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CORDIS

Cohomology and Singularities

Periodic Reporting for period 2 - CohoSing (Cohomology and Singularities)

Reporting period: 2021-04-01 to 2022-09-30

The project established the purity conjecture for flat cohomology and made significant advances on related questions, such as the Grothendieck--Serre conjecture. This has advanced the state of the art of our understanding of flat cohomology and of torsors under reductive group schemes.
During the reporting period the PI published the preprint with Peter Scholze “Purity for flat cohomology”, which is being refereed and is freely available as arXiv:1912.10932.

During the reporting period the PI published the preprint “The Bass--Quillen phenomenon for reductive group torsors”, which is freely available as arXiv:2204.08233. This completed the proof of a conjecture of Nisnevich about torsors under reductive groups over complements of smooth divisors.

During the reporting period the PI published the article “Grothendieck--Serre in the quasi-split unramified case” in “Forum of Mathematics, Pi”, which is freely available as arXiv:2009.05299. This resolved a major mixed characteristic case of the Grothendieck--Serre conjecture; this case had previously been considered widely open.

During the reporting period the PI published the survey article “Problems about torsors over regular rings (with an appendix by Yifei Zhao)” in “Acta Mathematica Vietnamica”, which is freely available as arXiv:2201.06424. This exposed the state of the art on problems that concerns torsors under reductive groups, especially, purity questions.

During the reporting period the PI’s published the article “Macaulayfication of Noetherian schemes”, which concerns another major part of the project (resolution of singularities), in “Duke Mathematical Journal” and is freely available as arXiv:1810.04493.

During the reporting period the PI published the article “Grothendieck--Lefschetz for vector bundles” in “Algebraic Geometry”, the joint article with Alexis Bouthier “Torsors on loop groups and the Hitchin fibration” that is accepted to “Annales Scientifiques de l’ENS”, and the preprint joint with Michalis Neururer and Abhishek Saha “The Manin constant and the modular degree” that is being refereed.

During the reporting period the PI gave numerous talks at conferences and seminars on the works mentioned above, as well as organized a working seminar with his students and postdocs around the themes of the project.

During the reporting period the PI hired Arnab Kundu and Shang Li as a PhD students and Hiroki Kato, Yifei Zhao, Zijian Yao, Kazuhiro Ito, Zhouhang Mao, and Siddharth Mathur as postdocs to work on the project.
The project introduced the use of techniques of derived algebraic geometry, specifically, animated rings, into the study of classical questions about cohomology. This opens the door to a wider use and adoption of these novel techniques.