Student seminar

## Goal

The goal of this series would be to:

- familiarise ourselves with the classical version of this abstract statement: understand Riemann–Roch for curves and understand Serre duality for projective varieties [AG];
- get some intuition for Riemann–Roch and Serre duality through lots of examples and applications;
- understand the statement of Grothendieck duality, see how it generalises the classical statements;
- 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);
- 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

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