AI 中文总结
研究引入可表示对称幺半$\infty$-范畴的Zariski框架统一谱理论，其与smashing框架有关，还探讨通过理想形成商的问题，引入相关性质，最后作为应用构造了$\mathbb{E}_\infty$-半环的商。
AI 中文摘要
我们引入了任何可表示对称幺半$\infty$-范畴的Zariski框架。这使我们能够统一高等代数中出现的几种谱理论。当范畴是紧生成时，Zariski框架是相干的，相关的谱空间恢复了交换环的经典Zariski谱和交换2-环的Balmer谱的Hochster对偶。此外，任何稳定可表示对称幺半$\infty$-范畴的smashing框架可与其对偶可化模的范畴的Zariski框架等同。此构造基于对称幺半$\infty$-范畴中的理想应理解为到单位对象的单态射这一原理。在适当的上下文中，该概念恢复了前面例子中出现的各种理想，包括厚理想和smashing理想，并且还适用于非稳定$\infty$-范畴的smashing理想。我们还研究了通过理想形成商的问题，这在高等代数环境中很微妙。为此，我们引入了带点$\infty$-范畴的两个性质，称为$\Sigma$-平凡性和$\Sigma$-正合性。这些条件确保了通过理想进行商运算表现良好。作为应用，我们构造了$\mathbb{E}_\infty$-半环的商。
英文摘要
We introduce the Zariski frame of any presentably symmetric monoidal $\infty$-category. This allows us to unify several spectral theories arising in higher algebra. The Zariski frame is coherent whenever the category is compactly generated, and the associated spectral space recovers both the classical Zariski spectrum of a commutative ring and the Hochster dual of the Balmer spectrum of a commutative $2$-ring. Moreover, the smashing frame of any stable presentably symmetric monoidal $\infty$-category can be identified with the Zariski frame of its category of dualizable modules. This construction is based on the principle that ideals in a symmetric monoidal $\infty$-category should be understood as monomorphisms into the unit object. In suitable contexts, this notion recovers the kinds of ideals appearing in the preceding examples, including thick ideals and smashing ideals, and it also accommodates the smashing ideals of non-stable $\infty$-categories. We also study the problem of forming quotients by ideals, which is subtle in the setting of higher algebra. To address this, we introduce two properties of pointed $\infty$-categories, called $Σ$-triviality and $Σ$-exactness. These conditions ensure that quotienting by ideals behaves well. As an application, we construct quotients of $\mathbb{E}_\infty$-semirings.
Comments48 pages, comments welcome!
