AI 中文总结
本文研究有限左正则带$B$与交换诺特环$R$对应的带代数$RB$的张量三角几何，明确其Balmer谱结构、局部化理想与稳定子集的双射关系，证明望远镜猜想对$D(RB)$成立，并构造反例说明广义Nerves of Steel猜想不成立。
AI 中文摘要
本文研究带代数$RB$的张量三角几何，其中$B$是有限左正则带，$R$是交换诺特环。我们证明$\text{Perf}(RB)$的Balmer谱同胚于$\text{Spec}\thinspace R\times L_B$，$L_B$是$B$的支撑格；还证明$D(RB)$的局部化理想与$\text{Spec}\thinspace R\times L_B$的稳定子集间存在双射，由此推出望远镜猜想对$D(RB)$成立；最后构造带$B$，使得非刚性张量三角范畴$\text{Perf}(RB)$不满足广义Nerves of Steel猜想。
英文摘要
In this paper, we study the tensor triangular geometry of band algebras $RB$, where $B$ is a finite left regular band and $R$ is a commutative Noetherian ring. We show that the Balmer spectrum of $\Perf(RB)$ is homeomorphic to $\Spec R\times L_B$, where $L_B$ is the support lattice of $B$. We also show that there is a bijection between the localizing ideals of $D(RB)$ and the stable subsets of $\Spec R\times L_B$. As a consequence, the telescope conjecture holds for $D(RB)$. Finally, we construct a band $B$ such that the generalized Nerves of Steel Conjecture fails for the non-rigid tensor triangulated category $\Perf(RB)$.
