0-Borel Fixed Ideals
0-Borel Fixed Ideals
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0-Borel 固定理想
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发表时间：
1996
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通讯作者：
I. Peeva
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作者：
I. Peeva
一个理想被称为波莱尔不动的，如果它在波莱尔群的作用下是不变的。对研究0-Borel不动理想的兴趣来自Bayer，Galligo和Stillman的定理（cf. [Ei，（15.20）]），证明了S中任何齐次理想的通有初始理想都是Borel不动的，并且如果k的特征为0，则每个Borel不动理想都是0-Borel不动的.当char k 6 = 0时，有一些Borel固定理想不是0-Borel固定的例子。0-Borel不动理想由Eliahou和Kervaire研究，他们在[EK]中描述了S/I的最小自由分解。分次或局部Noether环上极小自由归结上交换结合DG（=微分分次）代数结构的存在性是研究归结性质的有力工具.最好的例子是S/（y1，. . .，yi），对于S-正则序列y1，. . .，yi。Tate和Gulliksen的结果提供了另一个重要的例子：对于S中的任何齐次理想Q，k在S/Q上的最小分解。在这两种情况下的决议是一个自由交换除幂代数。一般来说，交换结合DG代数结构可能要复杂得多，或者可能根本不存在。后者甚至发生在有限的决议，如[Av]所示。然而，Avramov、Buchsbaum、Gover、Eisenbud、Kustin、米勒、Palmer和Srinivasan的结果在许多有趣的情况下建立了这样的结构。我们推广了[Sr，（3.6）]，证明了当Q是极大理想（x1，x2）的幂时，在S/Q的极小自由分解上存在这样的结构. . .，xr）。0-Borel固定理想和极大理想的幂的分解之间的主要区别在于极大理想的幂的分解是线性的，而0-Borel固定理想的分解是线性的，只有当理想的所有极小生成元具有相同的度。我们证明如下：
An ideal is called Borel fixed if it is invariant under the action of the Borel group. The interest in studying 0-Borel fixed ideals comes from a theorem of Bayer, Galligo, and Stillman (cf. [Ei,(15.20)]), which shows that the generic initial ideal of any homogeneous ideal in S is Borel-fixed, and further if the characteristic of k is 0 then every Borel-fixed ideal is 0-Borel fixed. When char k 6= 0 there are examples of Borel-fixed ideals which are not 0-Borel fixed. 0-Borel fixed ideals were studied by Eliahou and Kervaire, who described in [EK] the minimal free resolution of S/I. The existence of a commutative associative DG (= differential graded) algebra structure on a minimal free resolution over a graded or local Noetherian ring is a powerful tool for investigating the properties of the resolution. The best example is the Koszul resolution of S/(y1, . . . , yi) for an S-regular sequence y1, . . . , yi. Results of Tate and Gulliksen provide another important example: the minimal resolution of k over S/Q, for any homogeneous ideal Q in S. In both cases the resolution is a free commutative divided power algebra. In general a commutative associative DG algebra structure might be much more complicated or might not exist at all. The latter happens even for finite resolutions as shown in [Av]. However, results of Avramov, Buchsbaum, Gover, Eisenbud, Kustin, Miller, Palmer, and Srinivasan build such a structure in many interesting cases. We extend [Sr,(3.6)], which shows that such a structure exists on the minimal free resolution of S/Q when Q is a power of the maximal ideal (x1, . . . , xr). The main difference between the resolutions of a 0-Borel fixed ideal and of a power of the maximal ideal is that the resolution of a power of the maximal ideal is linear, whereas the resolution of a 0-Borel fixed ideal is linear only if all the minimal generators of the ideal have the same degree. We prove the following: