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This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra…
We study matrix factorizations of locally free coherent sheaves on a scheme. For a scheme that is projective over an affine scheme, we show that homomorphisms in the homotopy category of matrix factorizations may be computed as the…
We present a Langlands dual realization of the putative category of affine character sheaves. Namely, we calculate the categorical center and trace (also known as the Drinfeld center and trace, or categorical Hochschild cohomology and…
A relationship between curved differential algebras and corings is established and explored. In particular it is shown that the category of semi-free curved differential graded algebras is equivalent to the category of corings with…
In this paper, we examine the relation between certain subclasses of the classes of Gorenstein projective, Gorenstein flat and Gorenstein injective modules over a group algebra, which consist of the cofibrant, cofibrant-flat and fibrant…
Let $\mathbb{X}$ be a weighted projective line and $\operatorname{coh}\mathbb{X}$ the associated categoy of coherent sheaves. We classify the tilting complexes $T$ in $D^b(\operatorname{coh}\mathbb{X})$ such that $\tau^2 T\cong T$, where…
For the cluster category of a hereditary or a canonical algebra, equivalently for the cluster category of the hereditary category of coherent sheaves on a weighted projective line, we study the Grothendieck group with respect to an…
We study an example of a projective threefold with a non-isolated singularity and its derived category. The singular locus can be locally described as a line of surface nodes compounded with a threefold node at the origin. We construct a…
Many classes of projective algebraic varieties can be studied in terms of graded rings. Gorenstein graded rings in small codimension have been studied recently from an algebraic point of view, but the geometric meaning of the resulting…
To an abelian category A of homological dimension 1 satisfying certain finiteness conditions, one can associate an algebra, called the Hall algebra. Kapranov studied this algebra when A is the category of coherent sheaves over a smooth…
We study perverse sheaves of categories their connections to classical algebraic geometry. We show how perverse sheaves of categories encode naturally derived categories of coherent sheaves on $\mathbb{P}^1$ bundles, semiorthogonal…
We construct an isomorphism of graded Frobenius algebras between the orbifold Chow ring of weighted projective spaces and graded algebras of groups of roots of the unity.
We construct a new model structure on the category of dg presheaves over a topological space $X$, obtained through the right Bousfield localization of the local projective model structure. The motivation for this construction arises from…
We describe the structure of projective indecomposable modules for the subalgebra consisting of the elements of degree 0 in the hyperalgebra of the $r$-th Frobenius kernel for the algebraic group ${\rm SL}_2(k)$, using the primitive…
In these notes, an introduction to derived categories and derived functors is given. The main focus is the bounded derived category of coherent sheaves on a smooth projective variety.
In this review we discuss what is known about semiorthogonal decompositions of derived categories of algebraic varieties. We review existing constructions, especially the homological projective duality approach, and discuss some related…
This is the first in a series of papers that deals with duality statements such as Mukai-duality (T-duality, from algebraic geometry) and the Baum-Connes conjecture (from operator $K$-theory). These dualities are expressed in terms of…
We apply the technique of recollement to study the Gorenstein defect categories of triangular matrix algebras. First, we construct a left recollement of Gorenstein defect categories for a triangular matrix algebra under some conditions,…
Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of \v{S}penko and Van den Bergh, we construct equivalences between the derived categories of coherent sheaves…
We show that the Grothendieck groups of the categories of finitely-generated graded supermodules and finitely-generated projective graded supermodules over a tower of graded superalgebras satisfying certain natural conditions give rise to…