The main motivation of this paper is the study of bounded \(t\)-structures, after the key contributions of Antieau, Gepner and Heller who proved that obstructions to the existence of bounded \(t\)-structures on a stable \(\infty\)-category \(\mathscr{C}\) are controlled by the first negative \(K\)-group \(K_{-1}(\mathscr{C})\) [B. Antieau et al., Invent. Math. 216, No. 1, 241–300 (2019; Zbl 1430.18009)]. This work tackles the problem assuming that \(\mathscr{C}\) is endowed with a weight structure, a structure introduced in [M. V. Bondarko, J. \(K\)-Theory 6, No. 3, 387–504 (2010; Zbl 1303.18019)], similar (but not dual) to a \(t\)-structure, which axiomatizes the properties of naive truncations of chain complexes. This notion is closely related to the concept of regularity of \(\mathbb{E}_1\)-ring spectra, introduced in [C. Barwick and T. Lawson, “Regularity of structured ring spectra and localization in \(K\)-theory”, Preprint, arXiv:1402.6038].
In the first part, the author contributes to the topic of regular \(\mathbb{E}_1\)-ring spectra by proving the stability of regularity spectra under localizations (Proposition \(2.8\)) and the discreteness of bounded regular \(\mathbb{E}_1\)-ring spectra which are quasicommutative (Theorem \(2.11\)). The paper also provides a counterexample to the latter statement in the non-quasicommutative case (Construction \(2.12\)). The bulk of this section, however, is the (twofold) generalization of the concept of regularity to spectral stacks (Definition \(2.15\)): a spectral stack \(X\) is regular if there exists a regular atlas \(\operatorname{Spec}(R) \to X\), while it is homological regular if the standard \(t\)-structure on \(\operatorname{QCoh}(X)\) restricts to the subcategory of compact objects. While in general not equivalent, the two definitions agree if \(X\) is an affine spectral scheme or, with some additional hypothesis, if \(X\) is a quotient of a Noetherian connective \(\mathbb{E}_{\infty}\)-\(k\)-algebra under the action of a smooth affine group scheme (Theorem \(2.16\)).
In the second part, the author recalls the main definitions, properties and examples of weight structures on stable \(\infty\)-categories, and introduces the notion of adjacent structures (Definition \(3.9\)) - i.e., a weight structure \(\left(\mathscr{C}_{w\geqslant 0},\mathscr{C}_{w\leqslant 0}\right)\) and a \(t\)-structure \(\left(\mathscr{C}_{t\geqslant 0},\mathscr{C}_{t\leqslant 0}\right)\) such that \(\mathscr{C}_{w\geqslant 0}=\mathscr{C}_{t\geqslant 0}\). Arguing that the standard \(t\)-structure and the standard weight structure on \(\operatorname{Mod}^{\operatorname{perf}}_R\) are adjacent if and only if \(R\) is regular, the author conjectures that the existence of adjacent structures on a stable \(\infty\)-category \(\mathscr{C}\) should be a noncommutative analogue of regularity. Hence, it is proposed that if all the mapping spaces in the heart of the weight structure \(\operatorname{Hw}(\mathscr{C})\) are \(N\)-truncated for some fixed \(N\), the \(\infty\)-category \(\operatorname{Hw}(\mathscr{C})\) is actually discrete (Conjecture \(3.12\)). Finally, the author proves that if \(X\) is a quotient spectral stack satisfying the assumptions of Theorem \(2.16\), then Conjecture \(3.12\) holds for \(\operatorname{QCoh}(X)\).