Flat families by strongly stable ideals and a generalization of Gröbner bases

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• 作者：Francesca Cioffi^a ; ¹ ; ^{cioffifr@unina.it"" rel=""nofollow} ; ^{francesca.cioffi@unina.it"" rel=""nofollow} ; ; Margherita Roggero^b ; ^{margherita.roggero@unito.it"" rel=""nofollow} ;
• 关键词：Family of schemes ; Strongly stable ideal ; Gr&#xf6 ; bner basis ; Flatness
• 刊名：Journal of Symbolic Computation
• 出版年：2011
• 出版时间：September 2011
• 年：2011
• 卷：46
• 期：9
• 页码：1070-1084
• 全文大小：322 K

文摘

Let J be a strongly stable monomial ideal in S=K[x₀,…,x_n] and let be the family of all homogeneous ideals I in S such that the set of all terms outside J is a K-vector basis of the quotient S/I. We show that an ideal I belongs to if and only if it is generated by a special set of polynomials, the J-marked basis of I, that in some sense generalizes the notion of reduced Gröbner basis and its constructive capabilities. Indeed, although not every J-marked basis is a Gröbner basis with respect to some term order, a sort of reduced form modulo can be computed for every homogeneous polynomial, so that a J-marked basis can be characterized by a Buchberger-like criterion. Using J-marked bases, we prove that the family can be endowed, in a very natural way, with a structure of an affine scheme that turns out to be homogeneous with respect to a non-standard grading and flat in the origin (the point corresponding to J), thanks to properties ofJ-marked bases analogous to those of Gröbner bases about syzygies.