Classifying thick subcategories over a Koszul complex via the curved BGG correspondence

TL;DR

This paper classifies thick subcategories in derived categories of dg modules over Koszul complexes, unifying several known results and revealing invariance under Grothendieck duality.

Contribution

It provides a unified classification of thick subcategories over Koszul complexes, extending previous theorems and connecting to curved dg modules and the BGG correspondence.

Findings

Classification of thick subcategories over Koszul complexes.

Recovery of Stevenson’s theorem and exterior algebra classifications.

Lattice of thick subcategories is invariant under Grothendieck duality.

Abstract

In this work we classify the thick subcategories of the bounded derived category of dg modules over a Koszul complex on any list of elements in a regular ring. This simultaneously recovers a theorem of Stevenson when the list of elements is a regular sequence and the classification of thick subcategories for an exterior algebra over a field (via the BGG correspondence). One of the major ingredients is a classification of thick tensor submodules of perfect curved dg modules over a commutative noetherian graded ring concentrated in even degrees, recovering a theorem of Hopkins and Neeman. We give several consequences of the classification result over a Koszul complex, one being that the lattice of thick subcategories of the bounded derived category is fixed by Grothendieck duality.