Related papers: Degenerating $0$ in Triangulated Categories

We investigate the behavior of singularity categories and stable categories of Gorenstein projective modules along a morphism of rings. The natural context to approach the problem is via change of rings, that is, the classical adjoint…

The category of modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category C has been given two different descriptions: On the one hand, as shown by Osamu Iyama and Yuji Yoshino, it is equivalent to an…

In this paper, we explore when a locally finite triangulated category has dimension zero or finite representation type. We also study generation of derived categories by orthogonal subcategories.

We study the vanishing of cohomology in triangulated categories admitting a central ring action. In particular, we study vanishing gaps and symmetry.

For a finite quiver without sources or sinks, we prove that the homotopy category of acyclic complexes of injective modules over the corresponding finite dimensional algebra with radical square zero is triangle equivalent to the derived…

We systematically develop the theory of definable functors between compactly generated triangulated categories. Such functors preserve pure triangles, pure injective objects, and definable subcategories, and as such appear in a wide range…

We consider self-injective finite-dimensional graded algebras admitting a triangular decomposition. In a preceding paper, we have shown that the graded module category of such an algebra is a highest weight category and has tilting objects…

We give a general treatment of deformation theory from the point of view of homotopical algebra following Hinich, Manetti and Pridham. In particular, we show that any deformation functor in characteristic zero is controlled by a certain…

Recollements of triangulated categories may be seen as exact sequences of such categories. Iterated recollements of triangulated categories are analogues of geometric or topological stratifications and of composition series of algebraic…

We give a direct proof of the following known result: the Grothendieck group of a triangulated category with a silting subcategory is isomorphic to the split Grothendieck group of the silting subcategory. Moreover, we obtain its…

The three dimensional rotation group, SO(3), is a symmetry group of the normal hydrogen atom. Each reducible representation of this group can be associated with a degenerate energy level. If this atom is placed in an external magnetic…

We exhibit the Hodge degeneration from nonabelian Hodge theory as a $2$-fold delooping of the filtered loop space $E_2$-groupoid in formal moduli problems. This is an iterated groupoid object which in degree $1$ recovers the filtered circle…

In this paper we use A-infinity modules to study the derived category of a finite dimensional algebra over an algebraically closed field. We study varieties parameterising A-infinity modules. These varieties carry an action of an algebraic…

Let $\mathcal{D}$ be a Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated category with a rigid object $R$. Let $\Lambda=\operatorname{End}_{\mathcal{D}}R$ be the endomorphism algebra of $R$. We introduce the notion of mutation of maximal…

For a Grothendieck category C which, via a Z-generating sequence (O(n))_{n in Z}, is equivalent to the category of "quasi-coherent modules" over an associated Z-algebra A, we show that under suitable cohomological conditions "taking…

We introduce a notion of total acyclicity associated to a subcategory of an abelian category and consider the Gorenstein objects they define. These Gorenstein objects form a Frobenius category, whose induced stable category is equivalent to…

The homotopy category of complexes of projective left-modules over any reasonably nice ring is proved to be a compactly generated triangulated category, and a duality is given between its subcategory of compact objects and the finite…

Extriangulated categories give a simultaneous generalization of triangulated categories and exact categories. In this paper, we study silting subcategories of an extriangulated category. First, we show that a silting subcategory induces a…

A complete representation of 3D objects requires characterizing the space of deformations in an interpretable manner, from articulations of a single instance to changes in shape across categories. In this work, we improve on a prior…

The deformation cohomology of a tensor category controls deformations of its monoidal structure. Here we describe the deformation cohomology of tensor categories generated by one object (the so-called Schur-Weyl categories). Using this…