Derived invariants of commutative rings

这个项目的重点是交换代数中出现的派生范畴。 调查员和他的合作者带来承担的角度来看，三角形的范畴形成的代数拓扑建立新的结果，并发掘令人惊讶的连接，经典的不变量的模块交换环。另一个方面的拟议研究关注派生类别作为一个整体，并受到最近壮观的发展，在表示理论的有限维代数，由于鲁基耶：扩展了Konstevich、Bondal和货车den Bergh等人关于凝聚层的派生范畴的早期工作，Rouquier解决了一个长期存在的关于Auslander表示维的问题，也是Benson关于有限群模表示的一个重要猜想。为此，鲁基耶引入了派生范畴的维数概念，并证明了它与表示论和代数几何的相关性。一个关键的步骤Rouquier的工作是一个下界的导出范畴的外代数的向量空间的领域的特征2，这是一个例子，一个零维交换完全相交环。 该项目将研究一般交换环的衍生类别的维数，重点是局部完全相交。这带来了，并有助于研究，Hochschild上同调的交换代数。 所得结果将有助于阐明环的维数与经典不变量之间的关系。这里发展的技术将对表示论和代数几何的发展产生影响。人们早就认识到，由Verdier在60年代中期引入的代数几何和交换代数的导出范畴为做同调代数提供了一个方便的环境。然而，在过去的15年里，人们已经认识到，派生范畴，更一般地说，三角范畴，也非常适合于表达（和证明！）结果在数学和数学物理的不同科目。 这就产生了这样的效果，即在一个领域发展起来的方法往往深刻地影响了许多其他领域，并在它们之间产生了新的相互作用。一个主要组成部分的研究概述了这一建议，旨在利用一个基本概念所产生的代数拓扑结构，即“建设”的想法对象的一个给定的集合中的一个三角范畴，并使他们承担的问题，在交换代数。该项目有望产生影响的研究派生类别一般，特别是在几何，表示理论和代数拓扑。

The focus of this project is derived categories arising in commutative algebra. The investigator and his collaborators bring to bear a point of view on triangulated categories fashioned in algebraic topology to establish new results on, and unearth surprising connections among, classical invariants of modules over commutative rings. Another aspect of the proposed research concerns derived categories as a whole, and is inspired by recent spectacular developments in the representation theory of finite dimensional algebra, due to Rouquier: extending earlier work of Konstevich, Bondal and Van den Bergh, and others, on derived categories of coherent sheaves, Rouquier settled a long standing question concerning Auslander's representation dimension, and also an important conjecture of Benson concerning modular representations of finite groups. To this end, Rouquier introduced a notion of a dimension for derived categories, and demonstrated its relevance to representation theory and algebraic geometry. A crucial step Rouquier's work is a lower bound on the derived category of the exterior algebra on a vector space over a field of characteristic two, which is an example of a zero-dimensional commutative complete intersection ring. The project will investigate the dimension of derived categories of general commutative rings, with emphasis on locally complete intersections. This brings in, and contributes to the study of, Hochschild cohomology of commutative algebra. The results obtained will serve to clarify that relationship between dimension and classical invariants of rings. Techniques developed here would have an impact on developments in representation theory and algebraic geometry.It has long been recognized that derived categories, which were introduced by Verdier in the mid-sixties and in the context of algebraic geometry and commutative algebra, provide a convenient milieu for doing homological algebra. However, it has been realized over the last fifteen years that derived categories, and more generally, triangulated categories, are also exceptionally well-suited for expressing (and proving!) results in diverse subjects in mathematics and mathematical physics. This has had the effect that methods developed in one field have often profoundly influenced a host of others, and generated new interactions among them. One of the main components of the research outlined in this proposal seeks to exploit a elementary notion arising in algebraic topology, namely, the idea of `building' objects out of a given collections of objects in a triangulated category, and bring them to be bear on problems in commutative algebra. The project promises to have an impact on the study of derived categories in general, and, in particular, on geometry, representation theory, and algebraic topology.