Non-left-complete derived categories

TL;DR

This paper provides examples of abelian categories whose derived categories are not left-complete, highlighting cases like representations of the additive group over fields of positive characteristic.

Contribution

It introduces specific examples of abelian categories with non-left-complete derived categories, including the representation category of G_a over fields of characteristic p.

Findings

Derived categories of certain abelian categories are not left-complete.

The category of representations of G_a over a field of characteristic p is a key example.

These examples expand understanding of the limitations of derived category completeness.

Abstract

We give some examples of abelian categories A for which the derived category D(A) is not left-complete. Perhaps the most natural of these is where A is the category of representations of the additive group G_a over a field k of characteristic p>0.