Related papers: Relative singularity categories II: DG models
We determine the Ringel duals for all blocks in the parabolic versions of the BGG category O associated to a reductive finite dimensional Lie algebra. In particular we find that, contrary to the original category O and the specific…
We study singularity categories of exact categories with a focus on those associated to a complete hereditary cotorsion pair. As an application we identify a non-affine analogue of the singularity category of a Gorenstein local ring; with…
We study graded and ungraded singularity categories of some commutative Gorenstein toric singularities, namely, Veronese subrings of polynomial rings, and Segre products of some copies of polynomial rings. We show that the graded…
This paper can be thought of as an extended introduction to arXiv:0708.3398; nevertheless, most of its results are not covered by loc. cit. We consider the derived categories of DG-modules, DG-comodules, and DG-contramodules, the coderived…
Let $\Lambda = \left[\begin{array}{cc} A & 0 \\ M & B \end{array}\right] $ be an Artin algebra and $_BM_A$ a $B$-$A$-bimodule. We prove that there is a triangle equivalence $D_{sg}(\Lambda) \cong D_{sg}(A)\coprod D_{sg}(B)$ between the…
We investigate the algebro-geometric structure of a novel two-parameter quantum deformation which exhibits the nature of a semidirect or cross-product algebra built upon GL(2) x GL(1), and is related to several other known examples of…
We generalise recent results about quasi-Cartan, Cartan and diagonal subalgebras by introducing graded versions. We show that there is a correspondence between graded algebraic quasi-Cartan/ Cartan/ diagonal pairs and certain graded twisted…
The algebras of Kleinian type are finite dimensional semisimple rational algebras $A$ such that the group of units of an order in $A$ is commensurable with a direct product of Kleinian groups. We classify the Schur algebras of Kleinian type…
We analyze the structure of matter representations arising from codimension two singularities in F-theory, focusing on gauge groups SU(N). We give a detailed local description of the geometry associated with several types of singularities…
We develop the notion of deformation of a morphism in a left-proper model category. As an application we provide a geometric/homotopic description of deformations of commutative (non-positively) graded differential algebras over a local…
Categories over a field $k$ can be graded by different groups in a connected way; we consider morphisms between these gradings in order to define the fundamental grading group. We prove that this group is isomorphic to the fundamental group…
We compute the Krull-Gabriel dimension of the category of perfect complexes for finite dimensional algebras which are derived discrete.
These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new…
We introduce the notion of composition series of triangulated categories, which generalizes full exceptional sequences. The lengths of composition series yield invariants for triangulated categories. We study composition series of derived…
We formalize the main approach for showing Zariski descent-type statements for strong generation of triangulated categories associated to algebro-geometric objects. This recovers various known statements in the literature. As applications…
We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not smooth and proper, in which case degeneration is proved…
We construct and study a family of toric degenerations of the algebra of conformal blocks for a stable marked curve $(C, \vec{p})$ with structure group $SL_3(\mathbb{C}).$ We find that this algebra is Gorenstein. For the genus $0, 1$ cases…
In this paper we investigate the class of the connected graded algebras which are finitely generated in degree 1, which are finitely presented with relations of degrees greater or equal to 2 and which are of finite global dimension D and…
We classify three fold isolated quotient Gorenstein singularity $C^3/G$. These singularities are rigid, i.e. there is no non-trivial deformation, and we conjecture that they define 4d $\mathcal{N}=2$ SCFTs which do not have a Coulomb…
We develop a new framework of relative algebroids to address existence and classification problems of geometric structures subject to partial differential equations.