Homological Invariants of Modules Over Commutative Rings

摘要针对Iyengar DMS-0302892的研究，研究人员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，并开发了一些工具用于这一主题的进一步研究。该项目第一个方面的一个例子是拟研究Frobenius自同态的Betti数的渐近行为。所寻求的结果的范例是昆兹定理：当Frobenius自同态是平坦的时，非零特征的局部环恰恰是正则的。第二个例子涉及Koszul代数：研究人员将在最近与Herzog的工作的基础上，该工作的动机是Eisenbud，Floystad和Schreyer关于多项式环上的线性分解链与其Koszul对偶外代数上的模之间的关系。所提出的一些技巧是从有理同伦理论中衍生出来的，Avramov和Halperin的“镜子原理”是这一努力的指导思想。然而，管理它的许多关键结果并没有以所需的详细程度确定下来。Avramov和调查人员提出了一份手稿，填补了文献中的这一空白。该项目还将研究拓扑学家常用的细胞近似在交换环的背景下的作用。在19世纪下半叶，大卫·希尔伯特发现了称为簇的几何对象与其上定义的某些类型的函数之间的密切关系。后者形成了称为交换环的代数小工具。在上个世纪，交换环出现在组合学、拓扑学和其他数学分支中。它们还在密码学、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”中的技术来研究交换环。拓扑学一直是同调代数发展的主要力量，尽管它的一些根源可以追溯到几何。同调方法在解决交换代数中的问题方面被证明是非常有效的；反过来，这也为这门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.