Regular subcategories in bounded derived categories of affine schemes

TL;DR

This paper proves that the bounded derived category of coherent sheaves over an affine scheme has no proper regular subcategories and establishes lower bounds on the dimension of regular categories related to such schemes.

Contribution

It demonstrates the non-existence of proper regular subcategories in the derived category of an affine scheme and provides bounds on the dimension of regular categories with specific functors.

Findings

No proper regular subcategories in D^b(coh X) for connected affine schemes

Lower bounds on the dimension of regular categories with certain functors

Applications to cohomological annihilators and point-like objects

Abstract

Let $R$ be a commutative Noetherian ring such that $X=Spec R$ is connected. We prove that the category $D^b(coh X)$ contains no proper full triangulated subcategories which are regular. We also bound from below the dimension of a regular category $T$, if there exists a triangulated functor $T \to D^b(coh X)$ with certain properties. Applications are given to cohomological annihilator of $R$ and to point-like objects in $T$.