectrum of an algebra is not the spectrum of its perfection. In fact, in the flat and separable case the coperfection of an affine scheme is $\pi_{0}$ and may be non-separated. Here is an example.

** 4.5.1**

Lemma.* Let $R=\mathbb{F}_{p}[[u]]$ and consider the $R$-algebra*

[TABLE]

Then $A^{p^{\infty}}=R$ while $\pi_{0}(A/R)$ is the non-separated scheme obtained by glueing two copies of $\operatorname{Spec}(R)$ along the generic fibre.

Proof :

Let $X=\operatorname{Spec}A$, $S=\operatorname{Spec}R$. The fibre of $X\to S$ over the closed point has two connected components, while the generic fibre is connected. The two sections $s_{1},s_{2}\colon S\to X$, $s_{1}=\{x=u,y=1\}$ and $s_{2}=\{x=1,y=u\}$ meet all components of all fibres; it follows that the composition

[TABLE]

is given by glueing the two copies of $S$ along the generic fibre. Therefore $\pi_{0}(X/S)$ is non-separated. From 4.3.2 it follows that $A^{p^{\infty}}={\cal O}(\pi_{0}(X/S))$, which is equal to $R$. $\square$

The following is the most basic example of a non-perfect preperfection, that is, an $R$-algebra $A$ which is flat, separable, of finite presentation, for which the preperfection $A^{p^{\infty}/R}$ is not perfect. The ring $R$ is one-dimensional; we remark that, in accordance with Proposition 3.2.2, we need to choose $R$ with multiple branches. Since the preperfection is not perfect, it is natural to ask what happens if we take the preperfection once more. Here is the answer.

** 4.5.2**

Lemma.* Let $R=\mathbb{F}_{p}[[u,v]]/(uv)$ and $A=R[x,y,(x-y)^{-1}]/(xy-u)$. If $p\neq 2$, we have:*

• (1)

$A^{p^{\infty}}\simeq\frac{R[\alpha]}{(u\alpha,v^{2}-\alpha^{2})}$* mapping to $A$ by $\alpha\mapsto v\frac{x+y}{x-y}$,*

• (2)

$(A^{p^{\infty}})^{p^{\infty}}\simeq R$.

Notice that the restriction of $R\to A^{p^{\infty}}$ to the branch $\{u=0\}$ is $\mathbb{F}_{p}[[v]]\to\mathbb{F}_{p}[[v]][\alpha]/(v^{2}-\alpha^{2})$ which is not formally étale. Therefore $\phi$ itself is not formally étale and in particular not relatively perfect. The restriction $p\neq 2$ allows a simpler presentation of $A^{p^{\infty}}$ but is inessential.

Proof :

Once for all we set $k=\mathbb{F}_{p}$.

(1) Let $S=\operatorname{Spec}R$, $X=\operatorname{Spec}A$. The open complement $V=S\setminus\{s\}$ of the closed point of $S$ is affine, with ${\cal O}(V)=R_{u}\times R_{v}$. It is easy to see that ${\cal O}(\pi_{0}(X_{V}/V))=R_{u}\times R_{v}\times R_{v}$. The inclusion ${\cal O}(\pi_{0}(X_{V}/V)\hookrightarrow{\cal O}(X_{V})=A_{u}\times A_{v}$ maps the elements $(1,0,0)$, $(0,1,0)$, $(0,0,1)$ to $(1,0)$, $(0,\frac{x}{x-y})$, $(0,\frac{y}{y-x})$ respectively.

Applying the global sections functor to the pushout diagram of Lemma 3.3.5, and noticing that $\widehat{\pi}=\operatorname{Spec}(R\times R)$, we obtain a cartesian diagram

[TABLE]

The lower horizontal map sends $(f(u),g(v),h(v))$ to $(f(u),g(v),f(u),h(v))$. The right vertical map sends $(\alpha(u,v),\beta(u,v))$ to $(\alpha(u,0),\alpha(0,v),\beta(u,0),\beta(0,v)).$ It follows that the fibre product ${\cal O}(\pi_{0}(X/S))$ is the subring of $R_{u}\times R_{v}\times R_{v}$ generated as an $R$-submodule by $(1,1,1)$, $(u,0,0)$, $(0,v,0)$, $(0,0,v)$.

Since $p\neq 2$, we may choose instead $(1,1,1)$, $(u,0,0)$, $(0,v,v)$ and $(0,v,-v)$ as generators. We find that

[TABLE]

via the map $(u,0,0)\mapsto u$, $(0,v,v)\mapsto v$, $(0,v,-v)\mapsto\alpha$.

Finally, notice that the element $(0,v,-v)\in R_{u}\times R_{v}\times R_{v}$ is mapped to $v\frac{x+y}{x-y}$ in $A_{u}\times A_{v}$. This proves the claim.

(2) Let $B=A^{p^{\infty}}$. Notice first that any element of $B$ can be written uniquely as $f+g\alpha$, with $f\in R$ and $g\in R/u$. Therefore, any element of $B^{(p^{n})}=B\otimes_{R,F^{n}}R$ takes either the form $1\otimes f$ with $f\in R$ or $\alpha\otimes g$ with $g\in R/u^{p^{n}}$. In fact, the map of $R$-modules

[TABLE]

is an isomorphism, which we will use to rewrite the preperfection diagram of $B$. The $n$-th map in the diagram is $B^{(p^{n})}\to B^{(p^{n-1})}$ sending $1\otimes f$ to $1\otimes f$ and $\alpha\otimes g$ to $\alpha^{p}\otimes g=v^{p-1}\alpha\otimes g=\alpha\otimes v^{p^{n}-p^{n-1}}g$. Using the isomorphism of $R$-modules above, this becomes the map of $R$-modules

[TABLE]

sending $(f,g)$ to $(f,gv^{p^{n}-p^{n-1}})$. Consider now the preperfection diagram

[TABLE]

Let $H_{n}=G_{1}\circ\ldots\circ G_{n}\colon R\oplus R/u^{p^{n}}\to R\oplus R/u$ and let $(\ldots,a_{n},a_{n-1},\ldots,a_{0})$ be an element of the limit of the diagram. We can of course consider the limit in the category of $R$-modules, as it will automatically have an $R$-algebra structure making it into the limit in the category of $R$-algebras. Now, the image of $(f,g)\in R\oplus R/u^{p^{n}}$ via $H_{n}$ is $(f,gv^{p^{n}-1})$. Hence $a_{0}=(f_{0},g_{0})$ is such that for every $n\geqslant 1$, $g_{0}$ is in the ideal of $R/u$ generated by $v^{p^{n}-1}$. Therefore $g_{0}=0$. One can use the same argument to show that for every $a_{n}=(f_{n},g_{n})$, $g_{n}$ vanishes. Therefore the limit is simply the limit of the diagram:

[TABLE]

This shows that $B^{p^{\infty}}=R$. $\square$

5 Unramified $\operatorname{F}$-divided objects and

the étale fundamental pro-groupoid

In this section, we define the étale fundamental pro-groupoid $\mathscr{X}\to\Pi_{1}(\mathscr{X}/S)$ of a flat finitely presented algebraic stack and we prove Theorem A, namely that if moreover $\mathscr{X}/S$ is separable and $\mathscr{M}/S$ is a Deligne-Mumford stack, there is an isomorphism $\mathscr{H}\!om(\Pi_{1}(\mathscr{X}/S),\mathscr{M})\to\mathscr{H}\!om(\mathscr{X},\operatorname{Fdiv}(\mathscr{M}))$. As a first step, in 5.1 we build on Theorem 4.3.2 to prove this when $\mathscr{X}$ and $\mathscr{M}$ are algebraic spaces; in this case only the coarse moduli space $\pi_{0}(\mathscr{X}/S)$ appears in the source of the isomorphism. Then in 5.2 we introduce the étale fundamental pro-groupoid and its basic properties. Finally in LABEL:ss:coperfection_as_alg_stack we upgrade the result from algebraic spaces to algebraic stacks in the correct generality. In order to spare the reader unpleasant technicalities, some material on groupoid closures needed to handle $\Pi_{1}(\mathscr{X}/S)$ is relegated to LABEL:sec:groupoid_closure.

Note that as we observed in Remark 2.3.3, the canonical isomorphism

[TABLE]

allows an equivalent interpretation of the result in terms of the coperfection of $\mathscr{X}$. The interplay between the two viewpoints pervades the section, and the proofs.

5.1 The case of algebraic spaces

Let $S$ be an algebraic space of characteristic $p$ and $X$ a flat, finitely presented, separable $S$-algebraic space. The algebraic space $\pi_{0}(X/S)$ is relatively perfect over $S$. Therefore the natural morphism $\operatorname{Fdiv}(\pi_{0}(X/S))\longrightarrow\pi_{0}(X/S)$ is an isomorphism, and we obtain a natural morphism:

[TABLE]

** 5.1.1**

Theorem.* Let $S$ be a noetherian algebraic space of characteristic $p$. Let $X\to S$ be a flat, finitely presented, separable algebraic space. Let $M\to S$ be an arbitrary quasi-separated algebraic space. Then the natural morphism given by $\alpha\mapsto\operatorname{Fdiv}(\alpha)\circ\rho$*

[TABLE]

is a bifunctorial isomorphism of sheaves over $S$.

We make two remarks before giving the proof.

** 5.1.2**

Remarks. (1) In terms of coperfection, this theorem says that if $X\to S$ is a flat, finite type, separable morphism of noetherian algebraic $\mathbb{F}_{p}$-spaces then the inductive system of relative Frobenii

{X}$${X^{p/S}}$${X^{p^{2}/S}}$${\dots}$$\scriptstyle{\operatorname{F}_{X/S}}$$\scriptstyle{\operatorname{F}_{X^{p}/S}}

admits a colimit in the category of quasi-separated algebraic spaces over $S$; the colimit is the algebraic space $\pi_{0}(X/S)$, and is also a coperfection of $X\to S$.

(2) Point (1) is remarkable if we consider that for a noetherian ring $R$ and a flat, finite type separable algebra $R\to A$, taking the preperfection of $A$, i.e., the limit of relative Frobenius morphisms

{\dots}$${A^{p^{2}/R}}$${A^{p/R}}$${A}$$\scriptstyle{\operatorname{F}_{A^{p}/R}}$$\scriptstyle{\operatorname{F}_{A/R}}

does not guarantee to produce a perfect object, as illustrated in 4.5.2.

Proof :

Throughout, we write $\pi_{0}(X)$ instead of $\pi_{0}(X/S)$. Let $\rho_{0}:X\to\pi_{0}(X)$ be the natural map. Since $\pi_{0}(X)\to S$ is perfect, we have a canonical isomorphism $\underline{\operatorname{Hom}}(\pi_{0}(X/S),M)=\underline{\operatorname{Hom}}(\pi_{0}(X/S),\operatorname{Fdiv}(M))$ so the statement to be proven is that

[TABLE]

is a bifunctorial isomorphism of sheaves over $S$.

We start with easy observations. Obviously we can assume that $S$ is affine. Since the formation of the $\underline{\operatorname{Hom}}$ sheaves is compatible with base changes, it is enough to consider the sections over $S$ and prove that we have a bijection of $\operatorname{Hom}$ sets. Also, the injectivity part is clear because $\rho_{0}$ is an epimorphism of sheaves.

First we reduce to the case $M$ affine. We are free to fix a morphism $u\colon X\to M$ and prove that $\Phi$ induces a bijection between the subsets $\operatorname{Hom}_{u}(\pi_{0}(X/S),\operatorname{Fdiv}(M))$ and $\operatorname{Hom}_{u}(X,\operatorname{Fdiv}(M))$ of maps that induce the same $u$. Since $X\to S$ is quasicompact, the map $u$ factors through a quasicompact open subspace $M^{\prime}\subset M$, and all maps in the above $\operatorname{Hom}_{u}$ subsets factor through $\operatorname{Fdiv}(M^{\prime})$. Therefore, replacing $M$ by $M^{\prime}$ if necessary, we can assume that $M$ is quasicompact. Let $V\to M$ be an étale surjection with $V$ an affine scheme. Since $M$ is quasi-separated, then $g$ is finitely presented; hence the space X_{V}\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=X\times_{M}V is finitely presented, flat and separable over $S$. Now start from a map $f:X\to\operatorname{Fdiv}(M)$. Taking into account that \operatorname{Fdiv}(V)\xrightarrow{\,\smash{\raisebox{-2.15277pt}{\scriptstyle\sim}}\,}\operatorname{Fdiv}(M)\times_{M}V, see 2.3.2, point (v), by pullback along $V\to M$ we obtain a map $f_{V}:X_{V}\to\operatorname{Fdiv}(V)$. By assumption, since $V$ is affine the map

[TABLE]

is an isomorphism; hence $f_{V}$ factors uniquely via $\pi_{0}(X_{V})$. By the pushout property of Lemma 3.3.3, the diagram

[TABLE]

can be completed by a dashed arrow and the claim is proven.

Now we reduce to the case $X$ affine. Let $U\to X$ be an étale atlas with $U$ affine. Starting from a map $X\to\operatorname{Fdiv}(M)$, by assumption the composition $U\to X\to\operatorname{Fdiv}(M)$ factors through $\pi_{0}(U)$. Using once more the pushout of Lemma 3.3.3, the diagram

[TABLE]

can be completed by a dashed arrow and this completes the proof.

To conclude when $S,M,X$ are affine, let $X^{\operatorname{copf}}$ be the coperfection in the sense of sheaves as in 2.3.1, and compute:

[TABLE]

$\square$

5.2 The étale fundamental pro-groupoid

In this subsection, the étale fundamental pro-groupoid $\Pi_{1}(\mathscr{X}/S)$ of a flat finitely presented algebraic stacks $\mathscr{X}/S$ is defined as a 2-pro-object of the 2-category of algebraic stacks. Let us recall the definition of this concept. For more details, we refer to the paper [DD14].

** 5.2.1**

Definition. A nonempty 2-category $\mathcal{I}$ is 2-cofiltered if it satifies the following conditions:

1. (1)

Given two objects $i,j\in\mathcal{I}$, there is an object $k\in\mathcal{I}$ and arrows $k\to i$, $k\to j$; 2. (2)

Given two arrows $f,g:j\to i$, there is an arrow $h:k\to j$ and a 2-isomorphism $\alpha:fh\to gh$; 3. (3)

Given two 2-arrows $\alpha,\beta:f\to g$, where $f,g\in\mathscr{H}\!om_{\mathcal{I}}(j,i)$, there is an arrow $h:k\to j$ such that $\alpha h=\beta h$.

Clearly, a nonempty $1$-category is cofiltered if and only if it is 2-cofiltered when seen as a $2$-category.

** 5.2.2**

Definition. A 2-pro-object of a 2-category $\mathcal{C}$ is a 2-functor $F:\mathcal{I}\to\mathcal{C}$ from a small 2-cofiltered 2-category $\mathcal{I}$. The 2-category of 2-pro-objects of $\mathcal{C}$ is denoted by $2{\text{-}}{\rm Pro}(\mathcal{C})$. The category of morphisms between two 2-pro-objects $F:\mathcal{I}\to\mathcal{C}$ and $G:\mathcal{J}\to\mathcal{C}$ is

[TABLE]

where $\lim$ (resp. $\operatorname{colim}$) is the pseudolimit (resp. pseudocolimit) for strict 2-categories, cf. [DD14], Prop. 2.1.5. In particular, by a pro-algebraic stack we mean a 2-pro-object of the 2-category ${\rm\bf AlgStack}$ of algebraic stacks.

The index 2-category for defining $\Pi_{1}$ will be a 2-category of factorizations similar to that of Definition 3.1.1, with the difference that the étale part $\mathscr{E}\to S$ is allowed to be an algebraic stack rather than an algebraic space. For simplicity, we use again the notation ${\sf E}^{\operatorname{surj}}(\mathscr{X}/S)$ although to be fully consistent, the category defined in 3.1.1 should be denoted ${\sf E}^{\rm surj,rep}(\mathscr{X}/S)$ to indicate that $\mathscr{E}\to S$ is representable by algebraic spaces. No confusion is likely to occur since the former definition is not used anymore in the present section of the article.

** 5.2.3**

Definition. Let $\mathscr{X}/S$ be a flat finitely presented algebraic stack. We define ${\sf E}^{\rm surj}(\mathscr{X}/S)$ to be the following 2-category:

• •

objects are factorizations $\mathscr{X}\xrightarrow{h}\mathscr{E}\to S$ where $\mathscr{E}/S$ is an étale, finitely presented algebraic stack and $h$ is surjective;

• •

1-arrows $(\mathscr{X}\xrightarrow{h}\mathscr{E}\to S)\to(\mathscr{X}\xrightarrow{h^{\prime}}\mathscr{E}^{\prime}\to S)$ are pairs $(f,\alpha)$, with $f\colon\mathscr{E}\to\mathscr{E}^{\prime}$ and $\alpha\colon fh\to h^{\prime}$ giving a 2-commutative diagram:

[TABLE]

• •

2-arrows $(f,\alpha)\to(g,\beta)$ are $2$-morphisms $u\colon f\to g$ giving a commutative diagram:

[TABLE]

We emphasize that for a factorization $\mathscr{X}\to\mathscr{E}\to S$ in ${\sf E}^{\rm surj}(\mathscr{X}/S)$, the requirement that $\mathscr{E}\to S$ be quasi-separated will be crucial in the sequel, cf Remark LABEL:rmk:qs_importance(1). On the contrary, the condition of quasi-compactness of $\mathscr{E}\to S$ is automatic from the same property for $\mathscr{X}\to S$.

** 5.2.4**

Lemma.* Let $\mathscr{X}/S$ be a flat finitely presented algebraic stack. The 2-category ${\sf E}^{\rm surj}(\mathscr{X}/S)$ is small and 2-cofiltered. Moreover, it is equivalent to a 1-category.*

Proof :

Since $\mathscr{X}$ and $\mathscr{E}$ in ${\sf E}^{\rm surj}(\mathscr{X}/S)$ are all finitely presented, it is standard to deduce that ${\sf E}^{\rm surj}(\mathscr{X}/S)$ is a small 2-category. Moreover, ${\sf E}^{\rm surj}(\mathscr{X}/S)$ is nonempty, because it contains the image of $\mathscr{X}$ in $S$, which is open in $S$ hence étale over $S$. Next, we check the three conditions for 2-cofilteredness.

(1) Given two factorizations $h:\mathscr{X}\to\mathscr{E}$ and $h^{\prime}:\mathscr{X}\to\mathscr{E}^{\prime}$, there is the common refinement $\mathscr{X}\to\mathscr{E}\times_{S}\mathscr{E}^{\prime}$ and 2-commutative diagram

[TABLE]

Take the image $\mathscr{E}^{\prime\prime}$ of $\mathscr{X}\to\mathscr{E}\times\mathscr{E}^{\prime}$. Then $\mathscr{E}^{\prime\prime}$ is again an étale finitely presented $S$-stack and $h^{\prime\prime}:\mathscr{X}\to\mathscr{E}^{\prime\prime}$ is a common refinement of $h$ and $h^{\prime}$ in ${\sf E}^{\rm surj}(\mathscr{X}/S)$.

(2) Given two morphisms $(f,\alpha)$ and $(g,\beta)$

[TABLE]

we want to find a third morphism $(k,\gamma):\mathscr{E}^{\prime\prime}\to\mathscr{E}$ and a 2-isomorphism $u:fk\to gk$. For this we consider the 2-fibred product:

[TABLE]

Then $u$ is given by definition. Moreover, the morphisms $h:\mathscr{X}\to\mathscr{E}$ and $h^{\prime}:\mathscr{X}\to\mathscr{E}^{\prime}$ and the 2-commutativity isomorphisms

[TABLE]

provide a morphism $(h,h^{\prime}):\mathscr{X}\to\mathscr{E}^{\prime\prime}$.

(3) Given two morphisms $(f,\alpha)$, $(g,\beta)$ and two 2-morphisms $u,v:(f,\alpha)\to(g,\beta)$:

[TABLE]

we want to find a third morphism $(k,\gamma):\mathscr{E}^{\prime\prime}\to\mathscr{E}$ such that $uk=vk$. For this we view $f$ and $g$ as $\mathscr{E}$-valued points of the stack $\mathscr{E}^{\prime}$ and $u,v$ as sections of the Isom functor I\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=\underline{\operatorname{Isom}}_{\mathscr{E}}(f,g)\to\mathscr{E}, that is $u,v:\mathscr{E}\to\underline{\operatorname{Isom}}_{\mathscr{E}}(f,g)$. Since the diagonal of $\mathscr{E}^{\prime}$ is an étale morphism, the map $I\to\mathscr{E}$ is representable and étale, so its diagonal is an open immersion. We consider the fibred product:

[TABLE]

The 2-commutativity isomorphisms

[TABLE]

provide a morphism $\mathscr{X}\to I$. Moreover, the conditions $\beta\circ uh=\beta\circ vh=\alpha$ ensure that $(uh,vh)=(\beta^{-1}\alpha,\beta^{-1}\alpha)$, that is, we have a commutative square:

[TABLE]

We deduce a morphism $h^{\prime\prime}:\mathscr{X}\to\mathscr{E}^{\prime\prime}$. Moreover, since we have the diagram

[TABLE]

where the map $h$ is surjective, the vertical inclusion is in fact an isomorphism. Hence the two 2-morphisms $u,v$ are equalized by an isomorphism $k:\mathscr{E}^{\prime\prime}\to\mathscr{E}$. In particular, it means that for any such two morphisms $(f,\alpha)$ and $(g,\beta)$, there is at most one 2-isomorphism between them, thus ${\sf E}^{\rm surj}(\mathscr{X}/S)$ is equivalent to a 1-category. $\square$

** 5.2.5**

Definition. Let $\mathscr{X}/S$ be a flat finitely presented algebraic stack. We define the étale fundamental pro-groupoid $\Pi_{1}(\mathscr{X}/S)$ of $\mathscr{X}$ to be the pro-algebraic stack

[TABLE]

The pro-algebraic stack $\Pi_{1}(\mathscr{X}/S)$ is pro-étale by definition, and it comes with a canonical morphism $\mathscr{X}\to\Pi_{1}(\mathscr{X}/S)$ which is unique up to a unique 2-isomorphism. This object defines a 2-functor

[TABLE]

from the 2-category of flat finitely presented algebraic stacks over $S$ to the 2-category of pro-étale stacks over $S$. It is tautological from its definition that the 2-functor $\Pi_{1}(-/S)$ is pro-left adjoint to the inclusion ${\rm\bf EtStack}_{S}\hookrightarrow{\rm\bf FlStack}_{S}$. Finally, if $\mathscr{X}/S$ is moreover separable, the space of connected components $\pi_{0}(\mathscr{X}/S)$ is a member of the category ${\sf E}^{\rm surj}(\mathscr{X}/S)$. It follows that there is a morphism $\Pi_{1}(\mathscr{X}/S)\to\pi_{0}(\mathscr{X}/S)$ with target the constant 2-pro-object. This morphism is easily seen to be universal for morphisms from $\Pi_{1}(\mathscr{X}/S)$ to an étale algebraic space; we call it the coarse moduli space.

** 5.2.6**

$\Pi_{1}$** via smooth atlases.**

Now let us assume that $\mathscr{X}$ is separable. Let $U\to\mathscr{X}$ be a smooth atlas with $U$ finitely presented, and $R=U\times_{\mathscr{X}}U$. Note that, because of quasi-compactness and quasi-separation of $\mathscr{X}$, we can always choose $U\to\mathscr{X}$ to be quasi-compact and quasi-separated. Indeed, we can find a quasi-compact algebraic space $U_{0}$ as a smooth atlas of $\mathscr{X}$, then by [SP19] Tag 050Y, $U_{0}\to\mathscr{X}$ is quasi-compact. Taking an affine Zariski covering $U\to U_{0}$ provides an atlas which is quasi-compact and quasi-separated over $\mathscr{X}$. Now since $\mathscr{X}$ is finitely presented, a quasi-compact quasi-separated $U\to\mathscr{X}$ is also finitely presented, hence we can take $\pi_{0}$ of $U$ and $R$. In the sequel, for simplicity let us write $\pi_{0}(U)$ for $\pi_{0}(U/S)$. Therefore the groupoid presentation $R\rightrightarrows U$ of $\mathscr{X}$ induces a 2-commutative diagram

[TABLE]