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 étale cohomology. To accomplish this, we first revisit and extend work of Scheiderer on equivariant topos theory by functorially associating to a ∞-topos $\mathcal{X}$ with G-action a presentable stable ∞-category ${\mathrm{Sp}}^G(\mathcal{X})$, which recovers the ∞-category ${\mathrm{Sp}}^G$ of genuine G-spectra when $\mathcal{X}$ is the terminal G-∞-topos. Given a scheme X with $\frac{1}{2}\in {\mathcal{O}}_X$, our construction then specializes to produce an ∞-category ${\mathrm{Sp}}_b^{C_2}(X)$ of “b-sheaves with transfers” as b-sheaves of spectra on the small étale site of X equipped with certain transfers along the extension $X[i]\to X$; if X is the spectrum of a real closed field, then ${\mathrm{Sp}}_b^{C_2}(X)$ 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 ∞-category $\mathrm{SH}(X)$ of motivic spectra over X (in the sense of Morel-Voevodsky), and produce a natural equivalence of ∞-categories ${\mathrm{SH}}_b(X){}_p^{\wedge }\simeq {\mathrm{Sp}}_b^{C_2}(X){}_p^{\wedge }$ through amalgamating the étale and real étale 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 étale descent of the 2-complete b-motivic sphere spectrum, and construct a parametrized version of the $C_2$-Betti realization functor of Heller-Ormsby.

我们给出对的代数几何解释 $C_2$克劳斯·谢伊德（Claus Scheiderer）在研究电子同调学中的2扭转现象时引入了b-拓扑，从而建立了等价稳定同伦理论。为了实现这一点，我们首先通过功能性地关联到∞-topos来回顾和扩展Scheiderer在等变topos理论上的工作。$\mathcal{X}$具有G动作的可表示的稳定∞类${\mathrm{SP}}^G（\mathcal{X}）$，它恢复了∞类 ${\mathrm{SP}}^G$真正的G光谱何时$\mathcal{X}$是终端G -∞-topos。鉴于方案X与$\frac{\mathrm{1个}}{2}\in {\mathcal{Ø}}_X$，我们的构造便专门产生∞类 ${\mathrm{SP}}_b^{C_2}（X）$的“ b -sheaves与传输”为b上的小étale站点光谱-sheaves X配备有沿着延伸某些转让$X[\mathrm{一世}]\to X$; 如果X是一个实地封闭场的光谱，则${\mathrm{SP}}_b^{C_2}（X）$ 恢复 ${\mathrm{SP}}^{C_2}$。在一大类方案上，我们证明了p补全后，我们的结构组装成一个满足所有六个函子形式主义的前驱函子。然后我们介绍b-变量${\mathrm{SH}}_b（X）$ ∞类别 $\mathrm{SH}（X）$X上的动能谱（在Morel-Voevodsky的意义上），并产生自然等效的∞类${\mathrm{SH}}_b（X）{}_p^{\wedge }\simeq {\mathrm{SP}}_b^{C_2}（X）{}_p^{\wedge }$通过将汤姆·巴赫曼的étale和真实étale动机刚性定理融合在一起。这涉及纯粹的代数几何构造。$C_2$-泰特（Tate）建筑，这可能是独立利益。最后，在应用中，我们推导了一个“ b-刚度”定理，使用Segal猜想来显示2-完全b-势力球谱的étale下降，并构造该参数的参数化版本。$C_2$-Heller-Ormsby的贝蒂实现函子。