Ring-theoretic properties of augmented Iwasawa algebras

设p是质数。p进有理数的域Qp和它的整数环，p进整数Zp，是数论中有重要意义的环，作为p处通常整数或有理数的补全。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进解析群。试图理解这些群的表征理论在数论中有分支，并且与影响深远的朗兰兹纲领有关。上述p进解析群G具有由Zp和Qp上的拓扑导出的自然拓扑。G的表示遵从这种拓扑结构是很重要的，因此我们必须研究G的表示的一个受限子集。这意味着通常的群环k[G]的定义在这种情况下是不合适的。因此，我们必须形成一个“完成”的群环，kG。这里，k是特征p的有限域。G有两种情况我们必须考虑：G紧和G非紧。当G紧化时，环kG称为Iwasawa代数。当G是“在Zp上定义”的群时，例如GLn(Zp)，就会发生这种情况。所有Iwasawa代数共有的最重要的性质之一是Noetherian（所有理想都是有限生成的）。这是一个基本的环理论性质，在诺瑟环上已知的结果非常多。这有助于使伊wasawa代数的研究成为一个丰富而有趣的领域。当G是非紧化的情况更值得考虑——这种情况发生在G是Qp上定义的群时，例如GLn（Qp）。这类群在数论中经常出现，它们的表示理论是朗兰兹纲领的重要组成部分。不幸的是，在G是非紧化的情况下，环kG，称为增广Iwasawa代数，几乎在所有情况下都不是noether代数。这给将Iwasawa代数上的结果推广到增广Iwasawa代数上的结果带来了很大的困难，因为一旦知道一个环是诺瑟的，就可以使用更多的工具。然而，并不是所有人都输了。诺埃性的性质有一种自然的概括，称为相干性。如果每个有限生成的理想都是有限表示的（作为R模），则环R是相干的。任何诺埃尔环都是相干的，但反之则不成立，例如，如果R是一个有无限变量的多项式环，它将是相干的，但不是诺埃尔环。相干的概念是有用的，因为作为一种一般的哲学，关于诺瑟环上有限生成模块的陈述通常可以推广到关于相干环上有限呈现模块的陈述。此外，相干的定义可以推广到环上的模。相干模块具有允许在其研究中使用代数几何的技术和思想的特性。因此一个自然的问题出现了：所有增广Iwasawa代数都是相干的吗？可以证明，当然有一些小的例子。如果不是，G在什么条件下给出了kG的相干性和非相干性？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Iwasawa代数的其他环理论性质和不变量，例如，中心和Hochschild （co）同调。这个项目的结果有可能影响p进解析群的模表示理论，例如在Matthew Emerton的《p进约化群I和II的可容许表示的普通部分》和Jack Shotton最近的预印本《关于GL2的有限呈现光滑模p表示的范畴》(F)中可以看到。该项目属于EPSRC代数研究领域

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