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Cohomology and Singularities

Periodic Reporting for period 1 - CohoSing (Cohomology and Singularities)

Reporting period: 2019-10-01 to 2021-03-31

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 completed the joint article with Peter Scholze “Purity for flat cohomology”. The preprint is being refereed and is freely available as arXiv:1912.10932.

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

During the reporting period the PI prepared the preprint “Grothendieck--Serre in the quasi-split unramified case”, which is being refereed and 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 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.

During the reporting period the PI hired Arnab Kundu as a PhD student and Hiroki Kato, Yifei Zhao, Zijian Yao, and Kazuhiro Ito 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.