The structure and growth of Hochschild (co)homology

Project Information

Hochschild

Grant agreement ID: 101064551

DOI

10.3030/101064551

Project terminated on 14 August 2024

EC signature date 22 June 2022

Start date 1 October 2022

End date 30 September 2024

Funded under

• Marie Skłodowska-Curie Actions (MSCA)

Total cost

No data

EU contribution

€ 214 934,40

Coordinated by

KOBENHAVNS UNIVERSITET
Denmark

Periodic Reporting for period 1 - Hochschild (The structure and growth of Hochschild (co)homology)

Reporting period: 2022-10-01 to 2024-09-30

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The project produced fundamental research on Hochschild cohomology and related areas, and this research was disseminated at international conferences and workshops, and peer-reviewed and published in top quality journals.

One major achievement was a new relative theory of Koszul duality, developed in collaboration with Cameron, Letz, and Pollitz. This theory has many applications within commutative algebra and singularity theory, and made essential use of the tools developed earlier in the project. This work is currently under peer review. Another significant achievement is the development of the theory of proxy small objects. A preprint has been published on this work in collaboration with Iyengar and Stevenson. Following on from this work, significant new progress was made towards the growth problem for Hochschild cohomology in commutative algebra.

As another use of Hochschild cohomology in commutative algebra, the project greatly developed the theory of cohomological support varieties over a local ring in collaboration with Grifo and Pollitz. Many new results were proven using the theory developed earlier in the project, and these results have been peer reviewed and published in the Transactions of the American Mathematical society series B (the open access arm of Transactions of the American Mathematical society). The project also provided new fined-grained information on the Lie algebra structure of the first Hochschild cohomology group, in joint work with Rubio y Degrassi.

In a different direction, the project had applications to toric topology, revealing new links between cohomology operations on toric spaces and the commutative algebra of Stanley Reisner rings, and ultimately characterising equivariant formality of moment angle manifolds. This joint work with Amelotte has been peer reviewed and published in the Transactions of the American Mathematical society series B.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The relative theory of Koszul duality developed in the project has significantly improved our understanding of free resolutions in local commutative algebra, and how they relate to singularity theory. This was achieved by taking ideas from the project on the action of Hochschild cohomology operations on free resolutions.

The project has also significantly changed our understanding proxy small objects, identifying them with certain localisations of compactly generated triangulated categories. A number of new applications were presented in the paper containing these developments, and we expect this improvement in the state of the art to result in many new applications, especially towards the structure of Hochschild cohomology and the problem of defining noncommutative complete intersection rings.

One other major step forward taken in the project is in the theory of cohomological support varieties, which are defined using Hochschild cohomology. This part of the project produced results bounding the dimension of support varieties, both above and below, in terms of well-known homological invariants of rings and modules. With these bounds we were able, for example, to obtain a new structural understanding of the homotopy Lie algebra of a commutative local ring.