97 Citations
Derived noncommutative schemes, geometric realizations, and finite dimensional algebras
- MathematicsRussian Mathematical Surveys
- 2018
The main purpose of this paper is to describe various phenomena and certain constructions arising in the process of studying derived noncommutative schemes. Derived noncommutative schemes are defined…
The birational geometry of noncommutative surfaces.
- Mathematics
- 2019
We show that any commutative rationally ruled surface with a choice of anticanonical curve admits a 1-parameter family of noncommutative deformations parametrized by the Jacobian of the anticanonical…
Moduli spaces of semiorthogonal decompositions in families
- Mathematics
- 2020
For a smooth projective family of schemes we prove that a semiorthogonal decomposition of the bounded derived category of coherent sheaves of the fiber of a point uniquely deforms over an etale…
On stacky surfaces and noncommutative surfaces
- Mathematics
- 2022
. Let k be an algebraically closed field of characteristic ≥ 7 or zero. Let A be a tame order of global dimension 2 over a normal surface X over k such that Z( A ) = O X is locally a direct summand of…
Non-existence of exceptional collections on twisted flags and categorical representability via noncommutative motives
- Mathematics
- 2016
In this paper we prove that the finite product of Brauer--Severi varieties is categorically representable in dimension zero if and only if it admits a $k$-rational point if and only if it is rational…
Derived categories and birational geometry of Gushel-Mukai varieties
- Mathematics
- 2016
We study the derived categories of coherent sheaves on Gushel–Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques…
Derived categories of noncommutative quadrics and Hilbert schemes of points
- Mathematics
- 2016
A non-commutative deformation of quadric surface is usually taken to be a three-dimensional cubic Artin–Schelter regular algebra. In this paper we show that for such an algebra its bounded derived…
Moduli of noncommutative Hirzebruch surfaces
- Mathematics
- 2019
We introduce three non-compact moduli stacks parametrizing noncommutative deformations of Hirzebruch surfaces; the first is the moduli stack of locally free sheaf bimodules of rank 2, which appears…
Derived categories of Gushel–Mukai varieties
- MathematicsCompositio Mathematica
- 2018
We study the derived categories of coherent sheaves on Gushel–Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques…
References
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Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor
- Mathematics
- 2006
For a map f : X → Y of quasi-compact quasi-separated schemes, we discuss quasi-perfection, i.e., the right adjoint f of Rf∗ respects small direct sums. This is equivalent to the existence of a…
Generators and representability of functors in commutative and noncommutative geometry
- Mathematics
- 2002
We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is…
Higher Algebraic K-Theory of Schemes and of Derived Categories
- Mathematics
- 1990
In this paper we prove a localization theorem for the K-theory of commutative rings and of schemes, Theorem 7.4, relating the K-groups of a scheme, of an open subscheme, and of the category of those…
Ideals in triangulated categories: phantoms, ghosts and skeleta
- Mathematics
- 1998
We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties and that if I has these properties, then…
Some Algebras Associated to Automorphisms of Elliptic Curves
- Mathematics
- 2007
The main object of this paper is to relate a certain type of graded algebra, namely the regular algebras of dimension 3, to automorphisms of elliptic curves. Some of the results were announced in…
Uniqueness of enhancement for triangulated categories
- Mathematics
- 2010
The paper contains general results on the uniqueness of a DG enhancement for trian- gulated categories. As a consequence we obtain such uniqueness for the unbounded categories of quasi-coherent…
The homotopy theory of dg-categories and derived Morita theory
- Mathematics
- 2004
The main purpose of this work is to study the homotopy theory of dg-categories up to quasi-equivalences. Our main result is a description of the mapping spaces between two dg-categories C and D in…
Smoothness of equivariant derived categories
- Mathematics
- 2012
We introduce the notion of (homological) G‐smoothness for a complex G‐variety X, where G is a connected affine algebraic group. This is based on the notion of smoothness for dg algebras and uses a…