Olaf M. Schnürer

Matrix factorizations and semi-orthogonal decompositions for blowing-ups

We study categories of matrix factorizations. These categories are defined for any regular function on a suitable regular scheme. Our paper has two parts. In the first part we develop the foundations; for example we discuss derived direct and inverse image functors and dg enhancements. In the second part we prove that the category of matrix factorizations on the blowing-up of a suitable regular scheme X along a regular closed subscheme Y has a semi-orthogonal decomposition into admissible subcategories in terms of matrix factorizations on Y and X. This is the analog of a well-known theorem for bounded derived categories of coherent sheaves, and is an essential step in our forthcoming article which defines a Landau-Ginzburg motivic measure using categories of matrix factorizations. Finally we explain some applications.

Smoothness of equivariant derived categories

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 suitable enhancement of the G-equivariant derived category of X. If there are only finitely many G-orbits and all stabilizers are connected, we show that X is G-smooth if and only if all orbits O satisfy H^*(O; R)=R. On the way we prove several results concerning smoothness of dg categories over a graded commutative dg ring.

Perfect Derived Categories of Positively Graded DG Algebras

We investigate the perfect derived category

Six operations on dg enhancements of derived categories of sheaves

{{\rm dgPer}}(\mathcal{A})

Geometricity for derived categories of algebraic stacks

of a positively graded differential graded (dg) algebra

Matrix factorizations and motivic measures

\mathcal{A}

Proper base change for separated locally proper maps

whose degree zero part is a dg subalgebra and semisimple as a ring. We introduce an equivalent subcategory of

Six Operations on dg Enhancements of Derived Categories of Sheaves and Applications

{{{\rm dgPer}}}(\mathcal{A})

New enhancements of derived categories of coherent sheaves and applications

whose objects are easy to describe, define a t-structure on

Equivariant sheaves on flag varieties

{{{\rm dgPer}}}(\mathcal{A})