Spring School :

“Derived equivalences: representations and coherent sheaves”

CVJM Bildungsstätte Bundeshöhe,

Wuppertal, 3 March – 7 March 2008

Organizers: Lutz Hille (Bielefeld), Markus Reineke (Wuppertal)

Hosted and funded by:

Graduiertenkolleg „Darstellungstheorie und ihre Anwendungen in Mathematik und Physik” / BU Wuppertal

The Spring School:

The topic of the Spring School is the construction of derived equivalences between categories of coherent sheaves on algebraic varieties and modules over (non-commutative) algebras.

The Spring School is intended for Ph.D. students and (early) postdocs working on and interested in these fields.

The Spring School will follow the tradition of previous spring/summer schools in representation theory. Every participant chooses a subject of the program (see below) and gives a talk. The typical participant will not be an expert in the subject, but working in one of the fields indicated above and interested in jointly learning a new subject. There will be no special prerequisities except for standard knowledge in algebra, but participants are expected to prepare themselves and their talks in advance. It is recommended that this is done in small groups; in particular, the talks could/should be subdivided into smaller parts.

At the end of the week, there will be lectures by Alexey Bondal (University Aberdeen) on more recent developments in the field.

Program:

The program is divided into several sections. Advanced members might look to the last talks in each of the sections. Most talks use derived/triangulated categories. So we assume that each member knows about the basics. We also included some talks not using derived categories at all (indicated by "n-d"). We start the program with two talks reminding all participants on basic properties of derived/triangulated categories and also include some examples at the beginning. Section 1 is mainly devoted to the introduction and basic examples. In Section 2 we deal with the general theory on exceptional collections and tilting sheaves, respectively, tilting modules and tilting complexes. Most of the talks asume some basic knowledge in algebraic geometry (vector bundles, projective spaces, del Pezzo surfaces), whereas others are completely formal. We also include some examples from representation theory. Section 3 contains more concrete theory on various algebraic varieties.

Each participant should choose 2 to 3 talks he would like to give during the school. It might be a good idea to work in groups and to choose talks for a group of participants, We try to fix the speakers and the final program in December. There might be to many talks in the present program for the weak. Depending on the knowledge of participants and their wishes  we  will  delete some of talks.

For a first overview we refer to:

section 1: [Ha1]
section 2: [BO1],
section 3: [BO1]
section 4: [Br2], [HV]

for all talks we assume 45 min, otherwise we explicitely mention the time you may need

suggestions for joint preparation: 1.4 and 2.1; 1.1 and 2.2; 2.3 and 2.4 (also 2.5 or 2.7); 3.1 and 3.2; 2.1 and 3.3; 4.2
 
Section 1. Introduction

1.1 Torsten Hoge, Anja Hutschenreuter: Introduction to triangulated categories (2 talks, each 45 min)

definition of tringulated categories, derived categories as examples, further examples: the derived category for a hereditary category (sheaves in curves, modules over hereditary algebras)
ref: Happel's book [Ha] (or some citations therein) chpt. 1, see also IV.1

1.2 Boris Betzholz: The stable module category

the stable module category should be defined, some examples should be presented
ref: Happel's book [Ha] chpt. 1, sect. 2

1.3 Nikolay Dichev, Stefan Wolf: Happel's equivalence (60 or 90 min)

 Happels functor between the derived category and the stable module category over the repetitive algebra
ref: Happel's book [Ha] chpt. 2, Theorem 4.9 should be explained and the idea of the proof should be given

1.4 Gretar Amazeen: BGG-equivalences  and  equivalences of Beilinson

First examples of equivalnces between derived categories of coherent sheaves and module catgeories should be given
ref: [Be1], [BGG] (the notes are rather short, a more technical version of BGG can be found in Hille [Hi1], a modern version of Beilinson"s result can be found in [Baer] or [K1])

Section 2. Tilting theory

2.1 Yuming Liu, Qunhua Liu: Tilting bundles

Further examples of tilting bundles
[K1], [Ki], [R2],

2.2  Guodong Zhou: Tilting modules over hereditary algebras

[Ha1] chpt. IV (should not overlap with talk 1.1)

2.3 Bo Chen: Exceptional Sequences and the braid group action

mutations of exceptional sequences should be defined, the braid group relations should be checked
[GR],[Helix]: Gorodentsev p.57

2.4 Nicolas Pöttering: The braid group action for modules over hereditary algebras

[Ri], [C-B]

2.5 Dong Yang: Strongly exceptional sequences on P^2 and the Markov equation

the Markov equation should be explained, if there is more time: the equation for a quadric and the connection with cluster mutations can be mentioned

[R1] (for the additional part: [R2], [BBH],[R3])

2.6 Gregoire Dupont: Koszul Duality (60 - 90 min, could be divided into 2 talks)

this talk should be considered as an indendent example of a version of tilting, Koszul duality should be explained, if there is time one should mention  also the connection with [BGG]
[Helix]: Bondal p. 75, section 7, also [BGS] or [Hi1] could be used

2.7 Polynomial invariants for the braid group action

following the previous talk one should construct polynomials of Markov type for exceptional sequences of alrger length
[Hi2]

Section 3. Derived categories of coherent sheaves

3.1 Fourier-Mukai transforms (60 min)

an introduction to Fourier-Mukai transforms should be given
one may use [HV] and the references therein

3.2 Johannes Engel, Andreas Hochenegger: Orlov's theorem

Orlov's theorem on derived equivalnces between derived categories should be explained,
[Or]

3.3 Markus Hendler: Projective Space and Grassmannianns

this talk should extend the talk 2.1 wit  more  and advanced examples, one may use also new results of VandenBergh and Bridgeland
references of 2.1 and  [Br4],

3.4 Martin Bender, Stephanie Cupit: Toric varieties (60 - 90min, could be split into 2 talks)

this talk should be given by an expert in toric geometry

Section 4. New developements (for these talks the length may depend on the wishes of the speaker 45 - 90 min and the remaining time)

4.1 Mikhail Antipov: Flops

[Br1]

4.2 David Ploog: Bridgelands stability in triangulated categories (2 talks)

[Br3] and the classical notion of stability like in [Ru]

4.3 Markus Perling: The use of derived categories in physics

[St],[HHV] give some overview, further ideas are explained in [FHMSV], [HV]

4.4 Roland Olbricht: Van den Bergh's results

[VdB]

4.5 Helix theory

[GK], based on the talks in section 2 the theory of helices should be explained

References:

1. Baer, Dagmar Tilting sheaves in representation theory of algebras. Manuscripta Math. 60 (1988), no. 3, 323--347 [Baer]

2. Be\u\i linson, A. A. Coherent sheaves on $P\sp{n}$ and problems in linear algebra. (Russian) Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68--69. [Be2]

3. Be\u\i linson, A. A. The derived category of coherent sheaves on $P\sp n$. Selected translations. Selecta Math. Soviet. 3 (1983/84), no. 3, 233--237.[Be1]

4. Bern\v ste\u\i n, I. N.; Gel\cprime fand, I. M.; Gel\cprime fand, S. I. Algebraic vector bundles on $P\sp{n}$ and problems of linear algebra. (Russian) Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66--67. [BGG]

5. Bondal, A.; Orlov, D. Derived categories of coherent sheaves. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 47--56, Higher Ed. Press, Beijing, 2002. [BO1]

6. Bondal, Alexei; Orlov, Dmitri Reconstruction of a variety from the derived category and groups of autoequivalences. Compositio Math. 125 (2001), no. 3, 327--344. [BO2]

7. Bondal, A. I.; Polishchuk, A. E. Homological properties of associative algebras: the method of helices. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 2, 3--50; translation in Russian Acad. Sci. Izv. Math. 42 (1994), no. 2, 219--260. [BP]

8. Bondal, A. I. Helices, representations of quivers and Koszul algebras. Helices and vector bundles, 75--95, London Math. Soc. Lecture Note Ser., 148, Cambridge Univ. Press, Cambridge, 1990.

9. Bondal, A. I. Representations of associative algebras and coherent sheaves. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 25--44; translation in Math. USSR-Izv. 34 (1990), no. 1, 23--42.

10. Bridgeland, Tom Flops and derived categories. Invent. Math. 147 (2002), no. 3, 613--632. [Br1]

11. Bridgeland, Tom(4-SHEF-PM)
    Derived categories of coherent sheaves. (English summary) International Congress of Mathematicians. Vol. II, 563--582, Eur. Math. Soc., Zürich, 2006. [Br2]

12. Authors: Tom Bridgeland
    arXiv:math/0611510 [ps, pdf, other]
    Title: Spaces of stability conditions. [Br3]

13. Bridgeland, Tom t-structures on some local Calabi-Yau varieties. J. Algebra 289 (2005), no. 2, 453--483. [Br4]

14. Crawley-Boevey, William(4-OX)Exceptional sequences of representations of quivers [MR1206935 (94c:16017)]. (English summary) Representations of algebras (Ottawa, ON, 1992), 117--124. [C-B]

15. arXiv:hep-th/0505211 [ps, pdf, other]
    Title: Gauge Theories from Toric Geometry and Brane Tilings
    Authors: Sebastian Franco, Amihay Hanany, Dario Martelli, James Sparks, David Vegh, Brian Wecht, [FHMSV]

16. Gorodentsev, A. L.; Kuleshov, S. A. Helix theory. Mosc. Math. J. 4 (2004), no. 2, 377--440, 535. [GK]

17. Gorodentsev, A. L.; Rudakov, A. N. Exceptional vector bundles on projective spaces. Duke Math. J. 54 (1987), no. 1, 115--130.[GR]

18. Amihay Hanany, Christopher P. Herzog, David Vegh ,Title: Brane Tilings and Exceptional Collections 

    arXiv:hep-th/0602041 [ps, pdf, other], [HHV]

19. arXiv:hep-th/0511063 [ps, pdf, other]
    Title: Quivers, Tilings, Branes and Rhombi
    Authors: Amihay Hanany, David Vegh, [HV]

20. Happel, Dieter: Triangulated categories in the representation theory of finite-dimensional algebras. London Mathematical Society Lecture Note Series, 119. Cambridge University Press, Cambridge, 1988. x+208 pp. [Ha]

21. Hille, Lutz: Polynomial Invariants for Mutations, lecture at Bielefeld. [Hi2]

22. Hille, Lutz Consistent algebras and special tilting sequences. Math. Z. 220 (1995), no. 2, 189--205. [Hi1]

23. Lutz Hille, Michel Van den Bergh

    Fourier-Mukai Transforms, arXiv:math/0402043, [HV]

24. Kapranov, M. M. On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92 (1988), no. 3, 479--508. [K1]

25. Kapustin, A. N.; Orlov, D. O. Lectures on mirror symmetry, derived categories, and D-branes. (Russian) Uspekhi Mat. Nauk 59 (2004), no. 5(359), 101--134; translation in Russian Math. Surveys 59 (2004), no. 5, 907--940.

26. King, A. Tilting bundles on some rational surfaces(1997), http://www.maths.bath.ac.uk/~masadk/papers/. [Ki]

27. Orlov, D. O.(RS-AOS)
    Derived categories of coherent sheaves and equivalences between them. (Russian. Russian summary) Uspekhi Mat. Nauk 58 (2003), no. 3(351), 89--172; translation in Russian Math. Surveys 58 (2003), no. 3, 511--591. [Or]

28. Ringel, Claus Michael(D-BLF)
    The braid group action on the set of exceptional sequences of a hereditary Artin algebra. (English summary) Abelian group theory and related topics (Oberwolfach, 1993), 339--352.
    Contemp. Math., 171, Amer. Math. Soc., Providence, RI, 1994.[Ri]

29. Rudakov, A. N. Markov numbers and exceptional bundles on $P\sp 2$. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 1, 100--112, 240; translation in Math. USSR-Izv. 32 (1989), no. 1, 99--112. [R1]

30. Rudakov, A. N. Exceptional vector bundles on a quadric. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 4, 788--812, 896; translation in Math. USSR-Izv. 33 (1989), no. 1, 115--138. [R2]

31. Rudakov, A. N. Exceptional collections, mutations and helices. Helices and vector bundles, 1--6, London Math. Soc. Lecture Note Ser., 148, Cambridge Univ. Press, Cambridge, 1990. [Helix]

32. Rudakov, A. N.(2-MOSC)
    Integer-valued bilinear forms and vector bundles. (Russian) Mat. Sb. 180 (1989), no. 2, 187--194, 304; translation in
    Math. USSR-Sb. 66 (1990), no. 1, 189--197. [R3]

33. Jan Stienstra [ps, pdf, other] Hypergeometric Systems in two Variables, Quivers, Dimers and Dessins d'Enfants, arXiv:0711.0464

34. van den Bergh, Michel Non-commutative crepant resolutions. The legacy of Niels Henrik Abel, 749--770, Springer, Berlin, 2004. [VdB]

Schedule:

Monday, 3 March

12:30-14:00 Lunch

14:00-14:45 Hoge, Hutschenreuter part 1

15:00-15:45 Hoge, Hutschenreuter part 2

15:45-16:15 Coffee

16:15-17:15 Betzholz

17:30-18:00 Dichev, Wolf part 1

18:00-19:00 Dinner

Tuesday, 4 March

09:00-10:00 Dichev, Wolf part 2

10:00-10:30 Coffee

10:30-11:30 Amazeen

11:45-12:30 Liu, Liu part 1

12:30-14:00 Lunch

14:00-14:45 Liu, Liu part 2

15:00-16:00 Zhou

16:00-16:30 Coffee

16:30-18:00 Chen, Poettering

18:00-19:00 Dinner

Wednesday, 5 March

09:00-10:00 Yang

10:00-10:30 Coffee

10:30-11:30 Dupont

11:45-12:30 tba

12:30-13:30 Lunch

13:30-18:00 Free afternoon

18:00-19:00 Dinner

Thursday, 6 March

09:00-10:30 Engel, Hochenegger

10:30-11:00 Coffee

11:30-12:30 Bender, Cupit

12:30-14:00 Lunch

14:00-15:00 Hendler

15:15-16:15 Antipov

16:15-16:30 Coffee

16:30-18:00 Ploog

18:00-19:00 Dinner

Friday, 7 March

09:00-10:30 Perling

10:30-11:00 Coffee

11:00-12:30 Bondal part 1

12:30-13:30 Lunch

13:30-15:00 Bondal part 2

Participation:

The expected number of participants is about 20-25.

The funding Research Training Program plans to cover accomodation and full boarding. However, travel expenses can not be covered. If you want to participate in the Spring School, please send an email to M. Reineke (reineke “at” math.uni-wuppertal.de) , if possible indicating which topic you are willing to give a talk about.

The deadline for registration was 15 January 2008.

List of participants:

M. Reineke, 18 February 2008