A generalization of the Mitchell lemma: the Ulmer theorem and the Gabriel-Popescu theorem revisited
Let \(\mathcal U\) be a set of objects in a Grothendieck category \(\mathcal A\). Denote by \(\mathcal Gen({\mathcal U})\) the full subcategory of \(\mathcal A\) whose objects are the \(\mathcal U\)-generated objects, and by \({\mathcal A}_{\mathcal U}\) the class of objects \(M\) of \(\mathcal A\) such that the kernel of any morphism from a finite direct sum of objects in \(\mathcal U\) to \(M\) lies in \(\mathcal Gen({\mathcal U})\). Let \(({\mathcal U}^{\text{op}},\mathcal Ab)\) be the category of additive contravariant functors from \(\mathcal U\) to the category \(\mathcal Ab\) of abelian groups, with natural transformations as morphisms. Let \(T:{\mathcal A}\rightarrow ({\mathcal U}^{\text{op}},\mathcal Ab)\) be the functor defined by \(T(X)={\text{Hom}}_{\mathcal A}(-,X)_{|{\mathcal U}}\) on objects, and by \(T(f)={\text{Hom}}_{\mathcal A}(-,f)_{|{\mathcal U}}\) on morphisms. This functor has a left adjoint \(S\). The authors prove a generalization of \textit{B. Mitchell}'s lemma [J. Pure Appl. Algebra 20, 313--315 (1981; Zbl 0459.18006)] , and as a consequence they give a new proof of \textit{F. Ulmer}'s theorem [Invent. Math. 19, 331--336 (1973; Zbl 0257.18020)], which states that the functor \(S\) is exact if and only if \({\mathcal U}\subseteq {\mathcal A}_{\mathcal U}\). Also, they deduce a generalization of the Gabriel-Popescu theorem [\textit{N. Popescu} and \textit{P. Gabriel}, C. R. Acad. Sci., Paris 258, 4188--4190 (1964; Zbl 0126.03304)], by showing that if \(\mathcal U\) is a family of generators of \(\mathcal A\), then \(T\) is full and faithful and \(S\) is exact.
scientific article; zbMATH DE number 6132780