The Presentation Matrix Of The Iterated Extension Of Modules Of Finite Complexity Over Exterior Algebra

In this paper, we study the presentation matrix of the iterated extension of linear modules of complexity2over∧(V), which is an exterior algebra of3-dimensional vector space V.Let M be an∧-module, P1â†'P0â†'Mâ†'0is part of a minimal projec resolution of M. If we choose a basis of the free module Pt for t=0,1, then we associate to f a matrix A=(aij) with aij∈∧, and the matrix A=(aij) is called a presentation matrix of M. If M be an indecomposable Koszul∧-module of complexity one with cyclic length m, N be minimal Koszul A-module of complexity two with cyclic length n, then we have the presentation matrices of M, N are and respectively, a, b, aij∈V and a, b are linear independent.Let N be Koszul∧-module of complexity two,0(?)N1(?)N2(?)…(?)N Nr N be a fitration of N, we have M1be an indecomposable Koszul∧-module of complexity one with cyclic length n1, Mi(i≥2) be minimal Koszul∧-module of complexity two with cyclic length ni, such that M1is isomorphic to N1and Mt is isomorphic to Nt/Nt-1for i=2,…, r, then N is (r-1)th iterated extension of Mi for1≤i≤r. If we choose the basis of the free module in minimal projective resolution of N, we prove the matrix of ft(N) is and we prove Ki,j(1) has the following form for2≤j<j≤r.For j=j-2,i-3and j≥2,we also plove the simple form of Li,j(1),there are6cases for j=i-2and24cases for j=i-3,we provee every case respectively.