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，并开发了一些工具，用于进一步研究这个主题。该项目的第一个方面的一个例子是建议研究的渐近行为贝蒂数的Frobenius自同态。所寻求的结果的范例是昆兹定理：当Frobenius自同态是平坦的时，非零特征的局部环是正则的。 第二个例子涉及Koszul代数：调查人员将建立在最近的工作与赫尔佐格的动机是结果的艾森巴德，Floystad，和Schreyer之间的关系线性链的决议在一个多项式环和模块的Koszul对偶外代数。一些技术的建议是来自理性同伦理论，并在这方面的努力的指导灯是“镜子原则”的阿夫拉莫夫和Halperin。然而，许多关键的结果，支配它还没有固定下来，在所需的详细程度。阿夫拉莫夫和研究人员提出了一份手稿，填补了文献中的这一空白。该项目还将研究交换环中拓扑学家常用的细胞近似的作用。在世纪后半叶，大卫希尔伯特发现了称为簇的几何对象与定义在其上的某些类型的函数之间的密切关系。后者形成代数小工具称为交换环。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中得到了应用。这个项目旨在应用“同调代数”的技术来研究交换环。 拓扑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。同调方法在处理交换代数问题时被证明是非常有效的;反过来，这也为这个问题注入了新的思想。

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.