Iterated Linear Extension Of Linear Modules Over Exterior Algbras

Let V be a linear space over algebraic closed field k, a, b be linear indepen-dent vectors in V.â–³V is the exterior algebra over V. Linear module is said to be minimal linear modules with cyclic length m and complexity two if their representation matrix have the following form:In this paper, let V be an3-dimensional linear space over k. a, b, c be linear independent vectors in V. We will discuss the extension problems of3-linear modules M, L, I.Let the representation matrices of M,L,I are Fm(a,b),Fn(a,c),Fp(b,c) re-spectively. Let N be an linear extension of M by L, and J be an linear extension of N by I. Let the following exact sequence… P0(J)f0(J)â†'0is the minimal projective resolution of J.We can choose property bases of Pt(J) such that the matrix corresponding to ft(J) have tape of And in [21], Ct has already defined. In this paper, we discuss the representation matrices of F1(J).We show the following theorems:Theorem3.2Let M, L, I are linear modules overâ–³V, then we can choose property bases of Pi(J) for i≤2such that In chapter4, we give an isomorphism condition of two iterated extension modules.