# Ring-theoretic properties of augmented Iwasawa algebras

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

Let p be a prime number. The field of p-adic rationals Qp, and its ring of integers, the p-adic integers Zp, are rings that are significant in number theory, as a completion of the usual integers or rationals at p. Naturally matrix groups over these rings are also heavily studied, for example, the general linear group, special linear group, orthogonal group, symplectic group, as well as subgroups of such groups. These groups are called p-adic analytic groups. Attempting to understand the representation theory of these groups has ramifications in number theory and is connected to the far-reaching Langlands program.

The p-adic analytic groups G above have a natural topology induced by the topology on Zp and Qp. It is important that representations of G respect this topological structure, and hence we must study a restricted subset of the representations of G. This means that the usual definition of the group ring k[G] is inappropriate in this context. Hence we must form a "completion" of the group ring, kG. Here, k is a finite field of characteristic p.

There are two cases of G that we must consider: G compact and G non-compact. When G is compact, the ring kG is known as an Iwasawa algebra. This occurs when G is a group "defined over Zp", such as GLn(Zp). One of the most crucial properties that all Iwasawa algebras share is that of being Noetherian (all ideals are finitely generated). This is a basic ring-theoretic property, and an extremely large number of results on Noetherian rings are known. This contributes to making the study of Iwasawa algebras a rich and interesting field.

The case when G is non-compact is arguably more important to consider - this occurs for example when G is a group defined over Qp, for example GLn(Qp). Such groups appear frequently in number theory, and their representation theory is a crucial part of the Langlands program. Unfortunately, in the case when G is non-compact, the ring kG, called an augmented Iwasawa algebra, is in almost all cases not Noetherian. This presents significant difficulties in generalising results on Iwasawa algebras to results on augmented Iwasawa algebras, simply because so many more tools can be brought to bear once it is known a ring is Noetherian.

However, not all is lost. There is a natural generalisation of the property of Noetherianity, known as coherence. A ring R is coherent if every finitely-generated ideal is finitely-presented (as an R-module). Any Noetherian ring will be coherent, but the converse is not true, for example if R is a polynomial ring in an infinite number of variables, it will be coherent but not Noetherian. The notion of coherence is useful because, as a general philosophy, statements about finitely-generated modules over Noetherian rings can often be generalised to statements about finitely-presented modules over coherent rings. Moreover, the definition of coherence can be extended to modules over a ring. Coherent modules have properties that allow techniques and ideas from algebraic geometry to be used in their study.

Thus a natural question arises: are all augmented Iwasawa algebras coherent? It can be shown that certainly some small examples are. If not, what conditions on G give coherence and non-coherence of kG? This project will address these questions and hope to provide an answer. The project also aims to determine other ring-theoretic properties and invariants of augmented Iwasawa algebras, for example, the centre and the Hochschild (co)homology.

Results from this project have the potential to impact the modular representation theory of p-adic analytic groups, for example as seen in Matthew Emerton's Ordinary Parts of Admissible Representations of p-adic Reductive Groups I & II, and Jack Shotton's recent preprint On the Category of Finitely Presented Smooth Mod p Representations of GL2(F).

This project falls within the EPSRC Algebra research area

The p-adic analytic groups G above have a natural topology induced by the topology on Zp and Qp. It is important that representations of G respect this topological structure, and hence we must study a restricted subset of the representations of G. This means that the usual definition of the group ring k[G] is inappropriate in this context. Hence we must form a "completion" of the group ring, kG. Here, k is a finite field of characteristic p.

There are two cases of G that we must consider: G compact and G non-compact. When G is compact, the ring kG is known as an Iwasawa algebra. This occurs when G is a group "defined over Zp", such as GLn(Zp). One of the most crucial properties that all Iwasawa algebras share is that of being Noetherian (all ideals are finitely generated). This is a basic ring-theoretic property, and an extremely large number of results on Noetherian rings are known. This contributes to making the study of Iwasawa algebras a rich and interesting field.

The case when G is non-compact is arguably more important to consider - this occurs for example when G is a group defined over Qp, for example GLn(Qp). Such groups appear frequently in number theory, and their representation theory is a crucial part of the Langlands program. Unfortunately, in the case when G is non-compact, the ring kG, called an augmented Iwasawa algebra, is in almost all cases not Noetherian. This presents significant difficulties in generalising results on Iwasawa algebras to results on augmented Iwasawa algebras, simply because so many more tools can be brought to bear once it is known a ring is Noetherian.

However, not all is lost. There is a natural generalisation of the property of Noetherianity, known as coherence. A ring R is coherent if every finitely-generated ideal is finitely-presented (as an R-module). Any Noetherian ring will be coherent, but the converse is not true, for example if R is a polynomial ring in an infinite number of variables, it will be coherent but not Noetherian. The notion of coherence is useful because, as a general philosophy, statements about finitely-generated modules over Noetherian rings can often be generalised to statements about finitely-presented modules over coherent rings. Moreover, the definition of coherence can be extended to modules over a ring. Coherent modules have properties that allow techniques and ideas from algebraic geometry to be used in their study.

Thus a natural question arises: are all augmented Iwasawa algebras coherent? It can be shown that certainly some small examples are. If not, what conditions on G give coherence and non-coherence of kG? This project will address these questions and hope to provide an answer. The project also aims to determine other ring-theoretic properties and invariants of augmented Iwasawa algebras, for example, the centre and the Hochschild (co)homology.

Results from this project have the potential to impact the modular representation theory of p-adic analytic groups, for example as seen in Matthew Emerton's Ordinary Parts of Admissible Representations of p-adic Reductive Groups I & II, and Jack Shotton's recent preprint On the Category of Finitely Presented Smooth Mod p Representations of GL2(F).

This project falls within the EPSRC Algebra research area

## People |
## ORCID iD |

Konstantin Ardakov (Primary Supervisor) | |

James Timmins (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/R513295/1 | 01/10/2018 | 30/09/2023 | |||

2272759 | Studentship | EP/R513295/1 | 01/10/2019 | 31/03/2023 | James Timmins |