We consider the invariants Ker and Im for commutative squares in quasi-abelian categories. These invariants were introduced by Lambek for groups and then studied by Hilton and Nomura in categories exact in the sense of Buchsbaum.

Abstract We prove that the derived categories of abelian have unique enhancements—all them, unbounded, bounded, bounded above and below categories. The unseparated left completed a Grothendieck category are also shown to enhancements. Finally, we show complexes with quasi-coherent cohomology perfect enhancements for quasi-compact quasi-separated schemes.

Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. Some of them are already proved for hereditary abelian categories there. In the present paper, all basic results about tilting theory are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object T in a hereditary abelian category H, we verify that t...

For any abelian category C satsifying (AB5) over a separated, quasicompact scheme S , we construct a stack of 2-groups GL(C) over the flat site of S. We will give a concrete description of GL(C) when C is the category of quasi-coherent sheaves on a separated, quasi-compact scheme X over S. We will show that the tangent space gl(C) of GL(C) at the origin has a structure as a Lie 2-algebra.

This moduli space is Mh = Γ \Dh where Dh = GR/V is a homogeneous complex manifold on which Γ acts properly discountinuously. In the classical case n = 1 (polarized abelian varieties) or n = 2, h = 1, D is a bounded symmetric domain and Γ \D is a quasi-projective variety defined over a number field. In the non-classical case the situation is quite different. Given a smooth projective family X π ...

For a ringed space (X,O), we show that the deformations of the abelian category Mod(O) of sheaves of O-modules [11] are obtained from algebroid prestacks, as introduced by Kontsevich. In case X is a quasi-compact separated scheme the same is true for Qch(O), the category of quasi-coherent sheaves on X. It follows in particular that there is a deformation equivalence between Mod(O) and Qch(O).

We consider how, for quasi-degenerate neutrinos with tri-bi-maximal mixing at a high-energy scale, the mixing angles are affected by radiative running from high to low-energy scales in a supersymmetric theory. The limits on the high-energy scale that follow from consistency with the observed mixing are determined. We construct a model in which a non-Abelian discrete family symmetry leads both t...

We consider glass states of several disordered systems: vortices in impure superconductors, amorphous magnets, and nematic liquid crystals in random porous media. All these systems can be described by the random-field or random-anisotropy O(N) model. Even arbitrarily weak disorder destroys long range order in the O(N) model. We demonstrate that at weak disorder and low temperatures quasi-long r...

We establish braided tensor equivalences among module categories over the twisted quantum double of a finite group defined by an extension of a group G by an abelian group, with 3-cocycle inflated from a 3-cocycle on G. We also prove that the canonical ribbon structure of the module category of any twisted quantum double of a finite group is preserved by braided tensor equivalences. We give two...

We give the commensurability classifications of free products of finitely many finitely generated abelian groups. We show this coincides with the quasi-isometry classification and prove that this class of groups is quasiisometrically rigid. The following gives the complete commensurability and quasi-isometry classification of free products of finitely generated abelian groups. The quasi-isometr...