Asymptotic behaviour of Castelnuovo-Mumford regularity
Asymptotic behaviour of Castelnuovo-Mumford regularity
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Castelnuovo-Mumford 正则的渐近行为
DOI：
10.1090/s0002-9939-99-05020-0
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发表时间：
1999
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影响因子：
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通讯作者：
Vijay Kodiyalam
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作者：
Vijay Kodiyalam
设 S 是一个域上的多项式环。对于生成度最多为 P 的有级 S 模块，(i) 其第 n 次对称幂，(ii) 其第 n 次无扭对称幂，以及 (iii) 其第 n 次无扭对称幂的积分闭包中每一个的卡斯特努沃-蒙福德正则性都由 n 中的一个前导系数最多为 P 的线性函数限定。对于 S 的有级理想 I，I'的正则性是由所有足够大的 n 的线性函数给出的。让 S = k[xl,... , Xd] 是一个域 k 上的多项式环，具有通常的分级，即每个 xi 的度数为 1，让 m 表示 S 的最大分级理想。N 的卡斯特诺沃-芒福德正则性（记为 reg(N)）被定义为最小整数 m，使得对于每一个 j，N 的第 J 个协整都以 F -> o N --> 0 的度生成，其中对于一些整数 ai，Fi = 1 S(-aij)，我们将把这些整数称为 Fi 的捻度。那么，reg(N) < maxij {aij i}，如果解析度最小，则相等。关于这个不变量的其他等价定义和性质，请参见 [Snb]。对于 S 中的分级理想 I，I'的正则性作为 n 的函数的行为一直颇受关注。如果 I 定义了一个光滑复射影变，[BrtEinLzr, Proposition 1] 利用川俣-维韦格消失定理证明了 reg(In) < Pn + Q，其中 P 是 I 的最小生成子的最大度，Q 是用 I 的生成子度表示的常数。在 [GrmGmgPtt, Theorem 1.1] 和 [Chn, Theorem 1] 中证明了，如果 dim(R/I) < 1，那么对于所有 n E N，reg(In) < n. reg(I). 在 [Chn, Conjecture 1] 中，猜想这对任意级数理想都是真的。支持这一猜想的是 [Swn, Theorem 3.6] 的结果，即对于某个常数 P 和所有 n E N，reg(In) < Pn。对于单项式理想，[SmtSwn, Theorem 3.1]中明确地计算出了这样一个 P，[HoaTrn, Corollary 3.2]对其进行了改进。我们证明，在 S 和 N 如上所述的情况下，Symn(N) 及相关模块的正则性是由 n 的线性函数约束的，其前导系数为 1997 年 10 月 28 日编者收，1998 年 4 月 15 日修订。1991 数学主题分类。(?1999美国数学会
Let S be a polynomial ring over a field. For a graded S-module generated in degree at most P, the Castelnuovo-Mumford regularity of each of (i) its nth symmetric power, (ii) its nth torsion-free symmetric power and (iii) the integral closure of its nth torsion-free symmetric power is bounded above by a linear function in n with leading coefficient at most P. For a graded ideal I of S, the regularity of I' is given by a linear function of n for all sufficiently large n. The leading coefficient of this function is identified. Let S = k[xl,... , Xd] be a polynomial ring over a field k with its usual grading, i.e., each xi has degree 1, and let m denote the maximal graded ideal of S. Let N be a finitely generated non-zero graded S-module. The Castelnuovo-Mumford regularity of N, denoted reg(N), is defined to be the least integer m so that, for every j, the Jth syzygy of N is generated in degrees F -> o N --> 0 where Fi = 1 S(-aij) for some integers ai -which we will refer to as the twists of Fi. Then, reg(N) < maxij {aij i} with equality holding if the resolution is minimal. For other equivalent definitions and properties of this invariant, see [Snb]. For a graded ideal I in S, the behaviour of the regularity of I' as a function of n has been of some interest. If I defines a smooth complex projective variety, it is shown in [BrtEinLzr, Proposition 1] using the Kawamata-Viehweg vanishing theorem that reg(In) < Pn + Q where P is the maximal degree of a minimal generator of I and Q is a constant expressed in terms of the degrees of generators of I. In [GrmGmgPtt, Theorem 1.1] and in [Chn, Theorem 1] it is shown that if dim(R/I) < 1, then reg(In) < n. reg(I) for all n E N. In [Chn, Conjecture 1], this is conjectured to be true for an arbitrary graded ideal. Supporting this conjecture is the result of [Swn, Theorem 3.6] that reg(In) < Pn for some constant P and for all n E N. The method of proof makes it difficult to explicitly identify such a constant. For monomial ideals, such a P is explicitly calculated in [SmtSwn, Theorem 3.1] and improved upon in [HoaTrn, Corollary 3.2]. We show that with S and N as above, the regularity of Symn(N) and related modules is bounded above by a linear function of n with leading coefficient at Received by the editors October 28, 1997 and, in revised form, April 15, 1998. 1991 Mathematics Subject Classification. Primary 13D02; Secondary 13D40. (?1999 American Mathematical Society