Koszul Modules Of Complexity Two Over Exterior Algebra

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:...