On the (non)vanishing of some "derived" categories of curved dg algebras

TL;DR

This paper investigates various notions of derived categories for curved dg algebras, demonstrating that for certain examples, these categories are trivial or vanish, highlighting fundamental differences from classical derived categories.

Contribution

It shows that some concrete curved dg algebras have vanishing derived categories, revealing limitations of existing notions in the curved setting.

Findings

Derived categories vanish for specific curved dg algebras

The initial curved dg algebra's module category is precomplexes

Certain deformations of dg algebras also have vanishing derived categories

Abstract

Since curved dg algebras, and modules over them, have differentials whose square is not zero, these objects have no cohomology, and there is no classical derived category. For different purposes, different notions of "derived" categories have been introduced in the literature. In this note, we show that for some concrete curved dg algebras, these derived categories vanish. This happens for example for the initial curved dg algebra whose module category is the category of precomplexes, and for certain deformations of dg algebras.