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Register a new userThe six operations in equivariant motivic homotopy theory
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Authors
Marc Hoyois
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We introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks [X/G], where G is a linearly reductive linear algebraic group. We extend to this equivariant setting the main foundational results of motivic homotopy theory: the (unstable) purity and gluing theorems of Morel and Voevodsky and the (stable) ambidexterity theorem of Ayoub. Our proof of the latter is different than Ayoubs and is of interest even when G is trivial. Using these results, we construct a formalism of six operations for equivariant motivic spectra, and we deduce that any cohomology theory for G-schemes that is represented by an absolute motivic spectrum satisfies descent for the cdh topology.
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If $f:S to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal norm functor $f_otimes: mathcal H_*(S) tomathcal H_*(S)$, where $mathcal H_*(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite etale, we show that it stabilizes to a functor $f_otimes: mathcal{SH}(S) to mathcal{SH}(S)$, where $mathcal{SH}(S)$ is the $mathbb P^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendiecks Galois theory, with Betti realization, and with Voevodskys slice filtration; we prove that the norm functors categorify Rosts multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $Hmathbb Z$, the homotopy K-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $Hmathbb Z$ is a common refinement of Fulton and MacPhersons mutliplicative transfers on Chow groups and of Voevodskys power operations in motivic cohomology.
We give an algebro-geometric interpretation of $C_2$-equivariant stable homotopy theory by means of the $b$-topology introduced by Claus Scheiderer in his study of $2$-torsion phenomena in etale cohomology. To accomplish this, we first revisit and extend work of Scheiderer on equivariant topos theory by functorially associating to a $infty$-topos $mathscr{X}$ with $G$-action a presentable stable $infty$-category $mathrm{Sp}^G(mathscr{X})$, which recovers the $infty$-category $mathrm{Sp}^G$ of genuine $G$-spectra when $mathscr{X}$ is the terminal $G$-$infty$-topos. Given a scheme $X$ with $1/2 in mathcal{O}_X$, our construction then specializes to produce an $infty$-category $mathrm{Sp}^{C_2}_b(X)$ of $b$-sheaves with transfers as $b$-sheaves of spectra on the small etale site of $X$ equipped with certain transfers along the extension $X[i] rightarrow X$; if $X$ is the spectrum of a real closed field, then $mathrm{Sp}^{C_2}_b(X)$ recovers $mathrm{Sp}^{C_2}$. On a large class of schemes, we prove that, after $p$-completion, our construction assembles into a premotivic functor satisfying the full six functors formalism. We then introduce the $b$-variant $mathrm{SH}_b(X)$ of the $infty$-category $mathrm{SH}(X)$ of motivic spectra over $X$ (in the sense of Morel-Voevodsky), and produce a natural equivalence of $infty$-categories $mathrm{SH}_b(X)^{wedge}_p simeq mathrm{Sp}^{C_2}_b(X)^{wedge}_p$ through amalgamating the etale and real etale motivic rigidity theorems of Tom Bachmann. This involves a purely algebro-geometric construction of the $C_2$-Tate construction, which may be of independent interest. Finally, as applications, we deduce a $b$-rigidity theorem, use the Segal conjecture to show etale descent of the $2$-complete $b$-motivic sphere spectrum, and construct a parametrized version of the $C_2$-Betti realization functor of Heller-Ormsby.
We prove a topological invariance statement for the Morel-Voevodsky motivic homotopy category, up to inverting exponential characteristics of residue fields. This implies in particular that SH[1/p] of characteristic p>0 schemes is invariant under passing to perfections. Among other applications we prove Grothendieck-Verdier duality in this context.
We study generically split octonion algebras over schemes using techniques of ${mathbb A}^1$-homotopy theory. By combining affine representability results with techniques of obstruction theory, we establish classification results over smooth affine schemes of small dimension. In particular, for smooth affine schemes over algebraically closed fields, we show that generically split octonion algebras may be classified by characteristic classes including the second Chern class and another mod $3$ invariant. We review Zorns vector matrix construction of octonion algebras, generalized to rings by various authors, and show that generically split octonion algebras are always obtained from this construction over smooth affine schemes of low dimension. Finally, generalizing P. Gilles analysis of octonion algebras with trivial norm form, we observe that generically split octonion algebras with trivial associated spinor bundle are automatically split in low dimensions.
We construct many``low rank algebraic vector bundles on ``simple smooth affine varieties of high dimension. In a related direction, we study the existence of polynomial representatives of elements in the classical (unstable) homotopy groups of spheres. Using techniques of A^1-homotopy theory, we are able to produce ``motivic lifts of elements in classical homotopy groups of spheres; these lifts provide interesting polynomial maps of spheres and algebraic vector bundles.
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