Non-linear-Extension Of A Class Of Linear Modules Of Complexity Two Over Exterior Algebras

Exterior algebras, which are defined in a vector space V , are a class of veryimportant algebra. Exterior algebras and their modules have strong applicationbackground, and the extension of two modules is an elementary and importantpart of the study of modules.the minimal Koszul module of complexity two ?m?1Λ/(a,b) has presentationmatrices :we call it the minimal Koszul module of type (a,b) of complexity two with cycliclength m . we Let k be an algebraically closed field, V be an 3-dimensionallinear space over k, namely V = L(a,b,c), and a,b,c be linear independence inV .Λ=∧V is a exterior algebra over V. we make e?orts to study the non-linearof a koszulmodule M = ?m?1Λ/(a,b)extension of complexity two and a linearmodule L which is complexity two and of type (a,b) with cyclic length n . applythe method of representation matrix , firstly, we research on the representationmatrix of non-linear extension modules, the analyze terms of isomorphism.we have proved the following important theorems :1: If M L be as defined as above , N is the extension module of M by L, thenN is nonlinear.2: M, L, N be as defined as above, properly choose the bases of N, such3: let M,L be as defined as above, N1,N2 are the Koszul modules nonlinearextended from M by means of L, which satisfied the conditions were described inthe chapter 3. if there isμit,j,νit,j,ωit,j,e,e1in k and satisfies the condition: isomorphism .