Locally finitely presented Grothendieck categories with a flat generator

TL;DR

This paper investigates conditions under which locally finitely presented Grothendieck categories with flat generators also have enough projectives, providing counterexamples, connections to classical problems, and identifying classes where the property holds.

Contribution

It establishes an equivalence between having enough flat objects and exact products, offers a counterexample, and links the problem to classical ring theory and the Telescope Conjecture.

Findings

Counterexample showing the negative answer to the problem

Connection to Miller's classical ring-theoretical question

Identification of categories where the problem is affirmatively resolved

Abstract

A problem raised by Cuadra and Simson in 2007 asks whether any locally finitely presented Grothendieck category with enough flat objects also has enough projectives. In this paper, we start from a key observation: a locally finitely presented Grothendieck category has enough flat objects if, and only if, it has exact products. This enables several equivalent reformulations of the problem, allowing us to identify a counterexample (thus providing a negative solution to the problem), while also connecting it to a classical ring-theoretical question posed by Miller in 1975, and even to the Telescope Conjecture for compactly generated triangulated categories. Moreover, we describe several classes of Grothendieck categories where the problem can be answered affirmatively. For example, we show that a locally finitely presented Grothendieck category whose category of finitely presented objects is Krull--Schmidt has enough flats if, and only if, it is generated by a family of finitely generated projectives.