The Presentation Matrix Of Indecomposable Koszul Modules Of Complexity One Over Exterior Algebra

Exterior algebras are algebras with strong application background.It can be used in studying commutative algebras and coherent sheave categories over pro-jective space.But we haven't seen a systematic study of its representation the-ory.Recently Guo jin yun and Eisenbud have described the indecomposable modules of complexity one over exterior algebras respectively([l][2]),starting the studying of its representation theory.Suppose that the ground field k is algebraically closed.If M is an indecomposable shift Koszul module of complexity one over exterior algebra ,then M has a representation matrix of the formIn [1],Eisenbud arose a problem :if the homogeneous matrices as in Theorem3.3([l])(?) that square to zero,what is the relationship among the entries of the matrix?In this paper we study the presentation matrix of the indecomposable Koszul module of complexity one over exterior algebra .We describe the presentation matrix of M in space with lower levels and answer the question arose by Eisenbud partly.Let k be an algebraically closed field.V be an m-dimensional linear space over k,A = AV be the exterior algebra over V. M be a indecomposable Koszul A module of complexity one ,and we assumeis a minimal projective resolution of M.We discuss the matrix of f₁ and f₂,and we have the following main theorems.Theorem 3.1: Let V be a 2-dimensional linear space over k, for i=0,l,2 ,choos ing the basis of sA[i] suitably,then there exist linearly independent elements a,b in V such that the corresponding matrix of /1;ji have the following forms:( a b 0a bo "â– â– \sxsa b\ a )Theorem 3.2: Let V be a 3-dimensional linear space over k , choosing thebasis of sA[i] suitably,i=O,l ,then there exist linearly independent elements a,b,c in V such that the corresponding matrix of j\ has the following forma kib ai3 a koboa ksⁱb a )where a^ G L(b, c), i < j — 1, ki € k, 1 < i < s — 1.V(*)Theorem 3.3: Let V be a 3-dimensional linear space over k, choosing the basis of sA[i] suitably,i=O,l , according to theorem 3.2, the representation matrix A\ of M has the form *.Denoting atj = &?â– & + k^c.a, b, c is independent elements in V. if alyi+i ^ 0 for i — 1, ? â–  ? , s — 1 ,and that the first element among k'^ isn't 0 be fc;-,changing the bases of sA[0] , sA[l] and sA[2], then the matrix of f\ has the following forma b 0 a b00 c 0 ?â– ? 0 0 0 c ??? 0...