Categorical characterizations of regularity for algebraic stacks

Abstract

For a Noetherian scheme $X$ of finite Krull dimension, Neeman recently established two characterizations of the regularity of $X$ using strong generators and bounded $t$-structures on $\operatorname{Perf}(X)$. In this note, we obtain variants of Neeman's results for large classes of Noetherian algebraic stacks. An important intermediate step is the fact that $X$ is regular if and only if $\operatorname{Perf}(X)=D_{\operatorname{coh}}^b(X)$, which we establish for Noetherian algebraic stacks. Our approach also yields a criterion for the existence of classical generators for the bounded derived categories of coherent sheaves on algebraic stacks, generalizing previous results for commutative rings and schemes.