Regularity and bounded $t$-structures for algebraic stacks
Timothy De Deyn, Pat Lank, Kabeer Manali Rahul, Fei Peng
TL;DR
This work extends the homological characterization of regularity from schemes to Noetherian algebraic stacks by proving that regularity of a stack $\\mathcal{X}$ is equivalent to $Perf(\\mathcal{X}) = D^b_{coh}(\\mathcal{X})$, and introduces cohomological regularity to connect geometric and homological perspectives. It establishes a stacky Bondal–Van den Bergh type criterion via strong generation of $Perf(\\mathcal{X})$ under appropriate hypotheses, and a stacky Antieau–Gepner–Heller direction through bounded $t$-structures on $Perf_Z(\\mathcal{X})$ for closed subsets $Z$. The paper also proves openness results for the regular locus under Thomason-type hypotheses and develops a detailed framework tying generation, $t$-structures, and support to regularity on stacks. Together, these results transfer key scheme-level insights to a broad class of algebraic stacks, enabling robust categorical control over geometric regularity and its consequences for derived categories and their generators.
Abstract
Our work shows (the expected) cohomological characterization for regularity of (Noetherian) algebraic stacks; such a stack is regular if and only if all complexes with bounded and coherent cohomology are perfect. This naturally enables us to extend various statements known for schemes to algebraic stacks. In particular, the conjectures by Antieau--Gepner--Heller and Bondal--Van den Bergh, both resolved for schemes by Neeman, are proven for suitable algebraic stacks.