有限张量范畴的Benson--Etingof--Ostrik猜想
The Benson--Etingof--Ostrik conjecture for finite tensor categories
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中文总结 AI 辅助
本文证明了正特征代数闭域上每个有限对称张量范畴均存在到高阶Verlinde范畴的对称张量函子，验证了Benson-Etingof-Ostrik猜想，并改进了相关猜想，连接了受限Steinberg模与迭代广义上同调。
中文摘要 AI 辅助
我们证明了在正特征代数闭域上的每个有限对称张量范畴都容许一个到高阶Verlinde范畴的对称张量函子。这一猜想由Benson、Etingof和Ostrik针对任意中等增长的对称张量范畴提出。证明使用了作者在[CF1]中引入的高阶Frobenius函子。主要努力用于证明[CF1]中关于初等阿贝尔$p$-群的表示范畴中厚张量理想的一个猜想，该猜想蕴含高阶Frobenius函子是对称幺半的。事实上，我们证明了后一猜想的一个细化，该细化将张量积与受限Steinberg模联系起来，并关联到初等阿贝尔$p$-群的模的迭代广义上同调。
英文摘要
We prove that every finite symmetric tensor category over an algebraically closed field of positive characteristic admits a symmetric tensor functor to a higher Verlinde category. This was conjectured for arbitrary symmetric tensor categories of moderate growth by Benson, Etingof and Ostrik. The proof uses higher Frobenius functors, introduced by the authors in [CF1]. The main effort is directed towards a proof of a conjecture from [CF1] about thick tensor ideals in the representation category of elementary abelian $p$-groups, which implies that the higher Frobenius functors are symmetric monoidal. In fact, we prove a refinement of the latter conjecture that relates tensoring with restricted Steinberg modules to an iterated generalised cohomology of modules of elementary abelian $p$-groups.
