## 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

SHOWING 1-10 OF 54 REFERENCES

### 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…