The Representation Matrix And The Isomorphism Of The Extension Of Koszul Modules Of Complexity Two Over Exterior Algebra

Exterior algebra,with used-application background,can be used in differential geometry,tensor analysis,algebra geometry,topology,what's more, it's used in study-ing commutative algebras and the categories of coherent sheave over projective space,and so on.Exterior algebra can be used in so many areas,but there seems no system-atic study on their representation theory. Eisenbud([10]) has done some research about periodic module.With other scholars,professor Guo described Koszul mod-ules of complexity one by means of another method,which enriched the theory of tube category of tame algebra([15],[21]).At the same time, a series of research on Koszul module over exterior algebra have been done by Guo and other schol-ars([16],[21],[35],[36]).What's more,in [35],he brought in minimal Koszul module of complexity two,which is the translation of the syzygy module of cyclic Koszul mod-ule of complexity two,and its presentation matrix has the following formThe extension of two modules is an important and interesting part of the study of modules,and it also has an intimate relation with the computation of derivation and homological group. But the total research on tube category in tame hereditary algebra comes from the simple module of Kronecker algebra which has P1 variety extension.V is a linear space,Λ=ΛV is a exterior algebra over V. In this paper,we make efforts to research on the representation matrix of the extension of two minimal Koszul modules M=Ωm-1Λ/(a, b) and L=Ωn-1Λ/(a, c) of complexity two and the isomorphic matrix over exterior algebra,and a, b; a, c are the respectively linear independence vector pairs, with the presentation matrices of M,L are If 0→M→N→L→0 is an exact serie,N is a Koszul module,and then N is called an extension Koszul module of M by L,then with the presentation matrice of N isWe apply presentation matrix to research on extension modulus, and on these bases,we analyze the problems of isomorphism between N1 and N2,which are the Koszul modules extended from M by means of L,and we know that some conditions should be satisfied when there is an isomorphism between N1 and N2. Therefore,we have proved the following important theorems and corollary.Theorem:Let a, b, c be linear independence.For the extension module M has been definedbfore.We change the bases of the first two items of the projective reso-If N is the syzygy module ofΩN,which with the presentation matrices is and we continue to change the bases of the projective resolution,such that li,jt,ki,jt,i=1,2,…,n+2;j=1,2,…,m+1;t=1,2;are the none zero elements in field K.What's more,we know that some conditions should be satisfied when there is an isomorphism between N1 and N2.Theorem:Let K be an algebraically closed field,V be an m-dimensional linear space over K,Λ=ΛV be the exterior algebra over V.M,L be modules have been defined before.ThenN1,N2 are the Koszul modules extended from M by means of L,which satisfied the conditions were described in the chapter 3.If there are Let kn+1,1=1/e1sn+1,121+en'/e1k'n+1,1,then there is an isomorphism:N1(?)N2.