Negative K-groups of abelian categories

Homology and homotopy in semi-abelian categories

The theory of abelian categories proved very useful, providing an axiomatic framework for homology and cohomology of modules over a ring and, in particular, of abelian groups. For many years, a similar categorical framework has been lacking for non-abelian (co)homology, the subject of which includes the categories of non-abelian groups, of rings, Lie algebras, etc. The notion of semi-abelian category due to Janelidze, Marki and Tholen allows a unified study of many important homological properties of such categories. For instance, in any semi-abelian category, the classical diagram lemmas (the…

Reflexivity in Derived Categories

An adjoint pair of contravariant functors between abelian categories can be extended to the adjoint pair of their derived functors in the associated derived categories. We describe the reflexive complexes and interpret the achieved results in terms of objects of the initial abelian categories. In particular we prove that, for functors of any finite cohomological dimension, the objects of the initial abelian categories which are reflexive as stalk complexes form the largest class where a Cotilting Theorem in the sense of Colby and Fuller works.

On the second cohomology group in semi-abelian categories

We develop some new aspects of cohomology in the context of semi-abelian categories: we establish a Hochschild-Serre 5-term exact sequence extending the classical one for groups and Lie algebras; we prove that an object is perfect if and only if it admits a universal central extension; we show how the second Barr-Beck cohomology group classifies isomorphism classes of central extensions; we prove a universal coefficient theorem to explain the relationship with homology.

Abel范畴的n-平凡扩张中的Gorenstein对象

Gorenstein objects in the n-Trivial extensions of abelian categories

本文在Abel范畴中定义（强）Gorenstein投射与内射对象，研究其向n-平凡扩张范畴的性质迁移及刻画，并应用于结合环与三角矩阵环的模范畴。

G - functors arising from categorical group actions on abelian categories

A Mackey type decomposition for group actions on abelian categories is described. This allows us to define new Mackey functors which associates to any subgroup the $K$-theory of the corresponding equivariantized abelian category. In the case of an action by tensor autoequivalences the Mackey functor at the level of Grothendieck rings has a Green functor structure. As an application we give a description of the Grothendieck rings of equivariantized fusion categories under group actions by tensor autoequivalences on graded fusion categories.

Simplicial homotopy in semi-abelian categories

We study Quillen's model category structure for homotopy of simplicial objects in the context of Janelidze, Marki and Tholen's semi-abelian categories. This model structure exists as soon as the base category A is regular Mal'tsev and has enough regular projectives; then the fibrations are the Kan fibrations of simplicial objects in A. When, moreover, A is semi-abelian, weak equivalences and homology isomorphisms coincide.

Hereditary abelian model categegories

We discuss some recent developments in the theory of abelian model categories. The emphasis is on the hereditary condition and applications to homotopy categories of chain complexes and stable module categories.

On the abelianization of derived categories and a negative solution to Rosicky's problem

We prove for a large family of rings R that their lambda-pure global dimension is greater than one for each infinite regular cardinal lambda. This answers in negative a problem posed by Rosicky. The derived categories of such rings then do not satisfy the Adams lambda-representability for morphisms for any lambda. Equivalently, they are examples of well generated triangulated categories whose lambda-abelianization in the sense of Neeman is not a full functor for any lambda. In particular we show that given a compactly generated triangulated category, one may not be able to find a Rosicky funct…

Grothendieck groups of categories of abelian varieties

We compute the Grothendieck group of the category of abelian varieties over an algebraically closed field $k$. We also compute the Grothendieck group of the category of $A$-isotypic abelian varieties, for any simple abelian variety $A$, assuming $k$ has characteristic 0, and for any elliptic curve $A$ in any characteristic.

Homotopy invariance of higher K-theory for abelian categories

The main theorem in this paper is that the base change functor from a noetherian abelian category to its noetherian polynomial category induces an isomorphism on K-theory. The main theorem implies the well-known fact that A^1-homotopy invariance of K'-theory for noetherian schemes.

Gorenstein cohomology in abelian categories

We investigate relative cohomology functors on subcategories of abelian categories via Auslander-Buchweitz approximations and the resulting strict resolutions. We verify that certain comparison maps between these functors are isomorphisms and introduce a notion of perfection for this context. Our main theorem is a balance result for relative cohomology that simultaneously recovers theorems of Holm and the current authors as special cases.

Rigid objects, triangulated subfactors and abelian localizations

We study abelian localizations of triangulated categories induced by rigid contravariantly finite subcategories, and also triangulated structures on subfactor categories of triangulated categories. In this context we generalize recent results of Buan-Marsh and Iyama-Yoshino. We also extend basic results of Keller-Reiten concerning the Gorenstein and the Calabi-Yau property for categories arising from certain rigid, not necessarily cluster tilting, subcategories, as well as several results of the literature concerning the connections between 2-cluster tilting subcategories of triangulated categ…

Quotients of exact categories by pseudo-cluster tilting subcategories

We introduce the concept of a pseudo-cluster tilting subcategory from the viewpoint of the fact that the quotient of an exact category by a cluster tilting subcategory is an abelian category. We prove that the quotients in the case of pseudo-cluster tilting are always semi-abelian. In addition, it is abelian if and only if some self-orthogonal conditions are satisfied. We revisit the abelian quotient category of conflations by splitting ones, and get that there exists a unique exact substructure such that it is a cluster quotient.

Filtrations in abelian categories with a tilting object of homological dimension two

We consider filtrations of objects in an abelian category $\catA$ induced by a tilting object $T$ of homological dimension at most two. We define three disjoint subcategories with no maps between them in one direction, such that each object has a unique filtation with factors in these categories. This filtration coincides with the the classical two-step filtration induced by torsion pairs in dimension one. We also give a refined filtration, using the derived equivalence between the derived categories of $\catA$ and the module category of $End_\catA (T)^{op}$. The factors of this filtration con…

阿贝尔范畴中的等变K理论枚举不变量与穿墙公式

Equivariant K-theoretic enumerative invariants and wall-crossing formulae in abelian categories

本文将Joyce的同调泛穿墙公式提升至K理论并引入等变性，为模栈的等变K理论枚举不变量建立了通用穿墙框架，并指出拓扑K-同调构成等变乘性顶点代数。