We study regularity in the context of connective 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 connective ${{\mathcal{E}}_\infty}$-k-algebra and G is a linearly reductive group acting on R. Under reasonable assumptions, 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.

中文翻译：

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谱栈的规律性和权重心的离散性

我们在连接环光谱和光谱堆栈的背景下研究规律性。与此平行，我们在定义在域上的形式为 $X=[\operatorname{Spec} R/G]$ 的谱商堆栈上的紧致准相干滑轮类别上构建了一个权重结构，其中 R 是连接词 $ {{\mathcal{E}}_\infty}$-k-代数和 G 是作用于 R 上的线性约简群。在合理的假设下，我们证明 X 的规律性等价于 R 的规律性。我们还证明如果R 是有界的，这样的堆栈是离散的。这个结果可以用权重结构来解释，并暗示了一个普遍现象：对于具有兼容的有界权重结构的对称单面稳定 $\infty$-类别，满足强有界条件的相邻 t-结构的存在应该意味着离散性的重量心脏。