The Representation Matrix About Non-linear-Extension Of Modules Over Exterior Algebra

Exterior algebra,with strong application background,can be used in tensor al-gebra,differential geometry,topology,and so on.In 2002, Eisenbud studied the periodic resolutions over exterior algebras in [5].In 2006, professor Jinyun Guo described the Koszul modules of complexity one by means of another method which enriched the theory of tube category of tame al-gebra([13]).What's more,Guo and his students have done a series of research on finite complexity Koszul module over exterior algebra([10],[11],[12],[14],[29]).2009, Jing Guo studied on the extension of two minimal Koszul modules M=Ωm-1A/(a, b) and L=Ωn-1A/(a, b) of complexity two,and get some results about the presen-tation matrix and representation matrix theories.Qiang Fang ([9]) studied the non-linear-extension of periodic linear modules over exterior algebra of three dimensional space,have a result about representation matrix and representation matrix theorem.Studing the extension about exterior algebra is very interesting。In this paper, V= L(a, b, c),a,b,c are liner inbependence.we make efforts to study a koszul module M=Ωm-1A/(a, b) of complexity two and a module L's extension.The presentation matrices of M,L areIf O→M→N→L→O is a short exact serie, then N is called an extension module of M by L,with the presentation matrix of N isThe minimal projective resolution of N isThe representation matrix of ft(N) isWe will apply presentation matrix to study extension modules.And by these results we analyze the problems of isomorphism between N1 and N2,and we know that some conditions should be satisfied when there is a isomorphism between N1 and N2. In this paper,we proved the following important theorem.Theorem 3.1 Let module M=Ωm-1A/(a,b), and module L as defined above.We change the bases of the items of the projective resolution,such that C(t) with the presentation matrix isTheorem 3.2 Let module M,L, matrix C(t) be in article, then C(1),C(2) have the following relation: If Then...