Tate模作为凝聚模
Tate modules as condensed modules
AI总结：
本文证明任意离散环上的可数Tate模范畴完全忠实嵌入凝聚模范畴，并在有限型基环下刻画本质像，同时将结果推广至∞-范畴语境，通过反例说明有界性假设的必要性。
摘要：
我们证明了任意离散环上的可数Tate模范畴完全忠实地嵌入凝聚模范畴。若基环为有限型，我们刻画其本质像为由无限可数秩的自由模在直和、对偶和收缩下生成的子范畴。在∞-范畴语境下，在基环的相同假设下，我们建立了完美复形中具有一致有界tor振幅的可数Tate对象的∞-范畴到凝聚模的导出∞-范畴的完全忠实嵌入。有界性假设对于保证满忠实性是必要的，我们通过无界情形下的一个显式反例证明了这一点。
英文摘要：
We prove that the category of countable Tate modules over an arbitrary discrete ring embeds fully faithfully into that of condensed modules. If the base ring is of finite type, we characterize the essential image as generated by the free module of infinite countable rank under direct sums, duals and retracts. In the $\infty$-categorical context, under the same assumption on the base ring, we establish a fully faithful embedding of the $\infty$-category of countable Tate objects in perfect complexes, with uniformly bounded tor-amplitude, into the derived $\infty$-category of condensed modules. The boundedness assumption is necessary to ensure fullness, as we prove via an explicit counterexample in the unbounded case.
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