Regularity of Spectral Stacks and Discreteness of Weight-Hearts

Adeel A. Khan, V. Sosnilo
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引用次数: 8

Abstract

We study regularity in the context of ring spectra and spectral stacks. Parallel to that, we construct a weight structure on the category of compact quasi-coherent sheaves on spectral quotient stacks of the form $X=[\operatorname{Spec} R/G]$ defined over a field, where $R$ is a noetherian ${\mathcal{E}_{\infty}}$-$k$-algebra and $G$ is a linearly reductive group acting on $R$. In this context we show that regularity of $X$ is equivalent to regularity of $R$. We also show that if $R$ is bounded, such a stack is discrete. This result can be interpreted in terms of weight structures and suggests a general phenomenon: for a symmetric monoidal stable $\infty$-category with a compatible bounded weight structure, the existence of an adjacent t-structure satisfying a strong boundedness condition should imply discreteness of the weight-heart. We also prove a gluing result for weight structures and adjacent t-structures, in the setting of a semi-orthogonal decomposition of stable $\infty$-categories.
谱叠加的正则性与权重心的离散性
我们在环谱和谱堆的背景下研究了正则性。与此平行,我们在谱商堆栈上的紧拟相干束的范畴上构造了一个权结构,其形式为$X=[\operatorname{Spec} R/G]$,其中$R$是一个诺etherian ${\mathcal{E}_{\infty}}$ - $k$ -代数,$G$是作用于$R$的线性约化群。在这种情况下,我们证明$X$的正则性等价于$R$的正则性。我们还证明了如果$R$是有界的,那么这样的堆栈是离散的。这一结果可以用权结构来解释,并提出了一个普遍的现象:对于具有相容有界权结构的对称单轴稳定$\infty$ -类,满足强有界性条件的相邻t结构的存在应该意味着权心的离散性。我们还证明了在稳定$\infty$ -类别的半正交分解的情况下,权结构和相邻t结构的粘合结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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