Semi-Bousfield classes and nonmonotone perversities

Abstract

In the generality of a rigidly-compactly generated tensor triangulated category, we introduce semi-Bousfield classes in terms of the vanishing of the tensor product in positive degrees with respect to a fixed reasonable $t$-structure. We show that semi-Bousfield classes provide a common generalisation of Bousfield classes and compactly generated tensor-compatible $t$-structures. Then we specialise to the setting of the unbounded derived category $\mathcal{D}_{\mathrm{qc}}(X)$ of a Noetherian scheme $X$ and show that the stratification bijection naturally extends to an assignment which takes a (not necessarily monotone) perversity on $X$ to a semi-Bousfield class in $\mathcal{D}_{\mathrm{qc}}(X)$. If $X$ is regular, this assignment constitutes a stratification of the whole semi-Bousfield lattice, while in the singular case, its image consists precisely of those semi-Bousfield classes arising from objects of finite Tor-dimension. Restricting this bijection to monotone perversities recovers the recent classification of compactly generated tensor-compatible $t$-structures of Dubey and Sahoo, (arXiv:2204.05015).