Motivic homotopy theory of algebraic stacks (Q6566574)

Grothendieck and collaborators were the first to introduce the notion of a six functor formalism in the context of étale cohomology of schemes [\textit{M. Artin} (ed.) et al., Séminaire de géométrie algébrique du Bois-Marie 1963--1964. Théorie des topos et cohomologie étale des schémas (SGA 4). Un séminaire dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de P. Deligne, B. Saint-Donat. Tome 3. Exposés IX à XIX. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0245.00002)]. It became increasingly clear over the years the utility of such a framework in different geometric contexts. \textit{J. Ayoub} [Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I. Paris: Société Mathématique de France (2007; Zbl 1146.14001); Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. II. Paris: Société Mathématique de France (2007; Zbl 1153.14001)] developed a fully fledged six functor formalism associated to Morel-Voevodsky motivic homotopy category for schemes, and around the same time Laszlo and Olsson extended the Grothendieck's six functor formalism for the \(l\)-adic derived categories to algebraic stacks in [\textit{Y. Laszlo} and \textit{M. Olsson}, Publ. Math., Inst. Hautes Étud. Sci. 107, 109--168 (2008; Zbl 1191.14002); Publ. Math., Inst. Hautes Étud. Sci. 107, 169--210 (2008; Zbl 1191.14003)]. \textit{M. Hoyois} first [Adv. Math. 305, 197--279 (2017; Zbl 1400.14065)], and \textit{A. A. Khan} and \textit{C. Ravi} later [Adv. Math. 458, Part B, Article ID 109975, 104 p. (2024; Zbl 07945715)], extended Ayoub's work constructing \textit{genuine} motivic homotopy categories \(SH^{gen}(\mathcal X)\) for quotient and scalloped stacks respectively.\N\NAs suggested by its title, this paper focuses on yet another extension of \(SH(-)\) to a broad class of algebraic stacks: the so-called Borel motivic homotopy category \(SH_{\mathrm{ext}}\), independently also developed in [\textit{A. A. Khan} and \textit{C. Ravi}, Adv. Math. 458, Part B, Article ID 109975, 104 p. (2024; Zbl 07945715)]. While the genuine version \(SH^{\mathrm{gen}}\) of Hoyois and Khan-Ravi aligns more closely with the theory of genuine spectra in algebraic topology, the Borel extended category \(SH_{\mathrm{ext}}\) is conceptually closer to the construction of Laszlo-Olsson in the \(l\)-adic context. The main idea of the paper under review is to leverage the fact that algebraic stacks live in the \(\infty\)-category of Nisnevich sheaves \(\mathrm{Sh}_{\mathrm{Nis}}(\mathrm{Sch})\). Then one defines the category \(\mathrm{NisSt}\) of \textit{Nisnevich stacks} as those algebraic stacks admitting an atlas \(x: X \rightarrow \mathcal X\) that is a Nisnevich local epimorphism when it is seen in \(\mathrm{Sh}_{\mathrm{Nis}}(\mathrm{Sch})\). As noted in Example 2.1.6, a remarkably broad class of algebraic stacks belongs to \(\mathrm{NisSt}\).\N\NUsing right Kan extensions, one can extend any functor \(F: \mathrm{Sch}\rightarrow \mathcal{D}\) satisfying Nisnevich descent to a functor:\N\[\NF_{ext}: \mathrm{NisSt} \longrightarrow \mathcal D\N\]\N\noindent such that it satisfies descent along Nisnevich local epimorphisms. In particular, for \(F=SH\) one gets:\N\N\[\NSH_{ext}(\mathcal X) \cong\ \mathrm{lim}\Big(\begin{tikzcd} SH_{ext}(X) \arrow[r, shift right =2] \arrow[r,shift left =2] & SH_{ext}(X \times_{\mathcal X} X) \arrow[l,dotted] \arrow [r,shift right =4] \arrow[r] \arrow[r,shift left =4] & \cdots \arrow[l, shift left =2 ,dotted] \arrow[l, shift right =2, dotted] \end{tikzcd} \Big)\N\]\N\NThis construction immediatly provides four functors for the extended category \(SH_{ext}(\mathcal{X})\). Namely, for \(f: \mathcal{X}\rightarrow \mathcal{Y}\) one gets the usual pair of adjoint couples:\N\[\Nf^* \dashv f_*\N\]\N\N\[\N\otimes \dashv \mathcal{H}om(-,-)\N\]\NConstructing the exceptional functors, along with their relationships to these four functors, is more involved and follows the techniques recently developed in [\textit{J. Ayoub} et al., Forum Math. Sigma 10, Paper No. e61, 182 p. (2022; Zbl 1505.14051)]. The idea is to once again use Nisnevich descent to extend the functors \(SH_!^*\) and \(SH_*^!\) defined on pairs of schemes \((S,X)\) to pairs of Nisnevich stacks \((\mathcal S,\mathcal X)\) where \(\mathcal{X}\) is an \(\mathcal{S}\)-algebraic stack, with representable, locally finite type and separated structure map. Once the six-functor formalism is established, the author proves its fundamental properties and relations, including smooth and proper base change, localization fiber sequences, homotopy invariance for vector bundles, and purity for representable, smooth, separated, finite type morphisms.