Classification of homogeneous ideals in terms of the degrees of generic Grobner bases

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Let R be a polynomial ring in r indeterminates over an infinite field and M a homogeneous submodule of a finitely generated graded free module over R.We first made clear the relation between generic Grobner bases and Weierstrass bases of M. It can be summarized as follows. If one takes a Grobner basis of M with respect to the term over position order arising from the reverse lexicographic order in a suitable way, then it is also a Weierstrass basis of M.Next, we have given a proof to a fundamental theorm which will constitute a part of the core of our study in the long run.Theorem1 : : : Let p be an integer lying between 2 and r - 2. Suppose that M satisfies the following conditions (1) and (2).(1) For each integer i lying between r - p + 1 and r - l, the ith local cohomology of M vanishes.(2) M is reflexive over R.Then, there exists a homogeneous prime ideal I of R which fits into a long Bourbaki sequence with M. Conversely, if such a prime ideal I exists, then M satisfies conditions (1) and (2).Theorem2 : : : If M satisfies condition (1) above, then there is a homogeneous complete intersection f_1, . . . , f_{p - 2} of R satisfying the following conditions.(3) Let A be the factor ring of R defined by these p - 2 polynomials. Then, A is normal.(4) There are a finitely generated torsion-free graded module D over A and a homomorphism g from M to D over R such that g induces an isomorphism of the ith local cohomologies of M and D for each integer i lying between 0 and T - p + 1.