Ring-theoretic properties of augmented Iwasawa algebras

增广岩泽代数的环理论性质

• 批准号: 2272759
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2272759
• 关键词: Ring; theoretic; properties; augmented; Iwasawa

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

设p为素数。p-adic有理数域Qp和它的整数环，p-adic整数Zp，是数论中重要的环，作为通常的整数或有理数在p处的完备化。自然，这些环上的矩阵群也被大量研究，例如，一般线性群，特殊线性群，正交群，辛群，以及这些群的子群。这些群被称为p-adic解析群。试图理解这些群的表示理论在数论中有分支，并且与影响深远的朗兰兹计划有关。上面的p-adic解析群G具有由Zp和Qp上的拓扑导出的自然拓扑。重要的是，G的表示尊重这种拓扑结构，因此我们必须研究G的表示的一个限制子集。这意味着群环k[G]的通常定义在此上下文中是不合适的。因此，我们必须形成群环kG的“完备化”。这里，k是特征为p的有限域。我们必须考虑G的两种情况：G紧和G非紧。当G是紧的时，环kG称为岩泽代数。当G是一个“定义在Zp上”的群时，例如GLn（Zp），就会发生这种情况。所有岩泽代数共有的最重要的性质之一是诺特（所有理想都是等距生成的）。这是一个基本的环理论的性质，和一个非常大量的结果诺特环是已知的。这使得岩泽代数的研究成为一个丰富而有趣的领域。当G是非紧的情况下，可以说是更重要的考虑-这发生在例如当G是一个群定义在Qp，例如GLn（Qp）。这类群在数论中经常出现，它们的表示论是朗兰兹纲领的重要组成部分。不幸的是，在G是非紧的情况下，称为增广岩泽代数的环kG几乎在所有情况下都不是诺特环。这提出了显着的困难，推广结果岩泽代数的结果，增广岩泽代数，只是因为这么多的工具可以承担一旦它是已知的环是诺特。然而，并非一切都失去了。有一个自然概括的属性Noetherianity，被称为连贯性。环R是凝聚的，如果每个有限生成理想是有限表现的（作为R-模）。任何诺特环都是相干的，但匡威则不成立，例如如果R是无穷多个变量的多项式环，它将是相干的但不是诺特环。相干的概念是有用的，因为作为一般的哲学，关于诺特环上有限生成模的陈述通常可以推广到相干环上有限表现模的陈述。此外，相干性的定义可以扩展到环上的模。凝聚模的性质使得代数几何中的技巧和思想可以用于它们的研究。因此一个自然的问题出现了：所有的增广岩泽代数都是凝聚的吗？可以证明，确实有一些小的例子。如果不是，在G上什么条件给出kG的相干和非相干？这个项目将解决这些问题，并希望提供一个答案。该项目还旨在确定增广岩泽代数的其他环论性质和不变量，例如中心和Hochschild（上）同调。这个项目的结果有可能影响p-adic解析群的模表示理论，例如在Matthew Emerton的Ordinary Parts of Admissible Representations of p-adic Reductive Groups I & II，以及Jack肖顿最近的预印本On the Category of Approximately Presented Smooth Mod p Representations of GL 2（F）中可以看到。这个项目属于EPSRC代数研究领域的福尔斯