The Isomorphism Of A Class Of Iterated Extension Of Modules Over Exterior Algebra Of3-Dimensional Vector Space

let k be a algebraically closed field, and let V be a3-dimensional space over k.In this paper we study the isomorphism of a class of iterated extension of linear modules of complexity2over Λ.Let a, b, c be the base of V,n1>n2>n3>n4, and let Mi be minimal Koszul Λ-modules of complexity two with cyclic length ni with the representation matrix: If Koszul module N over Λ has a filtration0(?)CN1(?)N2(?)N3(?)N4of submodules, such that N1≌M1,and Nt/Nt-1≌Mt(2≤t≤r), we call N a iterated extension of Mi for1≤i≤r.Let N1and N2be two iterated extension of Mi for1≤i≤4, we know that the representation matrix of Nt for t=1,2have the following form:In this paper we study the isomorphism between N1and N2,we prove that N1and N2are isomorphic if and only if there is an invertible matrix which is over k,hij2=(hpqij)(ni+1)×(nj+1)are matrixes over k such that is invertible,where hij1is the first ni×nj part of Hij2and satisfy the following equalities:hi114k43(2)(j)=hi-1,j13-hi,j+112,1≤i≤n1+1,1≤j≤n3ω2k32(2)(j)=h1143h1134(φ12-h1,j+142)-h1144h1143(φ22-h1,j+132),1≤j≤n2,ω2k42(2)(j)=h1143(φ12-h1,j+132)-h1133(φ22-h1,j+142),1≤j≤n2, h1112ωk21(2)(j)=ω1δ11-h1112(hi-1,j21-hi,j+121)ρ121-h1112(δ21-h1,j+131)ρ221-h1112ρ321(δ31-h1,j+141),2≤i≤n2+1,1≤j≤n1, ω1k31(2)(j)=h1112ρ131(δ11-h1,j+121)+h1112ρ231(δ21-h1,j+131)+h1112ρ331(δ31-h1,j+141),1≤j≤n3, ω1k41(2)(j)=ρ141(δ11-h1,j+121)+ρ241(δ21-h1,j+131)+ρ341(δ31-h1,j+141),1≤j≤n1. where ω1=h1112h1123(h1132h1144-h1134h1142)+h1112h1124(h1133h1142-h1132h1143)+h1112h1112(h1134h1143-h1133h1144), ω2=h1143h1134-h1133h1144, ρ121=h1123h1132h1144+h1124h1133h1142-h1123h3411h1142-h1124h1132h1143, ρ221=h1124h1143-h1123h1144, ρ321=(h1112h1134-h1124h1132)(h1123h1134-h1124h1133), ρ131=(h1132h1144-h1134h1142), ρ231=(h1124h1142-h1112h1144), ρ331=(h1112h1134-h1124h1132), ρ141=(hi133h1142-h1132h1143), ρ241=(h1112h1143-h1123h1142), ρ341=(h1123h1132-h1112h1133),...