The Non-linear-Extension Of Periodic Linear Modules Over Exterior Algebra Of Three Dimensional Space

Exterior algebras are widely used in studying commutative algebras and the categories of coherent sheave over projective space.But there seems no systematic study on their representation theory and until the work of Eisenbud ([6]) and Guo ([13]) the describe the modules of complexity one (periodic modules) in dependently.Guo and his students also study the Koszul modules of other finite complexity recently.Guo etc. study the modules of finite complexity over the exterior algebras by their representation matrix.Suppose that k is an algebraically closed field.If M is an indecomposable Koszul module of complexity one over exterior algebra ,and assume thatis a minimal projective resolution of M,a representation matrix of M has the formLet k be an algebraically closed field. V be an 3-dimensional linear space over k,Λ=∧V be the exterior algebra over V. M be a indecomposable KoszulΛmodule of complexity one .We will discuss the presentation matrix of by choosing the basis of sΛ[i] suitably,i=0,1,then there exist a base a,b,c of V such that the representation matrix has the following form a_ij∈L(b,c),i < j - 1,1≤i≤n - 1,and L(a,b) is a subspace spanned by a,b.Let a, b be linearly independent elements in V , the linear module whosepresentation matrix of the form (?) is called uniserial linearmodule of type (a, b),and n is its length.In this paper,we discuss the non-linear extension of such modules and prove the following theorems:Theorem Let k be a field ,V be a 3-dimensional vector space over k,ΛV is a exterior algebra of V.Let a, b and a, c be two pair linearly independent vectors in V ,and let M, L be two uniserial linear modules of type (a, b) and (a, c) overΛVwith the cyclic length n,m respectively.There are at most P^2mn-1 families of modules N such that the following is an non linear extension...