Fundamental gerbes (Q1738873)

The paper under review centers on generalizations of the \textit{fundamental group scheme} defined by \textit{M. V. Nori} [Proc. Indian Acad. Sci., Math. Sci. 91, 73--122 (1982; Zbl 0586.14006)]. Recall that for a proper connected reduced scheme $X$ over a field $\kappa$ and a rational point $x_0 \in X(\kappa)$, the Nori fundamental group scheme $\pi(X,x_0)$ is a pro-finite group scheme over $\operatorname{Spec} \kappa$ whose base change over $X$ is the structure group of a torsor $P \to X$ with trivial fiber $P_{x_0}$, which is initial among such pro-finite group schemes in a certain sense. \par In previous work [J. Algebr. Geom. 24, No. 2, 311--353 (2015; Zbl 1349.14004)], the authors generalized Nori's construction in a number of ways, including allowing $X$ to be a fibered category (satisfying the property they introduced of being \textit{inflexible}), dropping the assumption that $X$ contains a rational point, and replacing $\pi(X,x_0)$ by the \textit{fundamental gerbe}, which is a pro-finite gerbe $\Pi_{X/\kappa}^{\mathrm{N}}$ over $\operatorname{Spec} \kappa$ equipped with a morphism $X \to \Pi_{X/\kappa}^{\mathrm{N}}$ which is initial among morphisms from $X$ to pro-finite gerbes. \par One of the main aims of the present paper is to further generalize the construction of $\Pi_{X/\kappa}^{\mathrm{N}}$ to allow more general classes of group schemes in the role played by finite group schemes. Thus, given $X$ and a class of group schemes $\mathscr{C}$, the authors define the \textit{$\mathscr{C}$-fundamental gerbe} to be a pro-$\mathscr{C}$-gerbe $\Pi_{X/\kappa}^{\mathscr{C}}$ (an inverse limit of gerbes whose points all have stabilizer groups in $\mathscr{C}$) equipped with a morphism $X \to \Pi_{X/\kappa}^{\mathscr{C}}$ which is initial among morphisms from $X$ to pro-$\mathscr{C}$-gerbes. \par For which classes $\mathscr{C}$ does the $\mathscr{C}$-fundamental gerbe exist, under reasonable hypotheses on $X$? To answer this, the authors introduce the key definition of a \textit{well-founded} class $\mathscr{C}$ of group schemes over fields, which means that $\mathscr{C}$ satisfies some natural stability conditions and is a subclass of the class of \textit{virtually nilpotent} groups $G$ (such a $G$ over a field $k$ is an affine group scheme of finite type whose base change over an algebraic closure $\overline{k}$ contains a nilpotent subgroup scheme of finite index). The first main result of the paper is that for any fibered category $X$ over $\kappa$ which is geometrically reduced, which satisfies a mild finiteness hypothesis (equivalent to being quasi-compact and quasi-separated when $X$ is a scheme), and such that $\mathrm{H}^0(X,\mathscr{O}_X) = \kappa$, the $\mathscr{C}$-fundamental gerbe exists for any well-founded $\mathscr{C}$. The authors furthermore assert a kind of converse: if $\Pi_{X/\kappa}^{\mathscr{C}}$ exists for any such $X$, then $\mathscr{C}$ must be well-founded. \par The class of all virtually nilpotent group schemes is itself a well-founded class, but it also contains many interesting well-founded subclasses, such as unipotent group schemes, affine finite type commutative group schemes, finite type group schemes of multiplicative type, and finite group schemes. In particular, when $\mathscr{C}$ is the class of unipotent groups, the construction of $\Pi_{X,\kappa}^{\mathrm{U}} := \Pi_{X/\kappa}^{\mathscr{C}}$ (the \textit{unipotent fundamental gerbe}) generalizes the unipotent fundamental group scheme $\pi_1^{\mathrm{U}}(X,x_0)$ of Nori, which was also constructed by him in [loc. cit.]. \par The second group of results in the paper concerns Tannakian interpretations of the $\mathscr{C}$-fundamental gerbe for various choices of $\mathscr{C}$. Let $\operatorname{Rep} \Pi_{X/\kappa}^{\mathscr{C}}$ denote the category of finite rank vector bundles on $\Pi_{X/\kappa}^{\mathscr{C}}$. Then $\operatorname{Rep} \Pi_{X/\kappa}^{\mathscr{C}}$ is a Tannakian category, and pullback along the universal morphism $X \to \Pi_{X/\kappa}^{\mathscr{C}}$ yields a functor \[ \operatorname{Rep} \Pi_{X/\kappa}^{\mathscr{C}} \to \mathrm{Vect}_X \tag{$*$}\] to the category of finite rank vector bundles on $X$. When $\mathscr{C}$ is the class of finite group schemes, the authors proved in their previous work [loc. cit.] that $(*)$ is fully faithful, with essential image consisting of the so-called essentially finite vector bundles on $X$; in Nori's original setting, this recovers his result that the category of $\pi(X,x_0)$-representations is equivalent to that of essentially finite vector bundles on $X$. In the present paper, when $\mathscr{C}$ is any well-founded subclass of the class of virtually unipotent groups (defined analogously to virtually nilpotent groups), the authors prove that $(*)$ is again fully faithful. When $\mathscr{C}$ is the class of unipotent groups, the authors show that the essential image of $(*)$ consists of the vector bundles on $X$ which are iterated extensions of trivial bundles; this generalizes the Tannakian description of $\pi_1^{\mathrm{U}}(X,x_0)$ given by Nori. When $\mathscr{C}$ is the class of virtually unipotent groups and $\kappa$ has characteristic zero, the authors show that the essential image of $(*)$ consists of the vector bundles on $X$ which are iterated extensions of essentially finite bundles (\textit{extended essentially finite locally free sheaves}, in the authors' terminology; these were defined as \textit{semi-finite bundles} by \textit{S. Otabe} [Commun. Algebra 45, No. 8, 3422--3448 (2017; Zbl 1408.14145)]). For the same $\mathscr{C}$ and $\kappa$ of positive characteristic, the authors show that the essential image consists of the bundles which become extended essentially finite after pullback by a sufficiently high power of the absolute Frobenius. \par In the final section of the paper, the authors take $\mathscr{C}$ to be the class of finite type groups of multiplicative type and give an alternative ``direct'' construction of the fundamental gerbe of multiplicative type $\Pi_{X,\kappa}^{\mathrm{MT}} := \Pi_{X/\kappa}^{\mathscr{C}}$, in terms of the Picard stack $\underline{\mathrm{Pic}}_{X/\kappa}$; this bypasses all the general machinery of the previous parts of the paper, and is valid for fibered categories $X$ as in the first main result described above, except the geometrically reduced hypothesis on $X$ can be dropped. The gerbe $\Pi_{X,\kappa}^{\mathrm{MT}}$ generalizes, and gives a conceptual interpretation of, the universal torsor of \textit{J.-L. Colliot-Thélène} and \textit{J.-J. Sansuc} [C. R. Acad. Sci., Paris, Sér. A 282, 1113--1116 (1976; Zbl 0337.14014); Duke Math. J. 54, 375--492 (1987; Zbl 0659.14028)]. Along the way, the authors prove a duality theorem for gerbes of multiplicative type generalizing the standard duality between group schemes of multiplicative type and abelian groups with a continuous action of $\mathrm{Gal}(\kappa^{\mathrm{sep}}/\kappa)$.