The derived category with respect to a generator

TL;DR

This paper introduces a new derived category $ ext{D}(G)$ for Grothendieck categories with a generator, establishing its properties, generation conditions, and recollement structures, with applications to various examples.

Contribution

It defines the derived category $ ext{D}(G)$ relative to a generator, proves it is well-generated, and explores its recollement structures, extending classical derived category theory.

Findings

$ ext{D}(G)$ is always well-generated.

When generators are finitely presented, $ ext{D}(G)$ is compactly generated.

Recollement structures analogous to Krause's are established.

Abstract

Consider a Grothendieck category $\mathcal{G}$ along with a choice of generator $G$, or equivalently a generating set $\{G_i\}$. We introduce the derived category $\mathcal{D}(G)$, which kills all $G$-acyclic complexes, by putting a suitable model structure on the category of chain complexes. It follows that the category $\mathcal{D}(G)$ is always a well-generated triangulated category. It is compactly generated whenever the generating set $\{G_i\}$ has each $G_i$ finitely presented, and in this case we show that two recollement situations hold. The first is when passing from the homotopy category $K(\mathcal{G})$ to $\mathcal{D}(G)$. The second is a $G$-derived analog to the recollement of Krause. We illustrate with several examples ranging from pure and clean derived categories to quasi-coherent sheaves on the projective line $P^1(k)$.