【作者】 杨优美；

【导师】 郭晋云；

【作者基本信息】 湖南师范大学 ， 基础数学， 2005， 硕士

【摘要】 外代数是一类有着很强应用背景的代数,在张量分析,微分形式的研究中有广泛的应用,随着研究的深入,在代数几何,微分几何,拓扑学等领域越来越多的出现了外代数。但其表示方面的研究我们未见有系统的理论。最近,郭晋云与Eisenbud分别用不同方法刻划了外代数上复杂度为1的不可分解模,其主要定理之一是每一复杂度为1的不可分解模具有由循环Koszul模构成的滤链,开始了其表示论的研究。 郭的主要方法是通过对一Koszul模的极小投射分解中的第一个映射所对应的矩阵的标准形式讨论相应Koszul模的结构。设k为代数闭域,V为k上m维向量空间,∧V为V上的外代数,复杂度为2的不可分解循环Koszul模有形式∧V╱(a,b),其中a,b是V中线性无关的向量。本文使用郭的方法,考虑复杂度为2的循环Koszul模的极小投射分解中出现的映射的标准形,从而得到其合冲模Ω~tM的结构,在这种情况下推广了郭和Eisenbud的结果。 首先,我们主要证明了: 预备定理3.3 设M=∧/(a,b)为外代数∧=∧V上复杂度为2的不可分解循环Koszul模,其中a,b为V中线性无关的向量,其极小投射分解为 …→P~t(M)(?)…→P~1(M)(?)P~0(M)(?)M→0适当选择P~tM(t≥1,)的基,则f_t对应的矩阵A_t具有双对角形式更多还原

【Abstract】 Exterior algebras are algebras with many applications.For example ,they are wildly used in tensor analysis and differntial forms. Recently,it appears more and more in algebra geometry,differential geometry ,topology and other fields. But we haven’t seen a systemation study of its representation theory. Recently, Guo and Eisenbud have independently studied its modules of complexity one in different ways,starting a study of its reprensention theory.To study the structure of a Koszul module, Guo considered the martrix correponding to the first map in its minimal projective resolution. Let k be an algebraically closed field, and let V be an m-dimensional vector space over k. Let AV be the exterior algebra of V. The indecomposable cyclic Koszul module of complexity two have the form M = AV/(a, b) ,where a and b are independent element in V. In this paper, using Guo’s method,we obtained some standard forms for the matrixes corresponding to the maps in a minimal projective resolution of M,and get some structured theorems of the syzygy modules Ω~tM.Our first technical results is:Preparation 3.3 Let M = A/(a, b) be a indecomposable Koszul module of complexity two over exterior algebra A = AV, with minimal projective resolution:where a and b are independent element in V. then for all t ≥1,by choosing the bases for P~t(M) suitably ,the matrix A_t corresponding to the map f_t has the bidiagonal formAs an preparation step, we proved:Proposion 4.2 Let A = AV be an exterior algebra of an m-dimention vector space V.let M = A/(a, b) be a indecomposable Koszul module of complexity two , then for t > 0 ,S7*M has a cyclic factor module L such that0 -> N -? fi’M -> L -+ 0is a relative extension ,L = A/(a, b), and N is a Koszul module of complexity one.We have following theorem for the syzygy module O’M:Theorem 4.6 Let M = A/(a,b) be a indecomposable Koszul module of complexity two over exterior algebra A = AV^, then for t > 0, WM has a cyclic filtration0 C N* C ? ? ? C N1 C N° = n’M such that N°/Nl ^ A/(a,6), andN’/N** = A/(a) for i > 0.In fact, for any basis {x, y} in the subspace L(a, b) of V , we have A/(a, b) = A/(x,y) as A modules. So in Preparation 3.3,we may choose the base of P{)andPl such that:By the same method of proving Preparation 3.3 , we prove the matrix At corresponded to the map ft also has the form:更多还原