Periodic Reporting for period 4 - CohoSing (Cohomology and Singularities)
Período documentado: 2024-04-01 hasta 2025-03-31
Flat cohomology is an important invariant of Noetherian schemes that encodes deep arithmetic properties, for instance, it is critically important for studying the Birch and Swinnerton-Dyer conjecture---one of the Millennium Problems---about the arithmetic of abelian varieties. This ERC project has established the so-called purity conjecture for flat cohomology, which is a generalization of earlier conjectures of Auslander--Goldman (1960), Grothendieck (1968), and Gabber (2004) that predicts that flat cohomology is insensitive to removing closed subschemes of small codimension, in other words, that the flat cohomology of a Noetherian scheme is the same as that any of its open subschemes granted that the closed complement of the open is sufficiently small. This is a first general result of its kind that has significantly broadened our understanding of arithmetic in positive and mixed characteristic settings.
One of the nonabelian analogues of purity is the Grothendieck--Serre conjecture (1958) about Zariski local triviality of generically trivial torsors under reductive groups. This prediction predates the aforementioned purity conjectures, and it has also greatly informed the subject, for instance, the special case of PGL_n-torsors, equivalently, of Severi--Brauer varieties was critical for Grothendieck's formulation of his purity conjecture for the Brauer group (of which purity for flat cohomology is a vast generalization). This ERC project has established significant new cases of the Grothendieck--Serre conjecture: the case of smooth varieties in mixed characteristic for reductive groups that contain sufficiently large parabolic subgroups. Prior to this project, the mixed characteristic case has been considered to be practically out of reach.
A central problem about Noetherian schemes is the resolution of singularities conjecture, which predicts that suitable blowing ups could make the scheme nonsingular. This ERC project has established the Cohen--Macaulay version of this resolution of singularities conjecture: by suitable blowing ups, the singularities of the Noetherian scheme in question can all be made Cohen--Macaulay, without changing the open locus of those singularities that were Cohen.. Macaulay to begin with. This variant of the resolution of singularities problem has been considered in the theory of moduli on varieties but was considered to be inaccessible with existing methods prior to this ERC project.
Overall, this ERC project has significantly advanced the state of the art of our understanding of flat cohomology, of torsors under reductive group schemes, and of the resolution of singularities.
One of the nonabelian analogues of purity is the Grothendieck--Serre conjecture (1958) about Zariski local triviality of generically trivial torsors under reductive groups. This prediction predates the aforementioned purity conjectures, and it has also greatly informed the subject, for instance, the special case of PGL_n-torsors, equivalently, of Severi--Brauer varieties was critical for Grothendieck's formulation of his purity conjecture for the Brauer group (of which purity for flat cohomology is a vast generalization). This ERC project has established significant new cases of the Grothendieck--Serre conjecture: the case of smooth varieties in mixed characteristic for reductive groups that contain sufficiently large parabolic subgroups. Prior to this project, the mixed characteristic case has been considered to be practically out of reach.
A central problem about Noetherian schemes is the resolution of singularities conjecture, which predicts that suitable blowing ups could make the scheme nonsingular. This ERC project has established the Cohen--Macaulay version of this resolution of singularities conjecture: by suitable blowing ups, the singularities of the Noetherian scheme in question can all be made Cohen--Macaulay, without changing the open locus of those singularities that were Cohen.. Macaulay to begin with. This variant of the resolution of singularities problem has been considered in the theory of moduli on varieties but was considered to be inaccessible with existing methods prior to this ERC project.
Overall, this ERC project has significantly advanced the state of the art of our understanding of flat cohomology, of torsors under reductive group schemes, and of the resolution of singularities.
The main results of this ERC project and their dissemination are the following.
1. The purity for flat cohomology published in the Annals of Mathematics in 2024.
2. Mixed characteristic cases of the Grothendieck--Serre conjecture published in Forum of Mathematics, Pi in 2022, as well as in the Journal of the European Mathematical Society in 2025.
3. Macaulayfication of Noetherian schemes published in the Duke Mathematical Journal in 2021.
These works, as well as all of the other outputs of this ERC project, were made freely available to the public on the arXiv.
1. The purity for flat cohomology published in the Annals of Mathematics in 2024.
2. Mixed characteristic cases of the Grothendieck--Serre conjecture published in Forum of Mathematics, Pi in 2022, as well as in the Journal of the European Mathematical Society in 2025.
3. Macaulayfication of Noetherian schemes published in the Duke Mathematical Journal in 2021.
These works, as well as all of the other outputs of this ERC project, were made freely available to the public on the arXiv.
The purity for flat cohomology that was established in the course of this ERC project has introduced novel derived techniques into arithmetic geometry, especially, into the study of the cohomology of Noetherian schemes. It was the first work in which animated ring appeared, and it was the first work where perfectoid and derived algebraic geometry techniques were used to study flat cohomology.
To establish the cases of the Grothendieck--Serre conjecture in the course of this ERC project, we have developed numerous broadly useful geometric techniques, such as the Gabber--Quillen presentation lemmas in mixed characteristic, patching techniques, and refinements as well as relative variants of Grothendieck--Harder style classification results for torsors over the projective line. These new methods have opened the doors for further investigations into
nonabelian purity phenomena, and many further problems in this area now appear to be within the realm of a plausible attack.
The Macaulayfication result published in the course of this ERC project and the techniques in commutative algebra that it has introduced have opened doors to further study of singularities in the pursuit of the resolution of singularities conjecture and in our understanding of the moduli of projective algebraic varieties.
To establish the cases of the Grothendieck--Serre conjecture in the course of this ERC project, we have developed numerous broadly useful geometric techniques, such as the Gabber--Quillen presentation lemmas in mixed characteristic, patching techniques, and refinements as well as relative variants of Grothendieck--Harder style classification results for torsors over the projective line. These new methods have opened the doors for further investigations into
nonabelian purity phenomena, and many further problems in this area now appear to be within the realm of a plausible attack.
The Macaulayfication result published in the course of this ERC project and the techniques in commutative algebra that it has introduced have opened doors to further study of singularities in the pursuit of the resolution of singularities conjecture and in our understanding of the moduli of projective algebraic varieties.