The Non-linear Secondary Extension Of Linear Modules Over An Exterior Algebra Of A Three-dimensional Vector Space

Let V be a three-dimensional vector space over a domain k, a, b, c is a base of V, Δ=∧V is a exterior algebra over V, let be a matrix over V, a, b are linear independent elements and m, t are positive integers. Linear modules which representation matrix is Fm1(a,b) are called minimal linear modules of complexity two with cyclic length m.Over the vector space, we discussed extension problems of three min-imal linear modules M, L, I, which representation matrices are Fm1(a, b), Fn1 (a,b), Fp1(a,c) respectively. Let N be an linear extension of M and L, J be a non-linear extension of N and I, the representation matrix of N is C1 is a matrix over V. The representation matrix of J is and the representation matrix of the syzygy module of J is In [20], the author skinned F1(N) and F2(N) in two cases m>n+1 and m<n+1. In this paper, we skinned D1 and D2 to discuss the structure of J.We have the following main theorem:Theorem3.2 Let M, L, I as mentioned above, we chose the bases of Pt(M)(?)Pt(L)(?)P4(I)(t=0,1,2) properly, such that the corresponding matrix of ft(J) is hij1,tij1,i=1,2,…,p+1,j=m+1,m+2,…,m+n are elements of k.Theorem4.3 Let V is a three dimensional vector space of domain k, V=L(a,b,c),N is a linear extension of M and L,J1,J2 are non-linear extensions of N and I,and they have the projective resolution as mentioned above,their representation matrices are then there is an isomorphism between J1 and J2.