Regularity of Spectral Stacks and Discreteness of Weight-Hearts

Abstract

Abstract 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.