Classifying thick subcategories over a Koszul complex via the curved BGG correspondence
Jian Liu, Josh Pollitz
TL;DR
The paper classifies thick subcategories of the bounded derived category of dg modules over a Koszul complex E on a list of elements in a regular ring, unifying Stevenson's complete-intersection case and exterior-algebra cases via a curved BGG correspondence. The authors establish a curved tensor-nilpotence framework and a support-theoretic classification for perfect curved dg A-modules perf(A,w), then transfer the classification to D^f(E) through an equivalence h: D^f(E) → perf(A,w). The main contributions include a lattice isomorphism between thick subcategories and specialization-closed subsets of the curved support supp(A,w), a curved version of Hopkins–Neeman–Thomason’s theory, and several corollaries about generators, duality, and symmetry in curvature settings. This yields a broad, intrinsic classification for Koszul complexes over regular rings, recovers known results in special cases, and provides a framework adaptable to curved algebra contexts with potential connections to Hochschild cohomology and singularity theory.
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.