Master Class on New Developments in Finite Generation of Cohomology
A celebrated result in the cohomology of finite group schemes is the Friedlander-Suslin theorem which states that the cohomology of a finite group scheme over a field is finitely generated. Recently, Wilberd van der Kallen proved a generalization of this result establishing the finite generation of cohomology of finite flat group schemes over Noetherian rings, based on earlier work with Franjou, Srinivas, and Touzé. This generalization opens the door to far-reaching consequences in a diverse range of areas. The aim of this master class is to understand van der Kallen's proof - and to do that, we will have several sets of lectures covering the relevant background material, the proof, as well as complementary results.
Location: Hybrid - on site at Bielefeld University and remotely on Zoom
Date: 25–27 September 2023
Registration Deadline: August 27, 2023 (for on site participation and arranged accommodation)
Organisers: Tobias Barthel (MPIM Bonn), Rudradip Biswas (Bielefeld University), Henning Krause (Bielefeld University), Julia Pevtsova (University of Washington, Seattle)
Speakers
- Antoine Touzé (Université de Lille)
- Vincent Franjou (Nantes Université)
- Paul Sobaje (Georgia Southern University)
- Peter Symonds (University of Manchester)
Relevant Material
Here are some relevant references for the topic of the master class:
- Vincent Franjou, Wilberd van der Kallen. Power reductivity over an arbitrary base. Doc. Math. Extra Vol., Andrei A. Suslin’s Sixtieth Birthday, 171–195 (2010). doi:10.4171/dms/7
- Eric M. Friedlander, Andrei Suslin. Cohomology of finite group schemes over a field. Invent math 127, 209–270 (1997). doi:10.1007/s002220050119
- Frank D. Grosshans. Contractions of the actions of reductive algebraic groups in arbitrary characteristic. Invent Math 107, 127–133 (1992). doi:10.1007/BF01231884
- Jens Carsten Jantzen. Representations of algebraic groups. Second edition. Mathematical Surveys and Monographs, 107. American Mathematical Society (2003). ISBN: 0-8218-3527-0
- Vasudevan Srinivas, Wilberd van der Kallen. Finite Schur filtration dimension for modules over an algebra with Schur filtration. Transformation Groups 14, 695–711 (2009). doi:10.1007/s00031-009-9054-0
- Antoine Touzé, Wilberd van der Kallen. Bifunctor cohomology and cohomological finite generation for reductive groups. Duke Math. J. 151 (2) 251–278 (2010). doi:10.1215/00127094-2009-065
- Antoine Touzé. Invariants, cohomologie et représentations fonctorielles des groupes algébriques. Spartacus-Idh (2022). ISBN: 978-2-36693-092-4
- Antoine Touzé. Universal classes for algebraic groups. Duke Math. J. 151 (2) 219–249 (2010). doi:10.1215/00127094-2009-064
- Wilberd van der Kallen. An integrality theorem of Grosshans over arbitrary base ring. Transformation Groups 19, 283–287 (2014). doi:10.1007/s00031-013-9241-x
- Wilberd van der Kallen. Good Grosshans filtration in a family. Autour des schémas en groupes. Vol. III, 111–129 (2015).
- Wilberd van der Kallen. Reductivity properties over an affine base. Indagationes Mathematicae, Volume 32, Issue 5, 961–967 (2021). doi:10.1016/j.indag.2020.09.009
- Wilberd van der Kallen. A Friedlander–Suslin theorem over a noetherian base ring. (2022). arXiv:2212.14600
- William C. Waterhouse. Introduction to affine group schemes. Graduate Texts in Mathematics, 66. Springer-Verlag (1979). doi:10.1007/978-1-4612-6217-6
The main reference is the recent preprint [11] by van der Kallen; its main result generalises a theorem by Friedlander–Suslin in [2]. The paper [11] is rather short and relies on various earlier and deep results by van der Kallen in combination with several coauthors. The most relevant of these earlier papers are included in the list. In addition one finds the text books by Jantzen [4] and Waterhouse [12] as basic references for the theory of finite group schemes and their representations.
The following are some relevant references for the talk by Peter Symonds:
- Peter Symonds. On the Castelnuovo-Mumford regularity of rings of polynomial invariants. Ann. of Math. (2) 174, no. 1, 499–517 (2011). doi:10.4007/annals.2011.174.1.14
- Peter Symonds. Group actions on rings and the Čech complex. Adv. Math. 240, 291–301 (2013). doi:10.1016/j.aim.2013.02.015
Accommodation
A limited number of hotel rooms
are available to junior participants
without sufficient funding to pay for accommodation from their own sources.
In order to be considered for one of these hotel rooms,
please tick the corresponding checkbox
on the registration form.
We recommend booking your accommodation early.
Funding
This master class is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via the TRR 358 "Integral Structures in Geometry and Representation Theory" (SFB-TRR 358/1 2023 – 491392403).
Contact
If you have any questions about the master class, please contact the organisers at birep.