A Grothendieck category with a noetherian generator and exact products that is not a module category

Abstract

We construct a Grothendieck category that has a noetherian generator, satisfies AB4*, and is not equivalent to a module category. This gives a negative answer to Djament's problem. The category is obtained as a Gabriel quotient of a module category over the endomorphism ring appearing in the work of Herbera, Př\'ıhoda, and Wiegand. The proof relies on the trace criterion established by Martini, Parra, Saor\'ın, and Virili.

arXiv comment4 pages