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

设R为无限域上R不定域中的一个多项式环，M为R上有限生成的分级自由模的齐次子模。我们首先明确了M的一般Grobner基与Weierstrass基之间的关系。如果对由反向字典顺序产生的项的位置顺序取M的Grobner基，那么它也是M的Weierstrass基。接下来，我们给出了一个基本定理的证明，这将构成我们长期研究的核心的一部分。定理1:：设p是介于2和r - 2之间的整数。设M满足下列条件(1)和(2)。(1)对于位于r - p + 1和r - 1之间的每一个整数i， M的第i个局部上同调消失。(2) M对R是自反的，则存在一个R的齐次素理想I，它与M拟合成一个长布尔巴基序列。反之，如果存在这样一个素理想I，则M满足条件(1)和(2)。定理2::：如果M满足上述条件(1)，则存在齐次完全交f_1，…， f_{p - 2} (R)满足以下条件。(3)设A是由这些p - 2多项式定义的R的因子环。那么A是正常的。(4)在a上存在一个有限生成的无扭转梯度模D和一个从M到D / R的同态g，使得g在0到T - p + 1之间的每一个整数i上导出M和D的第i个局部上同态。

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.