Mathematical Sciences: Homology and Representations Theory of Finite Dimensional Algebras and Classical Orders

本文主要研究有限维代数的同调问题。主要研究人员将考察衡量代数同调复杂性的几个维度，以获得更高合子的精确结构描述，并描述和利用重复的同调行为。将非常重视与相关领域的交叉连接。特别是，首席研究员将寻求替代塔西猜想关于离散赋值区域上的经典序的整体维度。本研究属于环论的一般领域。环是定义了加法和乘法的代数对象。虽然加法运算满足交换律，但乘法运算不是必须的。乘法不可交换的环的一个例子是整数上的n×n矩阵的集合。非对易环的研究已经成为代数的重要组成部分，因为它对数学和物理的其他分支的意义越来越大。

This research is concerned with the homological aspects of finite dimensional algebras. The principal investigator will examine several dimensions which measure the homological complexity of an algebra in order to obtain precise structural descriptions of higher syzygies, and to describe and exploit repetitive homological behavior. There will be a heavy emphasis on crossconnections to related areas. In particular, the principal investigator will seek replacements for the Tarsy conjecture on the global dimension of classical orders over discrete valuation domains. This research is in the general area of ring theory. A ring is an algebraic object having both an addition and a multiplication defined on it. Although the additive operation satisfies the commutative law, the multiplicative operation is not required to do so. An example of a ring for which multiplication is not commutative is the collection of nxn matrices over the integers. The study of noncommutative rings has become an important part of algebra because of its increasing significance to other branches of mathematics and physics.