The Non-linear Secondary Extension Of Linear Modules Over An Exterior Algebra

Suppose V is a three-dimensional vector space on the field k,{a,b,c} is basis V, ∧= ∧V is a exterior algebra over V. Letwhere m,t are positive integers. Let where, l is the number of rows c occurs in the first column.The complexity of the undecomposable ∧-modules with represation ma-trices F_m^t（a, b） and F_p^l（a, b, c） are two and one, respectively. Assume that M, L,I representation matrices are Fm1 （a,b）, F_n¹ï¼ˆa,b）, F_p^l（a,b,c） respective. Let N be a linear extension of M by L, and let J be a non-linear extension of N by I. The representation matrices of l The representation matrix of and the rep-resentation matrix of syzygy modules J This article discuss the the structure of J by explicitely giving D¹, D², and discuss when extensions of J₁ and J₂ of N and I are isomorphic.We first give characterization of D¹ and D²: The corresponding matrices are and If J₁ and J₂ are non-linear extension of N by I with the represation matrices of J respectively, where if there are 0≠e1∈k, such that forthen there is an isomorphism between J₁ and J₂.