Spring school 2009 :

local algebra

Nice, March 30th - April 3nd 2009

Speakers/Participants:
Tanja Becker (Nantes), Samuel Boissière (Nice), Frank Ditsche (Augsbourg), Antoine Ducros (Nice), Julia Eich (Augsbourg),
Yael Fleischmann (Mayence), Daniel Hanke (Mayence), Barbara Jung (Mayence), Heinrich Hartmann (Bonn),
Andreas Krug (Augsbourg), Christian Lehn (Mayence), Manfred Lehn (Mayence), Yasunari Nagai (Mayence),
Marc Nieper-Wisskirchen (Augsbourg), Rémy Oudompheng (Nice), Arvid Perego (Mayence), Michel Raibaut (Nice),
Soenke Rollenske (Bonn), Alessandra Sarti (Poitiers), Olivier Serman (Lille), Irene Sommer (Augsbourg),
Christoph Sorger (Nantes), Franz Vogler (Augsbourg), Anna Wissdorf (Mayence),
Constantin Wittenmeier (Augsbourg), Markus Zowislok (Mayence).

Accommodation:
Hôtel Mirabeau: 15 avenue Malausséna, 06000 Nice, http://www.hotel-mirabeau.net/
Villa Saint Hubert: 26 rue Michel Ange, 06100 Nice, http://www.villasainthubert.com/html_fr/presentation.htm

Schedule:
There will be 6 lectures of 45 minutes a day, scheduled as follows:
09h15 – 10h00 Lecture 1
10h00 – 10h30 Coffee break
10h30 – 11h15 Lecture 2
11h30 – 12h15 Lecture 3
12h15 – 14h15 Lunch (lunch at 13h)
14h15 – 15h00 Lecture 4
15h15 – 16h00 Lecture 5
16h00 – 16h30 Coffee break
16h30 – 17h15 Lecture 6

Notice: To avoid confusion, we agree to use the notation, conventions and definitions of Serre [S].

Monday

Morning: Basic tools

Lecture 1: Filtrations, gradings and completions.
Ref: [S] Chap. II.A (with sketches of proofs). For complements see [AM].
Speaker: Andreas Krug

Lecture 2: Hilbert-Samuel polynomials.
Ref: [S] Chap. II.B (with sketch of proofs of theorems 2 and 3)
Speaker: Julia Eich

Lecture 3: Dimension and normal rings.
Ref: [S] Chap. III.B and III.C (with sketch of proof of proposition 15)
Speaker: Markus Zowislok

Afternoon: Homological dimension and depth

Lecture 4: The Koszul complex
Ref: [S] Chap IV.A.1,2 (3) (with proof of proposition 3)
Speaker: Anna Wissdorf

Lecture 5: Depth and Cohen-Macaulay rings
Ref: [S] Chap IV.A.4 and IV.B
Speaker: Daniel Hanke

Lecture 6: Homological dimension and noetherian modules
Ref: [S] Chap IV.C
Speaker: Yael Fleischmann

Tuesday

Morning: Regular rings

Lecture 1: The Auslander-Buchsbaum formula relating depth and projective dimension.
Ref: [E95] Theorem 19.9 p.479 (with detailed proof)
Speaker: Tanja Becker

Lecture 2: Regular rings.
Ref: [S] Chap.IV.D (with proof of Theorem 9, but not of Corollary 4)
Speaker: Michel Raibaut

Lecture 3: The Auslander-Buchsbaum theorem : factoriality of regular rings.
Ref: [E95] Theorem 19.19 p.485 (with detailed proof). Other references : Bourbaki or initial proof in MR0094344 and MR0103906.
Speaker: Heinrich Hartmann

Afternoon: Free resolutions

Lecture 4: Minimal free resolution.
Ref : [S] Appendix I-1 and {E95] Theorem 20.2 (with proof).
Speaker: Franz Vogler

Lecture 5: Fitting ideals.
Ref : [E95] Chap.20.2 (with sketch of proofs and case of modules over a principal ideal domain). See also [BE77W] for complements.
Speaker: Constantin Wittenmeier

Lecture 6: What makes a complex exact?
Ref : [E95] Chap.20.3 (see also [BE73W] for complements)
Speaker: Marc Nieper-Wisskirchen

Wednesday

Morning: Sygyzies

Lecture 1: The Hilbert syzygy theorem.
Ref : [E95] Corollary 19.7 and 19.8 (with proof), [E00] and [E04] for geometric examples.
Speaker: Irene Sommer

Lecture 2: The Hilbert-Burch theorem.
Ref : [E00] theorem 3.2 (with sketch of proof and examples).
Speaker: Barbara Jung

Lecture 3: Duality and canonical modules.
Ref : [E95] Chap.21.1
Speaker: Rémy Oudompheng

Afternoon: Gorenstein rings

Lecture 4: Various definitions of Gorenstein rings.
Ref : [E95] Chap.21.2, 21.3, 21.7, 21.8 and [M] Chap.6.18.
Speaker: Antoine Ducros

Lecture 5: On the ubiquity of Gorenstein rings.
Ref : [B] (other definition, relation to previous definition, examples (in particular 5.1).
Speaker: Antoine Ducros

Lecture 6: Gorenstein rings of depth 2 and 3.
Ref : [BE77G] (Serre result and Theorem 4 with some idea of proof). See [BE77A] and [S60] for complements.
Speaker: Christian Lehn

Thursday

Morning: Canonical singularities

Lecture 1: Divisorial sheaves.
Ref: [R1] Appendix to §1 (with proofs). See [R2] for complements.
Speaker: Arvid Perego

Lecture 2: Definition of canonical singularities.
Ref: [R1] §0 and §1 (with proofs). See [R2] for complements.
Speaker: Alessandra Sarti

Lecture 3: Rational Gorenstein singularities in dimension 3.
Ref: [R1] §2 (with proofs). See [R2] for complements.
Speaker: Olivier Serman

Afternoon: Rational singularities

Lecture 4: Dualizing complexes I.
Ref: [H2] Chap. III (introductory). See [Hu] for complements and examples.
Speaker: Manfred Lehn

Lecture 5: Dualizing complexes II. See [Hu] for complements and examples.
Ref: [H2] Chap. V (introductory).
Speaker: Manfred Lehn

Lecture 6: Rationality of canonical singularities.
Ref: [El] (with sketch of proof).
Speaker: Yasunari Nagai

Friday

Morning: Factoriality

Lecture 1: Krull rings, factorial and Q-factorial rings and varieties.
Ref: [M]
Speaker: Samuel Boissière

Lecture 2: Reflexive sheaves.
Ref: [H3] §1 with proofs.
Speaker: Arvid Perego

Lecture 3: Product of locally factorial and Q-factorial varieties.
Ref: none!
Speaker: Olivier Serman

Good bye!

References:

[AM] M. F. Atiyah, I. G. Macdonald, Introduction to commutative algebra.
[B] H. Bass, On the ubiquity of Gorenstein rings, Math.Z. 82,8-28 (1963) (pdf file)
[BE73W] D. A. Buchsbaum and D. Eisenbud, What makes a complex exact?, J. Alg. 25, 259-268 (1973) (pdf file)
[BE73R] D. A. Buchsbaum and D. Eisenbud, Remarks on ideals and resolutions, Sistituto nazionale di alta matematica, symp. Math vol XI (pdf file)
[BE77A] D. A. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions, Amer. J. Math. 99, 447-485 (1977) (pdf file)
[BE77G] D. A. Buchsbaum and D. Eisenbud, Gorenstein ideals of height 3 (pdf file)
[BE77W] D. A. Buchsbaum and D. Eisenbud, What annihilates a module?, J. Alg. 47, 231-243 (1977) (pdf file)
[E95] D. Eisenbud, Commutative algebra with a view towards algebraic geometry, Springer.
[E00] D. Eisenbud, The geometry of Syzygies, Springer.
[E04] D. Eisenbud, Lectures on the Geometry of Syzygies, Trends in Commutative Algebra, MSRI Publ. 51 (2004) (pdf file)
[El] R. Elkik, Rationalité des singularités canoniques, Invent. Math. 64, 1-6 (1981) (pdf file)
[EG] D. Eisenbud and M. L. Green, Ideals of minors in free resolutions, Duke math. J. 75,339-352 (1994) (pdf file)
[F] W. Fulton, Intersection theory, Springer.
[H1] R. Hartshorne, Algebraic geometry, Springer
[H2] R. Hartshorne, Residues and duality, L.N.M. 20.
[H3] R. Hartshorne, Stable reflexive sheaves, Math. Annalen 254, 121-176 (1980) (pdf file)
[H4] R. Hartshorne, Generalized divisors on Gorenstein schemes, K-theory 8 (1994), 287-339 (pdf file)
[Hu] D. Huybrechs, Fourier-Mukai transforms in Algebraic Geometry, Oxford Sc. Pub.
[K] M. Kunte, Gorenstein modules of finite length, arXiv:0807.2956 (pdf file)
[L] S. Lang, Abelian varieties.
[M] H. Matsumura, Commutative ring theory.
[R1] M. Reid, Canonical 3-folds, Journées de Géométrie Algébrique d'Angers, 1979 (pdf file)
[R2] M. Reid, Young person's guide to canonical singularities, Proc. Sympos. Pure Math., Bowdoin 1985 (pdf file)
[S] J.-P. Serre, Local algebra, Springer
[S60] J.-P. Serre, Sur les modules projectifs, Séminaire Dubreil 1960-1961 (pdf file)

How to come to university: If you come by Train, the bus 23 (direction St. Maurice) stops at the train station and will take you close to university (get-off at "Vallot"). Walking from the train station is possible and takes about 25-30 min. If you come by Plane, the bus line 23 starts from Terminal 1and stops close to university (at Puget), while bus lines 99 stops at terminal 1and 2 and goes to the Train station.

Other informations: Plan of the Bus lines of Nice, a map of Nice, the airport, the city. Here is a map of the Campus (The math dept is W)