The Extension Of Modules Of Complexity Two Over Exterior Algebra

Exterior algebra, with strong application background, can be used in tensor analysis, differential geometry;algebra geometry, topology, and so on.Eisenbud([1]) have done some research about periodic module. With his students,Guo described Koszul modules of complexity one([2][3]) by means of anthermethod, which enriched the theory of tube category of tame algebra. What's more, Guo and his students have done a series of research on Koszul module over exterior algebra([3][4][5][6]), and in [5], he brought in minimal Koszul module of complexity two, which is the syzygy module of cyclic Koszul module of complexity two, and its presentation matrix has the following form.The 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 homologic group, but the total research on tube category in tame hereditary algebra comes from that the simple module of Kronecker algebra has P¹ variety extension.In this paper, we make efforts to researching on the extension of two minimal Koszul modules M =Ω^m-1Λ/(a, b) and L =Ω^n-1Λ/(a, b) of complexity two,withthe presentation matrices of M,L are A¹ =(?) and B¹ =(?) respectively.If 0→M→N→L→0 is an exact series, N is a Koszul module, and then N is called an extension Koszul module of M by L. We still apply presentation matrix to do research on extension modulus, and the computation contributes to a series of results. Based on 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.Therefore, we have proved the following important theorem.Theorem 4.4: Let k be an algebraically closed field. V be an m-dimensional linear space over k,Λ=∧V be the exterior algebra over V. M,L be as definedas above.Then the Koszul module extended from M by means of L is aP^{(n+2-m)(q-2)-l} variety when m≤n + 1.The following corollaries are interesting and got directly from Theorem 4.4.Corollary 4.5: M,L are defined as above. If m≤n+1, and N is an extension Koszul module of M by L. Then N' is an extension Koszul module ofΛ/(a, b) byΩ^n-m+1(Λ/(a,b)), s.t.N =Ω^m-1N'.