An example of a cocomplete abelian category that is not complete is constructed.

In this work, we s uggest a defnition for the category of mixed motives generated by the motive h1 (E) for E an elliptic curve without complex multiplication. We then compute the cohomology of this category. Modulo a strengthening of the Beilinson-Soulé conjecture, we show that the cohomology of our category agrees with the expected motivic cohomology groups. Finally for each pure motive (Symnh1 (E)) (–1) we construct families of nontrivial motives whose highest associated weight graded piece is (Symnh1 (E)) (–1).

In this survey paper we present some results relating the Goldie dimension, dual Krull dimension and subdirect irreducibility in modules, torsion theories, Grothendieck categories and lattices. Our interest in studying this topic is rooted in a nice module theoretical result of Carl Faith [Commun. Algebra27 (1999), 1807–1810], characterizing Noetherian modules M by means of the finiteness of the Goldie dimension of all its quotient modules and the ACC on its subdirectly irreducible submodules. Thus, we extend his result in a dual Krull dimension setting and consider its dualization, not only in modules, but also in upper continuous modular lattices, with applications to torsion theories and Grothendieck categories.

We prove, correct and extend several results of an earlier paper of ours (using and recalling several of our later papers) about the derived functors of projective limit in abelian categories. In particular we prove that if C is an abelian category satisfying the Grothendieck axioms AB3 and AB4* and having a set of generators then the first derived functor of projective limit vanishes on so-called Mittag-Leffler sequences in C. The recent examples given by Deligne and Neeman show that the condition that the category has a set of generators is necessary. The condition AB4* is also necessary, and indeed we give for each integer $m \geq 1$ an example of a Grothendieck category Cm and a Mittag-Leffler sequence in Cm for which the derived functors of its projective limit vanish in all positive degrees except m. This leads to a systematic study of derived functors of infinite products in Grothendieck categories. Several explicit examples of the applications of these functors are also studied.

In this paper we show that for a Grothendieck category $\mathcal{A}$ and a complex $E$ in $\mathbf{C}(\mathcal{A})$ there is an associated localization endofunctor $\ell$ in $\mathbf{D}(\mathcal{A})$. This means that $\ell$ is idempotent (in a natural way) and that the objects that go to 0 by $\ell$ are those of the smallest localizing (= triangulated and stable for coproducts) subcategory of $\mathbf{D}(\mathcal{A})$ that contains $E$. As applications, we construct $\text{K}$-injective resolutions for complexes of objects of $\mathcal{A}$ and derive Brown representability for $\mathbf{D}(\mathcal{A})$ from the known result for $\mathbf{D}(R-\mathbf{mod})$, where $R$ is a ring with unit.