Structure theory and rerpresentation theory of noncommutative rings

我们推广澳洲公式。然后得到一个短的精确序列，给出了连杆与对偶的关系。将此结果应用于可交换的Gorenstein局部环，得到一个有限生成模是极大Cohen-Macaulay当且仅当其连杆模是极大Cohen-Macaulay且是水平连接的。我们确定了一个满足Hasse原理的p群G。通过赫克代数的性质来研究其交换性。然后在G为p幂零和G的Sylow p子群H的阶等于p的假设下，确定了质数p和G的结构使Hecke代数可交换的条件，部分地解决了有限群模表示理论中最重要的问题之一Broue猜想。证明了当有限群的Sylow p-子群的阶为9时，主块上的Broue猜想成立。进一步证明了对于具有9阶缺陷群的非主块，当所考虑的群是一些重要的偶发有限单群时，Broue猜想是成立的。证明了当且仅当含有复合级数的右r模是循环的时，每个残环都是不正确的。然后证明了该性质在有限正则化扩展和Morita等价下是保持的。首先研究由结构系统定义的全矩阵代数，然后研究Frobenius全矩阵代数。这个代数与格伦斯坦平铺顺序有关。

We generalize Auslander formula. Then we get a short exact sequence which gives a relation between linkage and duality. Applying this result to commutative Gorenstein local rings, we get the fact that a finitely generated module is maximal Cohen-Macaulay if and only if its linkage module is maximal Cohen-Macaular and it is horizontally linkaged.We determine a p-group G satisfying Hasse principle. We study commutativity of Hecke algebra through its character. Then we determine the condition of the prime number p and of the structure of G so as the Hecke algebra to be commutative under the assumption that G is p-nilpotent and the order of H, a Sylow p-subgroup of G, is equal to p.We solved partly the Broue conjecture, one of the most important problem of modular representation theory of finite groups. We proved Broue conjecture is true for the principle block when the order of the Sylow p-subgroup of a finite group is equal to 9. Further, we proved the Broue conjecture is true for a non-principle block having a defect group of an order 9, whenever the groups under consideration are some important sporadic finite simple groups.We proved that every residue ring is not right Artinian if and only if every right R-module which has a composition series is cyclic. Then we showed that this property is preserved under a finite normalizing extension and Morita equivalence.We study a full matrix algebra defined by a structure system and then a Frobenius full matrix algebra. This algebra is related to a Gorenstein tiled order.