Hypersurface support for noncommutative complete intersections

TL;DR

This paper develops an infinite hypersurface support theory for noncommutative complete intersections, providing a new way to analyze their singularity categories and vanishing properties, inspired by classical commutative theories.

Contribution

It introduces an infinite hypersurface support framework for noncommutative complete intersections, extending classical support theories to a broader noncommutative setting.

Findings

Support theory characterizes vanishing objects in Sing(R)

Support detects non-vanishing objects in the singularity category

Framework applies to classify thick ideals in stable categories

Abstract

We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. By a noncommutative complete intersection we mean an algebra R which admits a smooth deformation $Q\to R$ by a Noetherian algebra $Q$ which is of finite global dimension. We show that hypersurface support defines a support theory for the big singularity category $Sing(R)$, and that the support of an object in $Sing(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz' support theory for (commutative) local complete intersections. In a companion piece, we employ hypersurface support, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.