Locally finitely presented Grothendieck categories with a flat generator
Lorenzo Martini, Carlos E. Parra, Manuel Saorín, Simone Virili
TL;DR
The paper tackles Cuadra–Simson’s question on when a locally finitely presented Grothendieck category with enough flat objects has enough projectives. It develops a colocalization framework via idempotent ideals and TTF triples to rephrase the problem in terms of associated module Giraud subcategories, proving both negative counterexamples (via Dubrovin–Puninski rings) and positive results in important special cases (commutative rings, semiregular endomorphism rings, and comodules). It also relates flatness to definable co-aisles and to the telescope conjecture, establishing equivalences between local finite presentability of G_I and the presence of projective generators, and provides concrete matrix- and Morita-type criteria in the ring case. The work clarifies when an AB*-4* category arising as a colocalization has enough projectives, producing a nuanced landscape with both counterexamples and robust affirmative regions for the CS problem. Overall, it connects purity, tensorial methods, and recollements to fundamental questions about generators, projectives, and homological algebra in Grothendieck contexts, with consequences for comodules and triangulated category conjectures.
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.