7-9 July 2011, Universität Stuttgart
Titles and abstracts
Leovigildo Alonso Tarrío
Compactly generated t-structures on schemes
It is an important result in the theory of Bousfield localization of
the derived category of a noetherian ring that they may be classified
by subsets of the spectrum. This theorem due to Hopkins and Neeman
opens a door of the link between stable homotopy and algebraic
geometry. The result can be extended to noetherian schemes provided
that we consider only localizations that are well behaved with the
monoidal structure of the category of quasi-coherent sheaves, a
condition that holds always in the affine case. This was obtained in
collaboration with Jeremías and Souto back in 2004 and afterwards
considered again by several authors. In the same paper compactly
generated localizations were characterized by those that are
associated to a stable for specialization subset (i.e. arbitrary union
of closed subsets).
For t-structures, as Stanley observed, such a classification is impossible because the collection of all t-structures on the derived category of a noetherian ring forms a proper class. A classification for the case of compactly generated t-structures in terms of filtration by stable for specialization subsets was given in collaboration with Jeremías and Saorín. In the talk we'll survey these results and report on work in progress about the classification of compactly generated t-structures on the derived category of quasi-coherent sheaves on a noetherian scheme.
Lidia Angeleri Hügel
Tilting and spectra of commutative noetherian rings
This is a report on joint work with David Pospí¨il, Jan ¦tovícek, and Jan Trlifaj. We give a complete classification of tilting modules and cotilting modules over a commutative noetherian ring R. In particular, we show that the equivalence classes of tilting modules of projective dimension at most one are parametrized by certain subsets of the Zariski spectrum of R. This is related to Gabriel's classification of hereditary torsion pairs in Mod-R and to the classification of recollements of the derived category D(Mod-R) due to Hopkins and Neeman.
Weight structures vs. t-structures
My talk is dedicated to weight structures (defined a few years ago, independently by me and by D. Pauksztello). Weight structures are natural counterparts of t-structures (for triangulated categories); they yield weight complexes, weight filtrations, and weight spectral sequences. Partial cases of the latter are: Deligne's weight spectral sequences (for singular and étale cohomology), coniveau spectral sequences, and Atiyah-Hirzebruch spectral sequences. I will describe some properties of weight structures, and mention two types of relations between them and t-structures: orthogonal structures and transversal structures. If a t-structure is orthogonal to a weight structure, then we obtain two distinct descriptions of certain ('weight') spectral sequences; in particular, one can describe Atiyah-Hirzebruch spectral sequences either using cellular filtrations of spectra or using the Postnikov t-structure for the stable homotopy category. If a weight structure is transversal to a t-structure, then there exists a certain 'weight' filtration for the heart of the latter that is strictly respected by morphisms. In particular, this is the case for complexes of polarizable Hodge structures, for Hodge modules, and (conjecturally) for mixed motives.
Stratifying derived module categories
The notion of recollement was introduced by Beilinson, Bernstein and Deligne. It provides a tool of deconstructing a triangulated category into smaller pieces, and can be viewed as an analogue of short exact sequence for modules. Iterated recollements then give rise to a stratification structure, as an analogue of composition series. This talk contains three parts. First I will discuss the question of existence and uniqueness of such a stratification, up to ordering and derived equivalence, for derived module categories. Then I will present a positive answer in the form of a Jordan Hoelder Theorem for piecewise hereditary algebras. Finally I will discuss derived simpleness of rings and algebras, meaning that the derived module category admits no nontrivial recollements. This talk is based on joint papers with Lidia Angeleri Huegel, Steffen Koenig and Dong Yang.
K-theory and t-structures
The K-theory of triangulated categories has been an object of
discussion for several years. Schlichting showed that there cannot
exist any higher K-theory of triangulated categories satisfying
desirable properties such as functoriality, additivity, localisation,
and agreement with Quillen's K-theory for derived categories.
Nevertheless, Neeman was able to define three K-theories for
triangulated categories, two of them functorial. All of them agree in
degree 0. He showed in a series of papers that in the presence of a
bounded non-degenerate t-structure his non-functorial K-theory agrees
with Quillen's K-theory of the heart. This is all that has been known
about these K-theories for some years. Neeman explicitly poses the
1-dimensional case as a problem in his survey on the K-theory of
triangulated categories "in order to show how embarrassingly little we
know" (sic). In this talk we will report on our proof that all these
K-theories agree in dimension one with Quillen's K-theory of the
heart, and show with examples that they may be non-isomorphic in that
dimension when there is not such a t-structure.
This is part of a joint work with Andrew Tonks (London Metropolitan) and Malte Witte (Heidelberg).
Cluster hearts and cluster tilting objects
Abstract as pdf-file.
Restriction of t-structures for dg algebras
We shall motivate and discuss the problem of restriction of t-structures on the derived category of a differential graded algebra to its perfect derived category.
Restrictions of t-structures to bounded subcategories
We shall address the problem of deciding when a t-structure, in particular a Bousfield localization, restricts to (one- or two-sided) bounded subcategories of certain triangulated categories. We shall pay special attention to the case when the ambient triangulated category is the derived category of a commutative Noetherian ring.
Idempotent completeness of homotopy categories, weight structures and weight complex functors
We prove that certain subcategories of the homotopy category of an additive category are idempotent complete. These results are used to confirm the standard example of a weight structure. Then we turn to weight complex functors. We discuss the existence of a strong weight complex functor if a filtered triangulated enhancement is available. The part about weight complex functors is based on M. Bondarko's work.
Tensor actions and hypersurfaces
We will give an overview of recent progress on classifying localizing subcategories of the stable derived category of a noetherian separated complete intersection scheme. Our main focus will be on the formalism of actions by tensor triangulated categories that this problem inspired and how one can obtain from this machinery a classification of localizing subcategories in the case of hypersurface rings. Time permitting we shall also discuss the classification in the non-affine and higher codimension cases.
Abstract methods for constructing t-structures
In this talk, I will mention two abstract methods for constructing t-structures. One of them, which is inherently an infinite construction and applies to many algebraic triangulated categories with infinite coproducts, uses Quillen's small object argument. The other is a simple observation which applies whenever one can obtain each object of a suspended subcategory from a fixed generator using a bounded number of extensions.
Adam-Christiaan van Roosmalen
t-Structures for hereditary categories
This is joint work with Don Stanley. It is known that an object in the
derived category of a hereditary category is uniquely determined by its
homologies. We use this strong connection to translate the classification
problem of aisles in the derived category to a classification problem in
the hereditary category, by 'cutting up' the aisles into its homologies.
The aim of this talk is to use this method to describe the following two main examples:
1) the t-structures on the bounded derived category of coherent sheaves of a smooth projective curve by using torsion theories, and
2) the t-structures on the bounded derived category of finite dimensional representations of a finite dimensional hereditary algebra by torsion theories and noncrossing partitions (using a result by Ingalls-Thomas and Igusa-Schiffler-Thomas).
Bounded t-structures for piecewise hereditary algebras
Using induction and restriction of t-structures with respect to recollements, in joint work with Qunhua Liu, we can describe which bounded t-structures on the derived category of a piecewise hereditary algebra can be assembled from t-structures on the corresponding derived simple factors. This talk will show such a description and report on some work in progress that shows an intimate relation between the hearts of such t-structures and the silting objects of the derived category - following recent results of Assem, Souto Salorio and Trepode on Ext-projective objects - thus generalising a well-known result of Keller and Vossieck for hereditary algebras of finite representation type.
f-categories and their applications
A large part of this talk will be devoted to recall the notions of f-structure and f-category, due to Beilinson, and to review their application, equally due to Beilinson, to triangulated categories. We then formulate a criterion which ensures the presence of an f-structure, and give concrete examples where this criterion is satisfied.