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Olaf M. Schnürer

Publications and source records attributed to Olaf M. Schnürer.

16 recordsLinked to original sources
arXiv2021

Unbounded derived categories of small and big modules: Is the natural functor fully faithful?

Consider the obvious functor from the unbounded derived category of all finitely generated modules over a left noetherian ring $R$ to the unbounded derived category of all modules. We answer the natural question whether this functor defines an equivalence onto the full subcategory of complexes with finitely generated cohomology modules in two special cases. If $R$ is a quasi-Frobenius ring of infinite global dimension, then this functor is not full. If $R$ has finite left global dimension, this functor is an equivalence. We also prove variants of the latter assertion for left coherent rings, for noetherian schemes and for locally noetherian Grothendieck categories.

math.CT↗
arXiv2019

Conservative descent for semi-orthogonal decompositions

Motivated by the local flavor of several well-known semi-orthogonal decompositions in algebraic geometry, we introduce a technique called conservative descent, which shows that it is enough to establish these decompositions locally. The decompositions we have in mind are those for projectivized vector bundles and blow-ups, due to Orlov, and root stacks, due to Ishii and Ueda. Our technique simplifies the proofs of these decompositions and establishes them in greater generality for arbitrary algebraic stacks.

math.AG↗
arXiv2019

Smoothness of Derived Categories of Algebras

We prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field, hereby answering a question of Iyama. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated as an algebra. These results are deduced from a general sufficient criterion for smoothness.

math.AG↗
arXiv2019

Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties

It is well known that the category of quasi-coherent sheaves on a gerbe banded by a diagonalizable group decomposes according to the characters of the group. We establish the corresponding decomposition of the unbounded derived category of complexes of sheaves with quasi-coherent cohomology. This generalizes earlier work by Lieblich for gerbes over schemes whereas our gerbes may live over arbitrary algebraic stacks. By combining this decomposition with the semi-orthogonal decomposition for a projectivized vector bundle, we deduce a semi-orthogonal decomposition of the derived category of a familiy of Brauer-Severi varieties whose components can be described in terms of twisted sheaves on the base. This reproves and generalizes a result of Bernardara.

math.AG↗
arXiv2016

Geometricity for derived categories of algebraic stacks

We prove that the dg category of perfect complexes on a smooth, proper Deligne-Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated categories, this means that the derived category of perfect complexes embeds as an admissible subcategory into the bounded derived category of coherent sheaves on a smooth, projective variety. The same holds for a smooth, projective, tame Artin stack over an arbitrary field.

math.AG↗
arXiv2016

Motivic vanishing cycles as a motivic measure

We show that the motivic vanishing cycles introduced by J. Denef and F. Loeser give rise to a motivic measure on the Grothendieck ring of varieties over the affine line. We discuss the relation of this motivic measure to the motivic measure we constructed earlier using categories of matrix factorizations.

math.AG↗
arXiv2015

New enhancements of derived categories of coherent sheaves and applications

We introduce new enhancements for the bounded derived category $D^b(Coh(X))$ of coherent sheaves on a suitable scheme $X$ and for its subcategory $Perf(X)$ of perfect complexes. They are used for translating Fourier-Mukai functors to functors between derived categories of dg algebras, for relating homological smoothness of $Perf(X)$ to geometric smoothness of $X,$ and for proving homological smoothness of $D^b(Coh(X)).$ Moreover, we characterize properness of $Perf(X)$ and $D^b(Coh(X))$ geometrically.

math.AG↗
arXiv2015

Matrix factorizations and motivic measures

This article is the continuation of [LS12]. We use categories of matrix factorizations to define a morphism of rings (= a Landau-Ginzburg motivic measure) from the (motivic) Grothendieck ring of varieties over $\mathbb{A}^1$ to the Grothendieck ring of saturated dg categories (with relations coming from semi-orthogonal decompositions into admissible subcategories). Our Landau-Ginzburg motivic measure is the analog for matrix factorizations of the motivic measure in [BLL04] whose definition involved bounded derived categories of coherent sheaves. On the way we prove smoothness and a Thom-Sebastiani theorem for enhancements of categories of matrix factorizations.

math.AG↗
arXiv2014

Proper base change for separated locally proper maps

We introduce and study the notion of a locally proper map between topological spaces. We show that fundamental constructions of sheaf theory, more precisely proper base change, projection formula, and Verdier duality, can be extended from continuous maps between locally compact Hausdorff spaces to separated locally proper maps between arbitrary topological spaces.

math.AT↗
arXiv2013

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.

math.AG↗
arXiv2013

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.

math.KT↗
arXiv2011

Homotopy categories and idempotent completeness, weight structures and weight complex functors

This article provides some basic results on weight structures, weight complex functors and homotopy categories. We prove that the full subcategories K(A)^{w < n}, K(A)^{w > n}, K(A)^- and K(A)^+ (of objects isomorphic to suitably bounded complexes) of the homotopy category K(A) of an additive category A are idempotent complete, which confirms that (K(A)^{w <= 0}, K(A)^{w >= 0}) is a weight structure on K(A). We discuss weight complex functors and provide full details of an argument sketched by M. Bondarko, which shows that if w is a bounded weight structure on a triangulated category T that has a filtered triangulated enhancement T' then there exists a strong weight complex functor T -> K(heart(w))^{anti}. Surprisingly, in order to carry out the proof, we need to impose an additional axiom on the filtered triangulated category T' which seems to be new.

math.CT↗
arXiv2010

Perfect Derived Categories of Positively Graded DG Algebras

We investigate the perfect derived category dgPer(A) of a positively graded differential graded (dg) algebra A whose degree zero part is a dg subalgebra and semisimple as a ring. We introduce an equivalent subcategory of dgPer(A) whose objects are easy to describe, define a t-structure on dgPer(A) and study its heart. We show that dgPer(A) is a Krull-Remak-Schmidt category. Then we consider the heart in the case that A is a Koszul ring with differential zero satisfying some finiteness conditions.

math.RT↗
arXiv2008

Equivariant Sheaves on Flag Varieties

We show that the Borel-equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of dg modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves. This proves a conjecture of Soergel and Lunts in the case of flag varieties.

math.RT↗
arXiv2006

Regular Connections on Principal Fiber Bundles over the Infinitesimal Punctured Disc

This paper concerns regular connections on trivial algebraic G-principal fiber bundles over the infinitesimal punctured disc, where G is a connected reductive linear algebraic group over an algebraically closed field of characteristic zero. We show that the pull-back of every regular connection to an appropriate covering of the infinitesimal punctured disc is gauge equivalent to a connection of the form X z^{-1}dz for some X in the Lie algebra of G. We may even arrange that the only rational eigenvalue of ad X is zero. Our results allow a classification of regular SL_n-connections up to gauge equivalence.

math.RT↗