# Regularity of spectral stacks and discreteness of weight-hearts

## TL;DR

This paper explores the regularity properties of spectral stacks and establishes a connection between regularity of the stack and the underlying algebra, also demonstrating conditions under which the stack's structure becomes discrete.

## Contribution

It constructs a weight structure on spectral quotient stacks and proves that regularity of the stack is equivalent to regularity of the algebra, also showing discreteness under boundedness.

## Key findings

- Regularity of spectral stacks is equivalent to algebra regularity.
- Bounded spectral stacks are shown to be discrete.
- A gluing result for weight structures and t-structures is established.

## 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 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.