Koszul对象和弱正则环的相关研究

【作者】 汪国军；

【导师】 李方；

【作者基本信息】 浙江大学 ， 基础数学， 2006， 博士

【摘要】 1．研究了极小马蹄型引理和Koszul对象的广义扩张封闭，我们主要得到： (a) 极小马蹄型引理成立的一些充分条件并研究了极小马蹄型引理的应用； (b) 突破经典扩张封闭概念，我们引入广义扩张封闭．在此基础上，我们研究了范畴K(A)，QK(A)，HK(A)和NBGr(A)的广义扩张封闭。 2．我们引入了一类非Koszul，非h-Koszul的2p(p≥1)齐次代数，即所谓的Koszul-型代数。虽然它与Koszul代数和h-Koszul代数有着本质的不同，但是它却具有许多类似于Koszul代数和h-Koszul代数的极好的同调性质。并研究了Koszul-型模范畴的广义扩张封闭和Koszul-型代数的单点扩张等问题。 3．将Koszul模，高阶Koszul模等(可以统称为Koszul对象)推广，我们介绍了f-模；研究了f-模的扩张封闭；给出了f-模的单点扩张。 4．将群分次正则环推广到群分次弱正则环；用新的方法研究了环的弱正则性，并得到半群环的弱正则性和T-弱正则性的一个等价刻画。更多还原

【Abstract】 Firstly, the minimal Horse-shoe Lemma and the generalized extension closure of Koszul objects are investigated, in this paper. Mainly, we have the following statements:1. We give some sufficient conditions for the minimal Horse-shoe Lemma holding. Moreover, some applications of the minimal Horse-shoe Lemma are given.2. The conception of generalization extension closure which is a wider conception than the classical extension closure is introduced. On this basis, we consider the generalization extension closure of the categories K(A), QK(A), HK(A) and NBGr(A).Secondly, we introduce the concept of Koszul-type algebra, which is a 2p homogeneous algebra and neither a Koszul algebra nor a h.-Koszul algebra, where p ≥ 1 is an integer. Although it is different from Koszul and high Koszul algebras, it does have a lot of perfect homological properties similar to Koszul and high Koszul algebras. Further more, the generalized extension closure of Koszul-type modules is discussed and the one point extension of Koszul-type algebras is investigated as well.Thirdly, we introduce the notion of f-module (algebra) which is the generalizations of Koszul module and high Koszul module (we usually call them Koszul objects); The extension closure of f-modules and the one point extension of f-algebras are discussed.Finally, as a generalization of the group-graded regular ring, the notion of group-graded weakly regular ring is introduced and we discuss it in a new way. In particular, an equivalent condition for the semigroup rings to be weakly regular(T-weakly regular) is given.更多还原