In this article one extends the classical theory of (intermediate) Jacobians to the “noncommutative world”. Concretely, one constructs a Q-linear additive Jacobian functor N 7→ J(N) from the category of noncommutative Chow motives to the category of abelian varieties up to isogeny, with the following properties: (i) the first de Rham cohomology group of J(N) agrees with the subspace of the odd ...

2 The category of small abelian categories and exact functors 4 2.1 Categorical properties of ABEX . . . . . . . . . . . . . . . . . . . 5 2.2 Pullbacks in ABEX . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 ABEX is finitely accessible . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Abelian categories as schemes . . . . . . . . . . . . . . . . . . . . 16 2.4.1 The functor of point...

We investigate how to characterize subcategories of abelian categories in terms intrinsic axioms. In particular, we find axioms which generating cogenerating functorially finite subcategories, precluster tilting and cluster categories. As a consequence prove that any d-abelian category is equivalent d-cluster subcategory an category, without assumption on the being projectively generated.

It is well-known that the category of comodules over a flat Hopf algebroid is abelian but typically fails to have enough projectives, and more generally, the category of graded comodules over a graded flat Hopf algebroid is abelian but typically fails to have enough projectives. In this short paper we prove that the category of connective graded comodules over a connective, graded, flat, finite...

The category of modules over a fixed quasigroup in the category of all quasigroups is equivalent to the category of representations of the fundamental groupoid of the Cayley diagram of the quasigroup in the category of abelian groups. The corresponding equivalent category of coverings, and the generalization to the right quasigroup case, are also described.

Recently, symmetric categorical groups are used for the study of the Brauer groups of symmetric monoidal categories. As a part of these efforts, some algebraic structures of the 2-category of symmetric categorical groups SCG are being investigated. In this paper, we consider a 2-categorical analogue of an abelian category, in such a way that it contains SCG as an example. As a main theorem, we ...

Any category A can be embedded in its right completion A. When A is small and abelian, this completion A is AB5 and the embedding is exact.

Let A be a left and right noetherian ring We denote by Amod the abelian category of nitely generated left A modules and by Aproj the category of nitely generated projective left A modules We de note by R A the Grothendieck group of Amod and by R A the Grothendieck group of Aproj If X is an object of Amod resp of Aproj we denote by X its representative in R A resp in R A We denote by modA the ab...

We exhibit a triangulated category T having both products and coproducts and a triangulated subcategory S ⊂ T which is both localizing and colocalizing, and for which neither a Bousfield localization nor a colocalization exists. It follows that neither the category S nor its dual satisfy Brown representability. Our example involves an abelian category whose derived category does not have small ...