# ANAGRAMS

Student seminar

## Goal

The goal of this series would be to:

1. familiarise ourselves with the classical version of this abstract statement: understand Riemann–Roch for curves and understand Serre duality for projective varieties [AG];
2. get some intuition for Riemann–Roch and Serre duality through lots of examples and applications;
3. understand the statement of Grothendieck duality, see how it generalises the classical statements;
4. understand some of the techniques and the overal setup of Hartshorne's proof (see [RD]) of Grothendieck duality (interesting from a geometric point of view);
5. understand some of the techniques and the overal setup of Murfet's [1] (and Neeman's [2]) proof of Grothendieck duality (interesting from an abstract point of view).

The final set of notes (i.e. the 4 lectures combined) is now also available.

## Schedule

January 8
Riemann–Roch and Serre duality
by Pieter Belmans, in G.017, at 10h00
notes
January 15
More on Riemann–Roch and Serre duality, with applications
by Pieter Belmans, in G.017, at 10h00
notes and the tool to compute dimensions of cohomology spaces
January 22
Derived categories and Grothendieck duality
by Pieter Belmans, in G.017, at 10h00
notes
January 29
Sketches of some of the proofs and applications
by Pieter Belmans, in G.017, at 10h00
notes

## References

[RD]
Hartshorne, Robin
Residues and duality
Springer-Verlag, 1966, Lecture Notes in Mathematics 20, pp. vii+423
[AG]
Hartshorne, Robin
Algebraic geometry
Springer, 1977, Graduate Texts in Mathematics 52, pp. xvi+496
[1]
Murfet, Daniel
The mock homotopy category of projectives and Grothendieck duality
Australian National University, 2007
[2]
Neeman, Amnon
The Grothendieck duality theorem via Bousfield's techniques and Brown representability
Journal of the American Mathematical Society, 1996, Vol. 9(1), pp. 205-236
[3]
Neeman, Amnon
Dualizing complexes---the modern way
Cycles, motives and Shimura varieties
Narosa Publishing House, 2010, pp. 419-447