标题： 高阶扎里斯基几何 显示英文标题

标题： Higher Zariski Geometry

摘要： 我们重新审视在稳定对称单oidal 幂等完备的$\infty$-范畴的背景下经典的张量三角几何构造，此后称为 2-环。 在此背景下，我们生成了一个扎里斯基拓扑、一个扎里斯基谱、一个局部 2-环空间的范畴（更一般地为$\infty$-拓），以及一个基于“由 Lurie 在\cite{LurieDAG5}中发展的带有几何结构的$\infty$-拓”框架的仿射谱-整体截面伴随关系。 利用 Kock 和 Pitsch 的工作，我们计算得出 2-环的扎里斯基谱的基础空间恢复了其同伦范畴的 Balmer 谱。 这些构造反映了经典扎里斯基几何中交换环（和交换环谱）的类似结构，并且我们还展示了经典扎里斯基几何与更高扎里斯基几何之间的额外相容性。 对于刚性的 2-环，我们证明 Balmer 和 Favi 的下降结果可以有协调增强。 作为推论，我们得到扎里斯基谱将刚性 2-环忠实满地嵌入到局部 2-环的$\infty$-拓中。 在附录中，我们证明了在刚性情况下关于望远镜猜想的“茎局部性原理”，扩展了 Hrbek 的早期工作。 显示英文摘要

摘要： We revisit the classical constructions of tensor-triangular geometry in the setting of stably symmetric monoidal idempotent-complete $\infty$-categories, henceforth referred to as 2-rings. In this setting, we produce a Zariski topology, a Zariski spectrum, a category of locally 2-ringed spaces (more generally $\infty$-topoi), and an affine spectrum-global sections adjunction, based on the framework of ``$\infty$-topoi with geometric structure'' as developed by Lurie in \cite{LurieDAG5}. Using work of Kock and Pitsch, we compute that the underlying space of the Zariski spectrum of a 2-ring recovers the Balmer spectrum of its homotopy category. These constructions mirror the analogous structures in the classical Zariski geometry of commutative rings (and commutative ring spectra), and we also demonstrate additional compatibility between classical Zariski and higher Zariski geometry. For rigid 2-rings, we show that the descent results of Balmer and Favi admit coherent enhancements. As a corollary, we obtain that the Zariski spectrum fully faithfully embeds rigid 2-rings into locally 2-ringed $\infty$-topoi. In an appendix, we prove a ``stalk-locality principle'' for the telescope conjecture in the rigid setting, extending earlier work of Hrbek.