Homological Invariants of Modules Over Commutative Rings

研究者和他的合作者解决了交换环上模的同调理论中的一些具体问题，并开发了一些用于进一步研究这一主题的工具。项目的第一个方面的一个例子是提议的研究Frobenius自同态的Betti数的渐近行为。所寻求的结果的范式是Kunz定理：当Frobenius自同态为平时，非零特征的局部环恰好是正则的。第二个例子与Koszul代数有关：研究者将以Herzog最近的工作为基础，该工作受到Eisenbud， Floystad和Schreyer关于多项式环上分辨率的线性链与其Koszul对偶外部代数上的模之间关系的结果的启发。一些被提出的技术来源于理性同伦理论，而在这一努力中的指路明灯是Avramov和Halperin的“镜子原理”。然而，许多决定它的关键结果并没有被确定在期望的细节水平上。阿夫拉莫夫和研究者提出了一份手稿来填补这一文献空白。该项目还将调查细胞近似的作用，主要是拓扑学家，在交换环的背景下。在19世纪下半叶，大卫·希尔伯特（David Hilbert）发现了一种密切的关系，即几何对象的变种（variation）与在其上定义的某些类型的函数之间的关系。后者形成代数小部件，称为交换环。在上个世纪，交换环在组合学、拓扑学和其他数学分支中出现。它们也在密码学、模式识别和理论物理等不同领域得到了应用。这个项目试图应用“同调代数”的技术来研究交换环。拓扑学一直是同态代数发展的主要力量，尽管它的一些根源可以追溯到几何。同调方法在处理交换代数问题中被证明是非常有效的；反过来，这又为这一学科注入了新的思想。

Abstract for the ward of Iyengar DMS-0302892 The investigator and his collaborators address a number of specific problems in the homological theory of modules over commutative rings, and develop some tools intended for further research in this topic. One example of the first aspect of the project is the proposed study of the asymptotic behaviour of Betti numbers of the Frobenius endomorphism. The paradigm for the results sought is Kunz's theorem: local ring of non-zero characteristic is regular precisely when the Frobenius endomorphism is flat. A second example concerns Koszul algebras: the investigator will build on recent work with Herzog that was motivated by results of Eisenbud, Floystad, and Schreyer on the relationship between linear strands of resolutions over a polynomial ring and modules over its Koszul dual exterior algebra. Some of the techniques proposed are derived from rational homotopy theory, and a guiding light in this endeavour is the "Looking glass principle" of Avramov and Halperin. However, many of the crucial results that govern it have not been pinned down in the desired level of detail. Avramov and the investigator propose a manuscript that fills this gap in the literature. The project will also investigate the role of cellular approximations, staple to topologists, in the context of commutative rings.In the second half of the 19th century, David Hilbert discovered a close relationship between geometric objects called varieties, and certain types of functions defined onthem. The latter form algebraic gadgets called commutative rings. Over the last century, commutative rings have arisen in combinatorics, topology, and other branches of mathematics. They have also found applications in diverse fields like cryptography, pattern recognition, and theoretical physics. This project seeks to apply techinques from 'homological algebra' to study commutative rings. Topology has been a major force in the development of homological algebra, although some of its roots can be traced to geometry. Homological methods have proved remarkably efficacious in tackling problems in commutative algebra; in turn, this has infused new ideas into the subject.