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HEIndividual fellowship2022–2024

Hochschild · The structure and growth of Hochschild (co)homology

Horizon Europe — Marie Skłodowska-Curie Actions

Duration
2022-10-01 → 2024-09-30
EU contribution
€214,934
Participants
1
Scheme
HORIZON-TMA-MSCA-PF-EF

Lines connect the coordinator with its partners.

Results in brief

The structure and growth of Hochschild (co)homology

Hochschild homology and cohomology are by now extremely important tools touching many areas of mathematics across algebra and geometry. There are many difficult open questions concerning their structure. The project combined methods from commutative algebra, representation theory and rational homotopy theory to improve our understanding of Hochschild homology and cohomology. At the project's core was the deep interplay between Hochschild cohomology and the cotangent complex, a bridge that was used in both directions. Methods originally developed in collaboration with Iyengar were used in the project to improve our knowledge on the cotangent complex. One focus of the project was the exponential growth of Hochschild cohomology for commutative rings that are not locally complete intersection. Significant progress was made on this open problem, shedding new light on Vigué-Poirrier's conjecture on rationally hyperbolic spaces, and on Gromov's closed geodesic problem in hyperbolic geometry. Another goal was to develop the theory of proxy small objects and their relation to non-commutative complete intersections through the action of Hochschild cohomology. The project also focused on a new theory of relative Koszul duality for homomorphisms of commutative rings, and the relation to the Hochschild cohomology action on free resolutions. Connections with toric topology were also investigated, revealing new links between cohomology operations on moment angle manifolds and the commutative algebra of Stanley Reisner rings.

Data: CORDIS, © European Union

Project objective

This project will combine methods from commutative algebra, representation theory and rational homotopy theory to improve our understanding of Hochschild homology and cohomology, especially the open problem of determining their growth. At the project's core is the deep interplay between Hochschild cohomology and the cotangent complex, a bridge that will be exploited in both directions. I will use techniques pioneered in his solution of Vasconcelos' conjecture, which were further developed in my work with Iyengar to drastically improve our knowledge on the cotangent complex. Concretely, the first objective is to show that non-complete intersection rings exhibit exponential growth in their Hochschild homology; through the theory of free loop spaces this will be applied to Vigu-Poirrier's conjecture on rationally hyperbolic spaces, and to Gromov's closed geodesic problem. Second, the same novel methods will also be used to shed light on the long out of reach Second Conjecture of Quillen on the cotangent complex. Third, I will develop the theory of natural operations on Hochschild cohomology, filling a gap in the state-of-the-art and adding a tool to be applied in the first two objectives. Each of these problems directly impacts our understanding of the homological behaviour of complete intersection rings, and will indirectly be used to develop and unify the theory of ""non-commutative complete intersection rings"" which mirror their behaviour. The proposed project will be hosted a world focal point for homotopical methods in algebra, and supervised by two leading experts in algebra and topology; it will raise my research profile to the top level, establishing my position as a leading figure at the intersection of commutative algebra, non-commutative algebra, and topology.""

Original text from CORDIS.

Participants

  • KOBENHAVNS UNIVERSITET · KOBENHAVNCoordinatorDenmark

Links

Data: CORDIS, © European Union