可定义函子与布朗-阿德斯可代表性
Definable functors and Brown--Adams representability
AI总结：
本文研究了环的导出范畴中布朗-阿德斯可代表性的条件，通过可定义函子的同构转换，证明了纯全局维数的关系，并给出了茶杯猜想的新证明及协调环的生成假设新结论。
摘要：
重新审视当环的导出范畴满足布朗-阿德斯可代表性时的条件，通过研究纯同调维数沿可定义函子的转移：证明对于任何环，导出范畴的纯全局维数至少等于环的纯全局维数；扩展了Beligiannis和Keller-Christensen-Neeman的结果。该结果通过构造“改变范畴”同构来获得，这些同构在可定义函子之间转换PExt群。相同的同构展示了何时可以转移布朗-阿德斯可代表性的性质。我们展示了如何利用这些方法来测试某些准 coherent sheaves 的导出范畴是否为布朗范畴。我们还研究了von Neumann正则环的导出范畴的结构，证明在许多情况下这些环控制布朗-阿德斯可代表性；这包括对茶杯猜想的新证明，以及一个新且简短的证明，即一个协调环满足Freyd的（强）生成假设当且仅当它是von Neumann正则环。
英文摘要：
The question of when the derived category of a ring satisfies Brown--Adams representability is revisited via studying the transfer of pure homological dimension along definable functors: it is shown that, for any ring, the pure global dimension of the derived category is at least the pure global dimension of the ring; expanding results of Beligiannis and Keller-Christensen-Neeman. This result is obtained by constructing `change of category' isomorphisms of PExt groups across definable functors. The same isomorphisms illustrate circumstances when one can transfer the property of Brown--Adams representability. We demonstrate how these methods can be used to test whether certain derived category of quasi-coherent sheaves are a Brown category. We also make an investigation into the structure of derived categories of von Neumann regular rings, which are shown in many cases to control Brown--Adams representability; this includes a new proof of the telescope conjecture, and a new and short proof that a coherent ring satisfies Freyd's (strong) generating hypothesis if and only if it is von Neumann regular.
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