A criterion for detectingm-regularity
A criterion for detectingm-regularity
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检测规则性的标准
DOI：
10.1007/bf01389151
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发表时间：
1987
影响因子：
3.1
通讯作者：
M. Stillman
中科院分区：
文献类型：
--
作者：
D. Bayer;M. Stillman
计算I的(第一)合式的一个算法独立于Spear[Spe77]和Schreyer[Sch80]：选择S的单项式上的一个序，然后构造一个由I的所有元素的前导项生成的单项理想in(1)，在GL(N)的一个单参数子群作用下可视为1的极限[BAY 82]，因此，在(1)中出现的是一个平面族的特殊纤维，它的一般纤维与I同构。它从一个著名的平面度标准[第76条]得出，In(L)的每个合子可以提升到I的一个合子；这样得到的合子集可以被修剪成I的极小合子的完全集。(I)中的随机理想是由Macaulay[Mac27]首先研究的；它的构造算法首先由Buchberger[Buc 65]，[Buc 76]给出。在[Hir]、[Bri 73]、[Ga174]、[Ga179]中研究了(I)，作为幂序列环的类似除法过程的一部分。在实践中使用这种合算法时出现了下列问题：在(L)中可以有比I的任何极小生成元或合子高的极小生成元和合子的次数，在这种情况下，这些高次的计算是不必要的；人们应该只在找到1的所有极小合子所必需的程度上计算in(1)的生成元和合子。为了修改syzygy算法以利用这一观察，人们想要一个确定何时找到i的所有极小合子的标准。目前，这个问题似乎很难解决。然而，对于所有的j，确定i的最小第j合度的范围的问题是容易处理的。回想一下，如果对所有j>；0([Mum66]，[Eigo 84])生成i的第j个合模，则i被定义为m-正则的。定义i的正则性reg(L)为对i是m-正则的最小m。我们有reg(in(1))>；reg(I)，因为正则性在平坦族中是上半连续的。当reg(in(I))>；reg(I)时，以度为单位的计算取消->reg(L)+1
An algorithm for computing the (first) syzygies of I is due independently to Spear [Spe77] and Schreyer [Sch80]: One chooses an ordering on the monomials of S, and then constructs a monomial ideal in (1) generated by the lead terms of all elements of I, in (l) can be viewed as the limit of 1 under the action of a 1-parameter subgroup of GL (n) on the Hilbert scheme [Bay 82], so in (1) occurs as the special fiber of a flat family whose general fiber is isomorphic to I. It follows from a well-known criterion for flatness [Art 76] that each syzygy of in (l) can be lifted to a syzygy of I; the set of syzygies thus obtained can be trimmed to give a complete set of minimal syzygies of I. The rnonomial ideal in (I) was first studied by Macaulay [Mac27]; an algorithm for its construction was first given by Buchberger [Buc 65],[Buc 76]. in (I) is studied in [Hir 64],[Bri 73],[Ga174],[Ga179] as part of an analogous division process for power series rings.The following problem arises in using this syzygy algorithm in practice: in (l) can have minimal generators and syzygies in degrees higher than any minimal generator or syzygy of I. In this situation, computations in these higher degrees are unnecessary; one should compute the generators and syzygies of in (1) in only those degrees necessary to find all minimal syzygies of 1. In order to modify the syzygy algorithm to take advantage of this observation, one would like a criterion for determining when all minimal syzygies of I have been found. This problem appears to be intractable at present. However, the question of bounding the degrees of the minimal jth syzygies of I, for all j, is tractable. Recall that I is defined to be m-regular if the jth syzygy module of I is generated in degrees< m+ j, for all j> 0 ([Mum66],[EiGo 84]). The regularity of I, reg (l), is defined to be the least m for which I is m-regular. We have reg (in (1))> reg (I), because regularity is upper-semicontinuous in flat families. When reg (in (I))> reg (I), computations in degrees> reg (l)+ 1 are un-