Homology Theory And Complex Of Polynomial Ideal With Several Generators

Monomial ideals are at the intersection of commutative algebra and combinatorial algebra. Many important problems in polynomial rings can be reduced to the study of monomial ideals. It became a standard method in commutative algebra to study invariants of arbitrary graded ideals in a polynomial ring by passing to their initial ideals with respect to some term order. By this process, one obtains a monomial ideals, but it is also accessible to combinatorial techniques and faster computational methods.To simplicial complex, one may attach monomial ideals and vicevisa. There is correspondence between them. And the map between the chain complex is boundary mapping. According to this, we try to give a expanded boundary mapping to study polynomial ideals with several generators. And also we can expand the complex and homology theory to polynomial ideals. And we all know this work is very important for our study.This paper gives a definition about expanded boundary mappings over polynomial ideals with several generators. It gives detailed proof about the existance and uniqueness of the definition, and, it computes the Kernel and Image under this definition, and further we give a brief description of the character of the homology group of polynomial ideals with several generators. At last we expand the Koszul complex.The main conclusions obtained in this paper are as follows:Definition3.1.1 Define a mapping as follows:For arbitrary (?), DefineHere we require D(k) = 0, D(x_i^α) =1. Then D is a liner mapping. So we call itexpanded boundary mapping.Theorem3.1.1 and theorem3.1.2 give detailed proof about the existance and uniqueness of the definition.Theorem3.2.2.1 For n = 2,S = k[x] = k[x₁,x₂ ], the Kernel, Image and the homology group under the definition of expanded boundary mapping are as follows:Theorem 3.2.2.2 For S = k[x₁,…, x_n], the Kernel, Image and the homology group under the definition of expanded boundary mapping are as follows:Theorem 3.3.1 For S = k[x₁,…, x_n], the dimension of the homology group under the definition of expanded boundary mapping is infinite.Theorem3.3.1.1 when D₂ acts on S/I = k[x₁,x₂]/1,x₂ >^m, the Kernel, Imageand the homology group under the definition of expanded boundary mapping are as follows:.Theorem3.3.2.1 when D_n acts on k[x₁,x₂…, x_n]/1,x₂,…, x_n >², the homology group under the definition of expanded boundary mapping are as follows:. Theorem3.3.2 For (?),the dimension of the homology group under the definition ofexpanded boundary mapping is (d - 2)ⁿ.